New Graph

             [3, 3, 1, 1, 7, 7, 5, 5], [6, 8, 8, 6, 2, 4, 4, 2]

 

 

π = [1, 1, 1, 1, 1, 1, 1, 1]

POSSIBLE RANKS

1 x 8
2 x 4

BASE DETERMINANT 4236243/134217728, .3156246990e-1

NullSpace of Δ

{2, 4, 5, 7}, {1, 3, 6, 8}

Nullspace of A

[{5, 7},{2, 4}] `,` [{6, 8},{1, 3}]

 

 
1 . Coloring, {}

R: [3, 3, 1, 1, 7, 7, 5, 5]    B: [6, 8, 8, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )``]`

For τ=1/2, [3, 1, 3, 1, 3, 1, 3, 1] . FixedPtCheck, [3, 1, 3, 1, 3, 1, 3, 1]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y1, y1, -y1, y1, -y1, y1, -y1, y1]
  p = s 2

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 2   p' = s 3   p' = s 4   p' = s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 6 1 vs 6 1 vs 6 1 vs 4 1 vs 4

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 2 .    order:   2

See Matrix
 

[y1, 0, y1, 0, y1, 0, y1, 0]

 

  p' = - s + s 2   p' = - s + s 3   p = s - s 2

Omega Rank for B :  cycles: {{4, 6}, {2, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[0, y1, 0, y1, 0, y1, 0, y1]

 

  p = - s + s 2   p = - s + s 3   p = - s + s 4


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, x1, x2, x3, x4, x5]
For A+2Δ :   [y1, -3 y1 - 3 y2 - y3 - 3 y4 - y5 - 3 y6 - y7, y2, y3, y4, y5, y6, y7]
For A-2Δ :   [-3 y4 - y5 - 3 y6 - y1 - 3 y2 - y3 - 3 y7, y4, y5, y6, y1, y2, y3, y7]

Range of {ΩΔi}: [-μ1, μ1, -μ1, μ1, -μ1, μ1, -μ1, μ1]

 
rank of M is 8 , rank of N is 6

M              \ ;   N

$ [ [0, 0, 2, 0, 2, 0, 2, 0] , [0, 0, 0, 2, 0, 2, 0, 2] , [2, 0, 0, 0, 2, 0, 2, 0] , [0, 2, 0, 0, 0, 2, 0, 2] , [2, 0, 2, 0, 0, 0, 2, 0] , [0, 2, 0, 2, 0, 0, 0, 2] , [2, 0, 2, 0, 2, 0, 0, 0] , [0, 2, 0, 2, 0, 2, 0, 0] ] $     $ [ [0, 1, 2, 1, 2, 2, 2, 2] , [1, 0, 1, 2, 2, 2, 2, 2] , [2, 1, 0, 1, 2, 2, 2, 2] , [1, 2, 1, 0, 2, 2, 2, 2] , [2, 2, 2, 2, 0, 1, 2, 1] , [2, 2, 2, 2, 1, 0, 1, 2] , [2, 2, 2, 2, 2, 1, 0, 1] , [2, 2, 2, 2, 1, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [1, -1, 1, -1, 1, -1, 1, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 16 , r'= 3/4

Ranges

Action of R on ranges, [[1], [1]]
Action of B on ranges, [[2], [2]]
β({1, 3, 5, 7}) = 1/2
β({2, 4, 6, 8}) = 1/2

ker N, [μ1, -μ1, μ1, -μ1, -μ2, μ2, -μ2, μ2]
Range of N
    [y3, y2, -y3 + y2 + y1, y1, y6, y5, y4, y6 - y5 + y4]

Partitions

Action of R on partitions, [[3], [3], [3], [3]]
Action of B on partitions, [[4], [4], [4], [4]]

α([{7, 8}, {5, 6}, {2, 3}, {1, 4}]) = 0/1
α([{1, 2}, {3, 4}, {6, 7}, {5, 8}]) = 0/1
α([{7, 8}, {1, 2}, {3, 4}, {5, 6}]) = 1/2
α([{2, 3}, {1, 4}, {6, 7}, {5, 8}]) = 1/2

b1 = {7, 8} ` , ` b2 = {1, 2} ` , ` b3 = {3, 4} ` , ` b4 = {5, 6} ` , ` b5 = {2, 3} ` , ` b6 = {1, 4} ` , ` b7 = {6, 7} ` , ` b8 = {5, 8}

Action of R and B on the blocks of the partitions: = [4, 3, 2, 1, 2, 3, 4, 1] [5, 8, 7, 6, 8, 7, 6, 5]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-4 partition graph.

`

Sandwich
Coloring {}
Rank4
R,B [3, 3, 1, 1, 7, 7, 5, 5], [6, 8, 8, 6, 2, 4, 4, 2]
π2 [0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0]
u2 [1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1] (dim 3)
wpp [2, 2, 2, 2, 2, 2, 2, 2]
π4 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u4 [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]

 

 
2 . Coloring, {2}

R: [3, 8, 1, 1, 7, 7, 5, 5]    B: [6, 3, 8, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , 4` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , -4` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [105, 57, 93, 47, 162, 38, 150, 66] . FixedPtCheck, [105, 57, 93, 47, 162, 38, 150, 66]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 1 .    order:   2

See Matrix
 

[-4 y1 + 3 y2, 0, y1, 0, y2, 0, -5 y1 + 4 y2 - y3, y3]

 

  p' = s 2 - s 4   p = - s 2 + s 4

Omega Rank for B :  cycles: {{2, 3, 8}, {4, 6}},   net cycles: 2 .    order:   6

See Matrix
 

[0, -y1 + 2 y3 - y2, y1, y3, 0, y3, 0, y2]

 

  p = s - s 4   p' = s - s 4

 » SYNC'D 15/2048 , 0.007324218750

 
3 . Coloring, {3}

R: [3, 3, 8, 1, 7, 7, 5, 5]    B: [6, 8, 1, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` - 1 + τ ` )`` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )` , -4` (` 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -4` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 4` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [105, 123, 171, 83, 333, 47, 285, 159] . FixedPtCheck, [105, 123, 171, 83, 333, 47, 285, 159]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 + y2 - y3 - y4, 0, y1, 0, y2, 0, y3, y4]

 

  p = - s 4 + s 5

Omega Rank for B :  cycles: {{4, 6}, {2, 8}},   net cycles: 1 .    order:   2

See Matrix
 

[y3, 3 y2 - 4 y1, 0, -y3 + 4 y2 - 5 y1, 0, y2, 0, y1]

 

  p' = - s 2 + s 4   p = s 2 - s 4

 » SYNC'D 5/512 , 0.009765625000

 
4 . Coloring, {4}

R: [3, 3, 1, 6, 7, 7, 5, 5]    B: [6, 8, 8, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )` , 4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [93, 47, 105, 57, 150, 66, 162, 38] . FixedPtCheck, [93, 47, 105, 57, 150, 66, 162, 38]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 1 .    order:   2

See Matrix
 

[y1, 0, -4 y1 + 3 y3, 0, -5 y1 - y2 + 4 y3, y2, y3, 0]

 

  p = - s 2 + s 4   p' = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 4, 6}, {2, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[2 y3 - y2 - y1, y3, 0, y2, 0, y1, 0, y3]

 

  p' = - s + s 4   p = - s + s 4

 » SYNC'D 15/2048 , 0.007324218750

 
5 . Coloring, {5}

R: [3, 3, 1, 1, 2, 7, 5, 5]    B: [6, 8, 8, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 4` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 4` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , -4` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [285, 159, 333, 47, 171, 83, 105, 123] . FixedPtCheck, [285, 159, 333, 47, 171, 83, 105, 123]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   4

See Matrix
 

[-y1 + y2 + y3 - y4, y1, y2, 0, y3, 0, y4, 0]

 

  p = - s 4 + s 5

Omega Rank for B :  cycles: {{2, 8}, {4, 6}},   net cycles: 1 .    order:   2

See Matrix
 

[0, y2, 0, y1, 0, -5 y2 + 4 y1 - y3, y3, -4 y2 + 3 y1]

 

  p' = s 2 - s 4   p = - s 2 + s 4

 » SYNC'D 5/512 , 0.009765625000

 
6 . Coloring, {6}

R: [3, 3, 1, 1, 7, 4, 5, 5]    B: [6, 8, 8, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 5 + τ ` )`` (` 1 + τ ` )` 2 , 4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )` , 4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 4` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [162, 38, 150, 66, 105, 57, 93, 47] . FixedPtCheck, [162, 38, 150, 66, 105, 57, 93, 47]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 1 .    order:   2

See Matrix
 

[y2, 0, 4 y2 - y1 - 5 y3, y1, 3 y2 - 4 y3, 0, y3, 0]

 

  p' = - s 2 + s 4   p = - s 2 + s 4

Omega Rank for B :  cycles: {{4, 6, 7}, {2, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[0, y2, 0, y3, 0, 2 y2 - y3 - y1, y1, y2]

 

  p' = s - s 4   p = - s + s 4

 » SYNC'D 15/2048 , 0.007324218750

 
7 . Coloring, {7}

R: [3, 3, 1, 1, 7, 7, 4, 5]    B: [6, 8, 8, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -4` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 4` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 4` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [333, 47, 285, 159, 105, 123, 171, 83] . FixedPtCheck, [333, 47, 285, 159, 105, 123, 171, 83]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 + y2 + y3 - y4, 0, y1, y2, y3, 0, y4, 0]

 

  p = - s 4 + s 5

Omega Rank for B :  cycles: {{2, 8}, {4, 6}},   net cycles: 1 .    order:   2

See Matrix
 

[0, y2, 0, y1, 4 y2 - 5 y1 - y3, 3 y2 - 4 y1, 0, y3]

 

  p' = s 2 - s 4   p = s 2 - s 4

 » SYNC'D 5/512 , 0.009765625000

 
8 . Coloring, {8}

R: [3, 3, 1, 1, 7, 7, 5, 2]    B: [6, 8, 8, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )``]`

For τ=1/2, [150, 66, 162, 38, 93, 47, 105, 57] . FixedPtCheck, [150, 66, 162, 38, 93, 47, 105, 57]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 1 .    order:   2

See Matrix
 

[-y3 + 4 y2 - 5 y1, y3, y2, 0, y1, 0, 3 y2 - 4 y1, 0]

 

  p = - s 2 + s 4   p' = - s 2 + s 4

Omega Rank for B :  cycles: {{2, 5, 8}, {4, 6}},   net cycles: 2 .    order:   6

See Matrix
 

[0, y2, 0, y1, -y2 + 2 y1 - y3, y1, 0, y3]

 

  p' = s - s 4   p = s - s 4

 » SYNC'D 15/2048 , 0.007324218750

 
9 . Coloring, {2, 3}

R: [3, 8, 8, 1, 7, 7, 5, 5]    B: [6, 3, 1, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` 1 + τ ` )`` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , -2` (` 1 + τ ` )` 2 ` (` - 5 - τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 2` (` 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 2` (` 5 - τ + 12τ 2 - τ 4 + τ 5 ` )`` (` - 1 + τ ` )` , 6` (` 1 + τ ` )` 2 ` (` - 5 - 3τ - 3τ 2 + 3τ 3 ` )` , -6` (` - 1 + τ ` )` 2 ` (` 5 + τ + 7τ 2 + 3τ 3 ` )` , 2` (` 1 + τ ` )`` (` - 5 - 2τ - 12τ 2 + 2τ 3 + τ 4 ` )` , -2` (` 1 + τ ` )` 3 ` (` - 5 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-249, -369, -279, -239, -990, -122, -834, -486] . FixedPtCheck, [249, 369, 279, 239, 990, 122, 834, 486]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

[y1, 0, y2, 0, y3, 0, y4, y5]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6}},   net cycles: 0 .    order:   4

[y2, y1, y3, y4, 0, y5, 0, 0]  

See Matrices
 

 » SYNC'D 87/2048 , 0.04248046875

 
10 . Coloring, {2, 4}

R: [3, 8, 1, 6, 7, 7, 5, 5]    B: [6, 3, 8, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , -2` (` 1 + τ ` )` , 2` (` - 1 + τ ` )` , -2` (` 1 + τ ` )` , 2` (` - 1 + τ ` )``]`

For τ=1/2, [-1, -1, -1, -1, -3, -1, -3, -1] . FixedPtCheck, [1, 1, 1, 1, 3, 1, 3, 1]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y1, y2, y1, y2, -y2, -y1, -y2, -y1]
  p' = s 2 - 2s 4 + 4s 5   p = s + 4s 5

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s + 4s 5   p' = s 3 - 2s 4 + 2s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 6 3 vs 7 3 vs 7 2 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 0 .    order:   2

See Matrix
 

[y1, 0, y1, 0, y2, 3 y1 - y2, y2, 3 y1 - y2]

 

  p = - s 2 + s 6   p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 4, 6}, {2, 3, 8}},   net cycles: 2 .    order:   3

See Matrix
 

[y2, y3, y2, y3, 0, y1, 0, y1]

 

  p = s - s 4   p' = s - s 4   p' = - s 2 + s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x2, x3, x4, 4 x1 - 2 x4 - 2 x3, 4 x2 + 4 x3 + 2 x4]
For A+2Δ :   [-y2, -y3 - 3 y1 - 3 y5, y2, y3, y1, -y4, y5, y4]
For A-2Δ :   [-y2, y1, y2, y3, -3 y1 - 3 y3 - y4, -y5, y4, y5]

Range of {ΩΔi}: [-μ2, -μ1, -μ2, -μ1, μ1, μ2, μ1, μ2]

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] ] $     $ [ [0, 1, 2, 1, 1, 1, 1, 1] , [1, 0, 1, 2, 1, 1, 1, 1] , [2, 1, 0, 1, 1, 1, 1, 1] , [1, 2, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 2, 1] , [1, 1, 1, 1, 1, 0, 1, 2] , [1, 1, 1, 1, 2, 1, 0, 1] , [1, 1, 1, 1, 1, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, -1, 0, -1, 1, 0, 1, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x2, x3, x1, x4, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[1], [4], [3], [3]]
Action of B on ranges, [[4], [1], [2], [2]]
β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

ker N, [μ3, μ2, μ3, μ2, μ1, %1, μ1, %1] %1 := -μ3 - μ2 - μ1
Range of N
    [y1, -y2 + y3 + y5, -y1 + y3 + y5, y2, -y4 + y3 + y5, y3, y4, y5]

Partitions

Action of R on partitions, [[6], [4], [3], [4], [2], [3], [2], [6]]
Action of B on partitions, [[8], [5], [1], [8], [1], [7], [7], [5]]

α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/8
α([{3, 4, 5, 6}, {1, 2, 7, 8}]) = 1/8
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/8
α([{1, 4, 7, 8}, {2, 3, 5, 6}]) = 1/8
α([{3, 4, 5, 8}, {1, 2, 6, 7}]) = 1/8
α([{1, 4, 5, 6}, {2, 3, 7, 8}]) = 1/8
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 1/8
α([{3, 4, 6, 7}, {1, 2, 5, 8}]) = 1/8

b1 = {3, 4, 5, 6} ` , ` b2 = {1, 4, 7, 8} ` , ` b3 = {3, 4, 7, 8} ` , ` b4 = {1, 4, 6, 7} ` , ` b5 = {1, 2, 7, 8} ` , ` b6 = {1, 2, 5, 6} ` , ` b7 = {3, 4, 5, 8} ` , ` b8 = {2, 3, 5, 8} ` , ` b9 = {2, 3, 5, 6} ` , ` b10 = {1, 2, 6, 7} ` , ` b11 = {1, 4, 5, 6} ` , ` b12 = {2, 3, 7, 8} ` , ` b13 = {3, 4, 6, 7} ` , ` b14 = {1, 2, 5, 8} ` , ` b15 = {1, 4, 5, 8} ` , ` b16 = {2, 3, 6, 7}

Action of R and B on the blocks of the partitions: = [2, 9, 6, 1, 9, 3, 5, 5, 2, 1, 3, 6, B, C, C, B] [A, D, 10, 4, 7, F, 10, 8, E, F, 4, 8, A, 7, D, E]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 4}
Rank2
R,B [3, 8, 1, 6, 7, 7, 5, 5], [6, 3, 8, 1, 2, 4, 4, 2]
π2 [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0]
u2 [1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
11 . Coloring, {2, 5}

R: [3, 8, 1, 1, 2, 7, 5, 5]    B: [6, 3, 8, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )` , 4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-111, -159, -123, -25, -162, -34, -66, -150] . FixedPtCheck, [111, 159, 123, 25, 162, 34, 66, 150]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 5 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[3 y1, 5 y1 + 5 y2 - 3 y3 - 3 y4 - 3 y5, 3 y2, 0, 3 y3, 0, 3 y4, 3 y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{4, 6}, {2, 3, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, y3, y3, y2, 0, 5 y3 - y2 - y1, y1, y3]

 

  p = s 2 - s 6   p' = - s 2 + s 4   p' = s 3 - s 5

 » SYNC'D 2641/131072 , 0.02014923096

 
12 . Coloring, {2, 6}

R: [3, 8, 1, 1, 7, 4, 5, 5]    B: [6, 3, 8, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 2` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 2` (` 5 - τ + 3τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 2` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 2` (` 5 - τ + 3τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [51, 19, 43, 25, 51, 19, 43, 25] . FixedPtCheck, [51, 19, 43, 25, 51, 19, 43, 25]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y1 - y2 - y3, y1, y2, y3, -y1 - y2 - y3, y1, y2, y3]
  p' = s 2 + 4s 5   p = s + 4s 4

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s + 4s 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 2 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 0 .    order:   2

See Matrix
 

[y2, 0, y1, y2 - y1, y2, 0, y1, y2 - y1]

 

  p' = s 2 - s 3   p' = - s 3 + s 4   p = s 2 - s 4   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 3, 8}, {4, 6, 7}},   net cycles: 2 .    order:   3

See Matrix
 

[0, y1, y2, y3, 0, y1, y2, y3]

 

  p' = s 2 - s 5   p' = s - s 4   p = s - s 4


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x3, x2, x1, 4 x3, 4 x2, 4 x1]
For A+2Δ :   [-y2, -y1, -y4, -y3, y2, y1, y4, y3]
For A-2Δ :   [-y1, -y2, -y3, -y4, y1, y2, y3, y4]

Range of {ΩΔi}: [%1, μ2, μ1, μ3, %1, μ2, μ1, μ3] %1 := -μ2 - μ1 - μ3

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] ] $     $ [ [0, 2, 2, 1, 4, 2, 2, 3] , [2, 0, 3, 2, 2, 4, 1, 2] , [2, 3, 0, 1, 2, 1, 4, 3] , [1, 2, 1, 0, 3, 2, 3, 4] , [4, 2, 2, 3, 0, 2, 2, 1] , [2, 4, 1, 2, 2, 0, 3, 2] , [2, 1, 4, 3, 2, 3, 0, 1] , [3, 2, 3, 4, 1, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [1, -1, 0, 0, 1, -1, 0, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [4], [1], [1]]
Action of B on ranges, [[2], [3], [4], [2]]
β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

ker N, [μ2, μ3, μ1, %1, μ2, μ3, μ1, %1] %1 := -μ3 - μ2 - μ1
Range of N
    [y1 + y5 - y2, y1 + y5 - y3, y1 - y4 + y5, y1, y2, y3, y4, y5]

Partitions

Action of R on partitions, [[1], [2], [1], [3], [2]]
Action of B on partitions, [[2], [4], [4], [5], [2]]

α([{2, 3, 4, 5}, {1, 6, 7, 8}]) = 1/8
α([{2, 5, 7, 8}, {1, 3, 4, 6}]) = 3/8
α([{3, 4, 5, 6}, {1, 2, 7, 8}]) = 1/8
α([{1, 2, 4, 7}, {3, 5, 6, 8}]) = 1/4
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/8

b1 = {2, 3, 4, 5} ` , ` b2 = {1, 6, 7, 8} ` , ` b3 = {2, 5, 7, 8} ` , ` b4 = {1, 3, 4, 6} ` , ` b5 = {3, 4, 5, 6} ` , ` b6 = {1, 2, 7, 8} ` , ` b7 = {1, 2, 4, 7} ` , ` b8 = {3, 5, 6, 8} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [2, 1, 3, 4, 2, 1, 5, 6, 4, 3] [3, 4, 8, 7, 7, 8, A, 9, 3, 4]
with invariant measure [1, 1, 3, 3, 1, 1, 2, 2, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 6}
Rank2
R,B [3, 8, 1, 1, 7, 4, 5, 5], [6, 3, 8, 6, 2, 7, 4, 2]
π2 [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
u2 [2, 2, 1, 4, 2, 2, 3, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 3, 3, 2, 3, 4, 2, 2, 1, 3, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
13 . Coloring, {2, 7}

R: [3, 8, 1, 1, 7, 7, 4, 5]    B: [6, 3, 8, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 4` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [255, 47, 203, 137, 102, 98, 150, 86] . FixedPtCheck, [255, 47, 203, 137, 102, 98, 150, 86]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 6 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   6

[y6, 0, y4, y5, y1, 0, y2, y3]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6}, {2, 3, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, 3 y3, -3 y3 + 5 y1 - 3 y2 + 5 y4 - 3 y5, 3 y1, 3 y2, 3 y4, 0, 3 y5]

 

  p = s 2 + s 3 - s 5 - s 6

 » SYNC'D 3891/65536 , 0.05937194824

 
14 . Coloring, {2, 8}

R: [3, 8, 1, 1, 7, 7, 5, 2]    B: [6, 3, 8, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , -2` (` - 1 + τ ` )` , 2` (` 1 + τ ` )` , -2` (` - 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )``]`

For τ=1/2, [3, 3, 3, 1, 3, 1, 3, 3] . FixedPtCheck, [3, 3, 3, 1, 3, 1, 3, 3]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y2, y3, -y2 - y3 - y1, y1, -y2 - y3 - y1, y1, y2, y3]
  p = s + 4s 4 + 8s 5 - 16s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s - 4s 3 - 4s 4 + 8s 5   p' = s 2 + 2s 3 - 4s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 2 vs 6 1 vs 6

Omega Rank for R :  cycles: {{1, 3}, {5, 7}, {2, 8}},   net cycles: 3 .    order:   2

See Matrix
 

[-y1 + 3 y2, y2, y1, 0, y1, 0, -y1 + 3 y2, y2]

 

  p' = - s + s 5   p = - s + s 5   p = - s + s 3   p' = - s + s 3

Omega Rank for B :  cycles: {{4, 6}, {2, 3, 5, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[0, y1, y1, 2 y1, y1, 2 y1, 0, y1]

 

  p = - s + s 2   p = - s + s 3   p = - s + s 4   p = - s + s 5   p = - s + s 6


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x2, x3, x1, -4 x3 + 2 x1 + 4 x2, 8 x2 - 4 x3, -4 x1 - 16 x2 + 8 x3]
For A+2Δ :   [-y4, -y3, y1, y2, -y1, -y2, y4, y3]
For A-2Δ :   [-y3, -y4, -y1, -y2, y1, y2, y3, y4]

Range of {ΩΔi}: [μ2, μ3, %1, μ1, %1, μ1, μ2, μ3] %1 := -μ2 - μ3 - μ1

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $     $ [ [0, 7, 6, 3, 5, 8, 11, 4] , [7, 0, 5, 6, 6, 5, 4, 11] , [6, 5, 0, 3, 11, 8, 5, 6] , [3, 6, 3, 0, 8, 11, 8, 5] , [5, 6, 11, 8, 0, 3, 6, 5] , [8, 5, 8, 11, 3, 0, 3, 6] , [11, 4, 5, 8, 6, 3, 0, 7] , [4, 11, 6, 5, 5, 6, 7, 0] ] $

Check is ΩΔN zero? true, πΔ= [1, 0, 0, -1, 0, -1, 1, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [2], [1], [1]]
Action of B on ranges, [[4], [3], [2], [4]]
β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

ker N, [%1, μ3, μ1, μ2, μ1, μ2, %1, μ3] %1 := -μ1 - μ3 - μ2
Range of N
    [y1, y1 - y4 + y5, y1 + y5 - y2, y1 + y5 - y3, y2, y3, y5, y4]

Partitions

Action of R on partitions, [[1], [1], [6], [5], [4], [6]]
Action of B on partitions, [[3], [4], [2], [5], [3], [5]]

α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 1/11
α([{3, 6, 7, 8}, {1, 2, 4, 5}]) = 1/11
α([{2, 3, 6, 7}, {1, 4, 5, 8}]) = 2/11
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 2/11
α([{2, 5, 6, 7}, {1, 3, 4, 8}]) = 3/11
α([{1, 5, 6, 8}, {2, 3, 4, 7}]) = 2/11

b1 = {2, 3, 6, 7} ` , ` b2 = {1, 4, 5, 8} ` , ` b3 = {1, 2, 5, 6} ` , ` b4 = {3, 4, 7, 8} ` , ` b5 = {2, 5, 6, 7} ` , ` b6 = {3, 6, 7, 8} ` , ` b7 = {1, 2, 4, 5} ` , ` b8 = {1, 3, 4, 8} ` , ` b9 = {1, 5, 6, 8} ` , ` b10 = {2, 3, 4, 7} ` , ` b11 = {1, 2, 3, 4} ` , ` b12 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [9, A, 4, 3, C, 3, 4, B, A, 9, 8, 5] [7, 6, 2, 1, 2, B, C, 1, 8, 5, 5, 8]
with invariant measure [2, 2, 1, 1, 3, 1, 1, 3, 2, 2, 2, 2]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 8}
Rank2
R,B [3, 8, 1, 1, 7, 7, 5, 2], [6, 3, 8, 6, 2, 4, 4, 5]
π2 [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u2 [7, 6, 3, 5, 8, 11, 4, 5, 6, 6, 5, 4, 11, 3, 11, 8, 5, 6, 8, 11, 8, 5, 3, 6, 5, 3, 6, 7] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
15 . Coloring, {3, 4}

R: [3, 3, 8, 6, 7, 7, 5, 5]    B: [6, 8, 1, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 2` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )` , 2` (` - 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -2` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , -6` (` 5 - 4τ + 3τ 2 ` )`` (` 1 + τ ` )` 2 , 6` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` , -2` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 2` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-47, -93, -105, -83, -270, -74, -258, -102] . FixedPtCheck, [47, 93, 105, 83, 270, 74, 258, 102]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 0, 2 y2, 0, y1, y2, y3, y4]

 

  p = s 3 - s 5

Omega Rank for B :  cycles: {{2, 8}, {1, 4, 6}},   net cycles: 2 .    order:   6

See Matrix
 

[3 y3, 3 y1, 0, 3 y2, 0, -3 y3 + 5 y1 - 3 y2 + 5 y4, 0, 3 y4]

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 1/16 , 0.06250000000

 
16 . Coloring, {3, 5}

R: [3, 3, 8, 1, 2, 7, 5, 5]    B: [6, 8, 1, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` - 1 + τ ` )`` (` 1 + τ ` )` , 2` (` 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 , 2` (` - 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 , 2` (` - 1 + τ ` )` 2 , -2` (` - 1 + τ ` )`` (` 1 + τ ` )` , 2` (` 1 + τ ` )` 2 `]`

For τ=1/2, [3, 9, 9, 1, 9, 1, 3, 9] . FixedPtCheck, [3, 9, 9, 1, 9, 1, 3, 9]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y2, y3, -y2 - y3 - y1, y1, -y2 - y3 - y1, y1, y2, y3]
  p = s - 4s 4 - 8s 5

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s - 4s 4 - 8s 5   p' = s 2 + 2s 3 + 4s 4 + 4s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 2 vs 6 2 vs 6

Omega Rank for R :  cycles: {{2, 3, 5, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, y1 + y2, 0, y1 + y2, 0, y1, y2]

 

  p' = - s 2 + s 5   p' = - s 2 + s 4   p' = - s 2 + s 3   p = s 2 - s 3

Omega Rank for B :  cycles: {{4, 6}, {2, 8}},   net cycles: 0 .    order:   2

See Matrix
 

[3 y1 - y2, y1, 0, y2, 0, y2, 3 y1 - y2, y1]

 

  p' = s 3 - s 4   p = s 2 - s 5   p' = s 2 - s 4   p' = - s 4 + s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x2, x3, -4 x1 + 2 x3, -4 x2 - 8 x1 + 4 x3, -8 x2 + 4 x3]
For A+2Δ :   [-y3, -y4, -y1, -y2, y1, y2, y3, y4]
For A-2Δ :   [-y3, -y4, -y1, -y2, y1, y2, y3, y4]

Range of {ΩΔi}: [μ2, μ3, %1, μ1, %1, μ1, μ2, μ3] %1 := -μ2 - μ1 - μ3

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $     $ [ [0, 2, 4, 2, 3, 5, 7, 5] , [2, 0, 4, 4, 3, 3, 5, 7] , [4, 4, 0, 5, 7, 2, 3, 3] , [2, 4, 5, 0, 2, 7, 5, 3] , [3, 3, 7, 2, 0, 5, 4, 4] , [5, 3, 2, 7, 5, 0, 2, 4] , [7, 5, 3, 5, 4, 2, 0, 2] , [5, 7, 3, 3, 4, 4, 2, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, 0, 1, -1, 1, -1, 0, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x5, x3, x4, x8, x6, x7]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [3], [2], [1]]
Action of B on ranges, [[4], [2], [1], [4]]
β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

ker N, [μ3, μ2, %1, μ1, %1, μ1, μ3, μ2] %1 := -μ3 - μ2 - μ1
Range of N
    [y2 + y4 - y3, -y5 + y2 + y4, y2 + y4 - y1, y2, y1, y4, y3, y5]

Partitions

Action of R on partitions, [[2], [5], [5], [6], [3], [3]]
Action of B on partitions, [[1], [4], [3], [1], [4], [3]]

α([{2, 3, 6, 7}, {1, 4, 5, 8}]) = 1/7
α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 1/14
α([{3, 6, 7, 8}, {1, 2, 4, 5}]) = 5/14
α([{2, 5, 6, 7}, {1, 3, 4, 8}]) = 1/7
α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 3/14
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/14

b1 = {2, 3, 6, 7} ` , ` b2 = {1, 4, 5, 8} ` , ` b3 = {1, 2, 3, 6} ` , ` b4 = {1, 2, 5, 6} ` , ` b5 = {3, 4, 7, 8} ` , ` b6 = {2, 5, 6, 7} ` , ` b7 = {3, 6, 7, 8} ` , ` b8 = {1, 2, 4, 5} ` , ` b9 = {1, 3, 4, 8} ` , ` b10 = {4, 5, 7, 8} ` , ` b11 = {1, 2, 3, 4} ` , ` b12 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [4, 5, 8, A, 3, C, 3, A, B, 7, 8, 7] [2, 1, 9, 9, 6, 2, 8, 7, 1, 6, 7, 8]
with invariant measure [2, 2, 3, 1, 1, 2, 5, 5, 2, 3, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {3, 5}
Rank2
R,B [3, 3, 8, 1, 2, 7, 5, 5], [6, 8, 1, 6, 7, 4, 4, 2]
π2 [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u2 [2, 4, 2, 3, 5, 7, 5, 4, 4, 3, 3, 5, 7, 5, 7, 2, 3, 3, 2, 7, 5, 3, 5, 4, 4, 2, 4, 2] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
17 . Coloring, {3, 6}

R: [3, 3, 8, 1, 7, 4, 5, 5]    B: [6, 8, 1, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 4` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [102, 98, 150, 86, 255, 47, 203, 137] . FixedPtCheck, [102, 98, 150, 86, 255, 47, 203, 137]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 6 vs 6 5 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y1, 0, y2, y3, y4, 0, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6, 7}, {2, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - 3 y2 - 3 y3 - 3 y4 + 5 y5, 3 y1, 0, 3 y2, 0, 3 y3, 3 y4, 3 y5]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 3891/65536 , 0.05937194824

 
18 . Coloring, {3, 7}

R: [3, 3, 8, 1, 7, 7, 4, 5]    B: [6, 8, 1, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , 2` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -2` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , 2` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-51, -25, -57, -49, -51, -25, -57, -49] . FixedPtCheck, [51, 25, 57, 49, 51, 25, 57, 49]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y1, -y1 - y2 - y3, y2, y3, y1, -y1 - y2 - y3, y2, y3]
  p = s - 4s 4

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s - 4s 4   p' = s 2 - 4s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 3 vs 6 2 vs 6

Omega Rank for R :  cycles: {{1, 3, 4, 5, 7, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y1, 0, y2, y3, y1, 0, y2, y3]

 

  p' = - s 2 + s 5   p = s - s 4   p' = - s + s 4

Omega Rank for B :  cycles: {{4, 6}, {2, 8}},   net cycles: 0 .    order:   2

See Matrix
 

[y1 - y2, y1, 0, y2, y1 - y2, y1, 0, y2]

 

  p = - s 2 + s 4   p = - s 2 + s 5   p = - s 2 + s 3   p = - s 2 + s 6


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x3, x2, x1, -4 x3, -4 x2, -4 x1]
For A+2Δ :   [-y1, -y2, -y3, -y4, y1, y2, y3, y4]
For A-2Δ :   [-y1, -y2, -y3, -y4, y1, y2, y3, y4]

Range of {ΩΔi}: [μ3, %1, μ1, μ2, μ3, %1, μ1, μ2] %1 := -μ3 - μ1 - μ2

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] ] $     $ [ [0, 1, 2, 1, 4, 3, 2, 3] , [1, 0, 2, 2, 3, 4, 2, 2] , [2, 2, 0, 3, 2, 2, 4, 1] , [1, 2, 3, 0, 3, 2, 1, 4] , [4, 3, 2, 3, 0, 1, 2, 1] , [3, 4, 2, 2, 1, 0, 2, 2] , [2, 2, 4, 1, 2, 2, 0, 3] , [3, 2, 1, 4, 1, 2, 3, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, -1, 1, 0, 0, -1, 1, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x7, x6, x5, x4, x3, x2, x1, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [3], [4], [1]]
Action of B on ranges, [[2], [4], [1], [2]]
β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

ker N, [%1, μ3, μ2, μ1, %1, μ3, μ2, μ1] %1 := -μ3 - μ2 - μ1
Range of N
    [y5, y3, y4, y2, y1, -y3 + y5 + y1, -y4 + y5 + y1, y5 - y2 + y1]

Partitions

Action of R on partitions, [[2], [5], [2], [5], [1]]
Action of B on partitions, [[2], [2], [3], [3], [4]]

α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/8
α([{3, 5, 6, 8}, {1, 2, 4, 7}]) = 3/8
α([{2, 5, 7, 8}, {1, 3, 4, 6}]) = 1/8
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 1/8
α([{1, 2, 3, 8}, {4, 5, 6, 7}]) = 1/4

b1 = {1, 4, 6, 7} ` , ` b2 = {1, 2, 3, 8} ` , ` b3 = {4, 5, 6, 7} ` , ` b4 = {3, 5, 6, 8} ` , ` b5 = {2, 5, 7, 8} ` , ` b6 = {1, 2, 4, 7} ` , ` b7 = {1, 3, 4, 6} ` , ` b8 = {1, 2, 3, 4} ` , ` b9 = {5, 6, 7, 8} ` , ` b10 = {2, 3, 5, 8}

Action of R and B on the blocks of the partitions: = [3, 8, 9, 2, 4, 3, 6, 6, 4, 2] [7, A, 1, 6, 5, 4, 7, 4, 6, 5]
with invariant measure [1, 2, 2, 3, 1, 3, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {3, 7}
Rank2
R,B [3, 3, 8, 1, 7, 7, 4, 5], [6, 8, 1, 6, 2, 4, 5, 2]
π2 [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
u2 [1, 2, 1, 4, 3, 2, 3, 2, 2, 3, 4, 2, 2, 3, 2, 2, 4, 1, 3, 2, 1, 4, 1, 2, 1, 2, 2, 3] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
19 . Coloring, {3, 8}

R: [3, 3, 8, 1, 7, 7, 5, 2]    B: [6, 8, 1, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )` , 4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-66, -150, -162, -34, -123, -25, -111, -159] . FixedPtCheck, [66, 150, 162, 34, 123, 25, 111, 159]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 5 vs 6 3 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-3 y1 - 3 y2 + 5 y3 + 5 y5 - 3 y4, 3 y1, 3 y2, 0, 3 y3, 0, 3 y5, 3 y4]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{4, 6}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y3, y2, 0, y1, y2, -y3 - y1 + 5 y2, 0, y2]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6

 » SYNC'D 2641/131072 , 0.02014923096

 
20 . Coloring, {4, 5}

R: [3, 3, 1, 6, 2, 7, 5, 5]    B: [6, 8, 8, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [203, 137, 255, 47, 150, 86, 102, 98] . FixedPtCheck, [203, 137, 255, 47, 150, 86, 102, 98]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 6 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   6

[y6, y5, y4, 0, y3, y2, y1, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 8}, {1, 4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - 3 y2 - 3 y3 - 3 y4 + 5 y5, 3 y1, 0, 3 y2, 0, 3 y3, 3 y4, 3 y5]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 3891/65536 , 0.05937194824

 
21 . Coloring, {4, 6}

R: [3, 3, 1, 6, 7, 4, 5, 5]    B: [6, 8, 8, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 1 + τ ` )` , -2` (` - 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , -2` (` - 1 + τ ` )``]`

For τ=1/2, [3, 1, 3, 3, 3, 3, 3, 1] . FixedPtCheck, [3, 1, 3, 3, 3, 3, 3, 1]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y3 - y2 - y1, y1, y3, y2, y3, y2, -y3 - y2 - y1, y1]
  p' = s 2 + 2s 3 - 4s 5   p' = s - 4s 3 - 4s 4 + 8s 5   p = s + 4s 4 + 8s 5 - 16s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 2 vs 6 1 vs 6

Omega Rank for R :  cycles: {{1, 3}, {5, 7}, {4, 6}},   net cycles: 3 .    order:   2

See Matrix
 

[-y1 + 3 y2, 0, y1, y2, y1, y2, -y1 + 3 y2, 0]

 

  p' = s - s 5   p' = s 2 - s 4   p' = s 3 - s 5   p = s - s 5

Omega Rank for B :  cycles: {{1, 4, 6, 7}, {2, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, 2 y1, 0, y1, 0, y1, y1, 2 y1]

 

  p = - s + s 2   p = - s + s 3   p = - s + s 4   p = - s + s 5   p = - s + s 6


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x2, x3, 4 x1 + 2 x3 - 4 x2, 8 x1 - 4 x2, -4 x3 - 16 x1 + 8 x2]
For A+2Δ :   [-y2, -y1, y4, -y3, -y4, y3, y2, y1]
For A-2Δ :   [-y2, -y3, -y1, -y4, y1, y4, y2, y3]

Range of {ΩΔi}: [%1, μ3, μ2, μ1, μ2, μ1, %1, μ3] %1 := -μ3 - μ2 - μ1

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $     $ [ [0, 3, 6, 5, 5, 6, 11, 8] , [3, 0, 3, 6, 8, 5, 8, 11] , [6, 3, 0, 7, 11, 4, 5, 8] , [5, 6, 7, 0, 4, 11, 6, 5] , [5, 8, 11, 4, 0, 7, 6, 3] , [6, 5, 4, 11, 7, 0, 5, 6] , [11, 8, 5, 6, 6, 5, 0, 3] , [8, 11, 8, 5, 3, 6, 3, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, -1, 1, 0, 1, 0, 0, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x4, x3, x2, x1, x8, x7, x5, x6]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [3], [1], [4]]
Action of B on ranges, [[4], [2], [2], [1]]
β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

ker N, [μ2, μ3, %1, μ1, %1, μ1, μ2, μ3] %1 := -μ2 - μ1 - μ3
Range of N
    [y1 + y3 - y4, y1 + y3 - y5, -y2 + y1 + y3, y1, y2, y3, y4, y5]

Partitions

Action of R on partitions, [[5], [2], [2], [4], [1], [4]]
Action of B on partitions, [[3], [1], [6], [3], [1], [5]]

α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 3/11
α([{1, 2, 4, 5}, {3, 6, 7, 8}]) = 2/11
α([{2, 3, 6, 7}, {1, 4, 5, 8}]) = 2/11
α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 1/11
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 2/11
α([{2, 3, 4, 7}, {1, 5, 6, 8}]) = 1/11

b1 = {1, 2, 3, 6} ` , ` b2 = {1, 2, 5, 6} ` , ` b3 = {2, 3, 6, 7} ` , ` b4 = {3, 4, 7, 8} ` , ` b5 = {1, 4, 5, 8} ` , ` b6 = {4, 5, 7, 8} ` , ` b7 = {1, 2, 4, 5} ` , ` b8 = {3, 6, 7, 8} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {5, 6, 7, 8} ` , ` b11 = {2, 3, 4, 7} ` , ` b12 = {1, 5, 6, 8}

Action of R and B on the blocks of the partitions: = [9, 4, 7, 2, 8, A, 8, 7, 1, 6, 2, 4] [5, 5, C, 3, B, 3, 6, 1, 6, 1, A, 9]
with invariant measure [3, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {4, 6}
Rank2
R,B [3, 3, 1, 6, 7, 4, 5, 5], [6, 8, 8, 1, 2, 7, 4, 2]
π2 [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u2 [3, 6, 5, 5, 6, 11, 8, 3, 6, 8, 5, 8, 11, 7, 11, 4, 5, 8, 4, 11, 6, 5, 7, 6, 3, 5, 6, 3] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
22 . Coloring, {4, 7}

R: [3, 3, 1, 6, 7, 7, 4, 5]    B: [6, 8, 8, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )` , 4` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )` , 4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-123, -25, -111, -159, -66, -150, -162, -34] . FixedPtCheck, [123, 25, 111, 159, 66, 150, 162, 34]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 5 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[3 y1, 0, 3 y5, 3 y4, 3 y3, 5 y1 + 5 y5 - 3 y4 - 3 y3 - 3 y2, 3 y2, 0]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{2, 8}, {1, 4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, 5 y2 - y1 - y3, 0, y2, y1, y2, 0, y3]

 

  p' = s 3 - s 5   p' = s 2 - s 4   p = s 2 - s 6

 » SYNC'D 2641/131072 , 0.02014923096

 
23 . Coloring, {4, 8}

R: [3, 3, 1, 6, 7, 7, 5, 2]    B: [6, 8, 8, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 5 - τ + 3τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 2` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 2` (` 5 - τ + 3τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 2` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [43, 25, 51, 19, 43, 25, 51, 19] . FixedPtCheck, [43, 25, 51, 19, 43, 25, 51, 19]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y3, y1, y2, -y3 - y1 - y2, y3, y1, y2, -y3 - y1 - y2]
  p' = s 2 + 4s 5   p = s + 4s 4

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s + 4s 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 2 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 0 .    order:   2

See Matrix
 

[y2, -y2 + y1, y1, 0, y2, -y2 + y1, y1, 0]

 

  p' = s 2 - s 5   p' = s 4 - s 5   p' = s 3 - s 5   p = s 2 - s 6

Omega Rank for B :  cycles: {{2, 5, 8}, {1, 4, 6}},   net cycles: 2 .    order:   3

See Matrix
 

[y1, y3, 0, y2, y1, y3, 0, y2]

 

  p = - s + s 4   p' = - s + s 4   p' = - s 2 + s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x2, x3, 4 x1, 4 x2, 4 x3]
For A+2Δ :   [-y2, -y3, -y4, -y1, y2, y3, y4, y1]
For A-2Δ :   [-y1, -y2, -y3, -y4, y1, y2, y3, y4]

Range of {ΩΔi}: [%1, μ1, μ2, μ3, %1, μ1, μ2, μ3] %1 := -μ1 - μ2 - μ3

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] ] $     $ [ [0, 1, 2, 3, 4, 3, 2, 1] , [1, 0, 1, 2, 3, 4, 3, 2] , [2, 1, 0, 2, 2, 3, 4, 2] , [3, 2, 2, 0, 1, 2, 2, 4] , [4, 3, 2, 1, 0, 1, 2, 3] , [3, 4, 3, 2, 1, 0, 1, 2] , [2, 3, 4, 2, 2, 1, 0, 2] , [1, 2, 2, 4, 3, 2, 2, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, 0, 1, -1, 0, 0, 1, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x2, x1, x4, x3, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [3], [1], [2]]
Action of B on ranges, [[2], [4], [4], [1]]
β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

ker N, [%1, μ2, μ1, μ3, %1, μ2, μ1, μ3] %1 := -μ1 - μ3 - μ2
Range of N
    [y2 + y5 - y1, y2 + y5 - y3, y2, y2 + y5 - y4, y1, y3, y5, y4]

Partitions

Action of R on partitions, [[1], [2], [2], [3], [1]]
Action of B on partitions, [[4], [1], [4], [5], [1]]

α([{4, 5, 6, 7}, {1, 2, 3, 8}]) = 3/8
α([{1, 2, 4, 7}, {3, 5, 6, 8}]) = 1/8
α([{3, 4, 5, 6}, {1, 2, 7, 8}]) = 1/8
α([{2, 3, 4, 5}, {1, 6, 7, 8}]) = 1/4
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/8

b1 = {1, 2, 4, 7} ` , ` b2 = {4, 5, 6, 7} ` , ` b3 = {1, 2, 3, 8} ` , ` b4 = {3, 5, 6, 8} ` , ` b5 = {3, 4, 5, 6} ` , ` b6 = {1, 2, 7, 8} ` , ` b7 = {2, 3, 4, 5} ` , ` b8 = {1, 6, 7, 8} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [4, 2, 3, 1, 1, 4, 6, 5, 3, 2] [2, 8, 7, 3, 8, 7, A, 9, 2, 3]
with invariant measure [1, 3, 3, 1, 1, 1, 2, 2, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {4, 8}
Rank2
R,B [3, 3, 1, 6, 7, 7, 5, 2], [6, 8, 8, 1, 2, 4, 4, 5]
π2 [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
u2 [1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 3, 2, 2, 2, 3, 4, 2, 1, 2, 2, 4, 1, 2, 3, 1, 2, 2] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
24 . Coloring, {5, 6}

R: [3, 3, 1, 1, 2, 4, 5, 5]    B: [6, 8, 8, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 2` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -6` (` 1 + τ ` )` 2 ` (` 5 - 4τ + 3τ 2 ` )` , 6` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` , 2` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -2` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , -2` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 2` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-258, -102, -270, -74, -105, -83, -47, -93] . FixedPtCheck, [258, 102, 270, 74, 105, 83, 47, 93]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y3, y4, y1, 2 y1, 0, 0, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 8}, {4, 6, 7}},   net cycles: 2 .    order:   6

See Matrix
 

[0, 3 y2, 0, 5 y2 - 3 y1 - 3 y3 + 5 y4, 0, 3 y1, 3 y3, 3 y4]

 

  p = s + s 2 - s 4 - s 5

 » SYNC'D 1/16 , 0.06250000000

 
25 . Coloring, {5, 7}

R: [3, 3, 1, 1, 2, 7, 4, 5]    B: [6, 8, 8, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` 1 + τ ` )` , 2` (` - 1 + τ ` )` , -2` (` 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )``]`

For τ=1/2, [-3, -1, -3, -1, -1, -1, -1, -1] . FixedPtCheck, [3, 1, 3, 1, 1, 1, 1, 1]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y2, -y1, -y2, -y1, y1, y2, y1, y2]
  p = s - 4s 5

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s - 4s 5   p' = s 3 - 2s 5   p' = s 2 - 2s 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 6 3 vs 7 3 vs 7 3 vs 6 2 vs 6

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y1, y2, y1, y3, 0, y3, 0]

 

  p = s 3 - s 4   p' = - s 3 + s 4   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{5, 7}, {4, 6}, {2, 8}},   net cycles: 3 .    order:   2

See Matrix
 

[0, 3 y1 - y2, 0, 3 y1 - y2, y1, y2, y1, y2]

 

  p' = s 3 - s 5   p = s - s 5   p' = s - s 5   p' = s 2 - s 4


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x3, x1, x2, x4, -4 x3 - 2 x2, -4 x1 - 2 x4]
For A+2Δ :   [y1, y2, y4, -y2, -y5, -3 y1 - 3 y4 - y3, y5, y3]
For A-2Δ :   [-y1 - 3 y3 - 3 y5, -y2, y1, y2, -y4, y3, y4, y5]

Range of {ΩΔi}: [μ2, -μ1, μ2, -μ1, μ1, -μ2, μ1, -μ2]

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] ] $     $ [ [0, 1, 2, 1, 1, 1, 1, 1] , [1, 0, 1, 2, 1, 1, 1, 1] , [2, 1, 0, 1, 1, 1, 1, 1] , [1, 2, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 2, 1] , [1, 1, 1, 1, 1, 0, 1, 2] , [1, 1, 1, 1, 2, 1, 0, 1] , [1, 1, 1, 1, 1, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [1, 0, 1, 0, 0, -1, 0, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x8, x7, x5, x6, x4, x2, x3, x1]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[1], [1], [2], [3]]
Action of B on ranges, [[4], [4], [3], [2]]
β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

ker N, [%1, μ1, %1, μ1, μ2, μ3, μ2, μ3] %1 := -μ1 - μ2 - μ3
Range of N
    [y3 + y5 - y2, -y1 + y3 + y5, y2, y1, -y4 + y3 + y5, y3, y4, y5]

Partitions

Action of R on partitions, [[5], [4], [6], [1], [5], [1], [4], [6]]
Action of B on partitions, [[3], [8], [3], [8], [2], [7], [7], [2]]

α([{1, 2, 6, 7}, {3, 4, 5, 8}]) = 1/8
α([{2, 3, 5, 8}, {1, 4, 6, 7}]) = 1/8
α([{2, 3, 6, 7}, {1, 4, 5, 8}]) = 1/8
α([{1, 2, 5, 8}, {3, 4, 6, 7}]) = 1/8
α([{3, 4, 5, 6}, {1, 2, 7, 8}]) = 1/8
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/8
α([{1, 4, 7, 8}, {2, 3, 5, 6}]) = 1/8
α([{2, 3, 7, 8}, {1, 4, 5, 6}]) = 1/8

b1 = {1, 2, 5, 8} ` , ` b2 = {2, 3, 5, 8} ` , ` b3 = {2, 3, 6, 7} ` , ` b4 = {1, 4, 5, 8} ` , ` b5 = {2, 3, 7, 8} ` , ` b6 = {1, 4, 5, 6} ` , ` b7 = {1, 4, 6, 7} ` , ` b8 = {3, 4, 5, 6} ` , ` b9 = {1, 2, 7, 8} ` , ` b10 = {1, 4, 7, 8} ` , ` b11 = {3, 4, 7, 8} ` , ` b12 = {1, 2, 5, 6} ` , ` b13 = {2, 3, 5, 6} ` , ` b14 = {1, 2, 6, 7} ` , ` b15 = {3, 4, 5, 8} ` , ` b16 = {3, 4, 6, 7}

Action of R and B on the blocks of the partitions: = [F, 1, C, B, C, B, 10, 9, 8, 10, E, F, 1, 8, 9, E] [5, 5, 4, 3, 2, 7, 6, 7, 2, D, D, A, A, 4, 3, 6]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {5, 7}
Rank2
R,B [3, 3, 1, 1, 2, 7, 4, 5], [6, 8, 8, 6, 7, 4, 5, 2]
π2 [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0]
u2 [1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
26 . Coloring, {5, 8}

R: [3, 3, 1, 1, 2, 7, 5, 2]    B: [6, 8, 8, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 1 + τ ` )`` (` - 5 - 2τ - 12τ 2 + 2τ 3 + τ 4 ` )` , -2` (` - 1 + τ ` )`` (` - 5 + τ ` )`` (` 1 + τ ` )` 3 , 6` (` - 5 - 3τ - 3τ 2 + 3τ 3 ` )`` (` 1 + τ ` )` 2 , -6` (` - 1 + τ ` )` 2 ` (` 5 + τ + 7τ 2 + 3τ 3 ` )` , 2` (` - 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 2` (` 5 - τ + 12τ 2 - τ 4 + τ 5 ` )`` (` - 1 + τ ` )` , -2` (` - 1 + τ ` )`` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , -2` (` - 5 - τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-834, -486, -990, -122, -279, -239, -249, -369] . FixedPtCheck, [834, 486, 990, 122, 279, 239, 249, 369]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   4

[y1, y2, y4, 0, y5, 0, y3, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6}},   net cycles: 0 .    order:   4

[0, 0, 0, y5, y4, y2, y3, y1]  

See Matrices
 

 » SYNC'D 87/2048 , 0.04248046875

 
27 . Coloring, {6, 7}

R: [3, 3, 1, 1, 7, 4, 4, 5]    B: [6, 8, 8, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `6` (` - 5 - 3τ - 3τ 2 + 3τ 3 ` )`` (` 1 + τ ` )` 2 , -6` (` 5 + τ + 7τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` 2 , 2` (` 1 + τ ` )`` (` - 5 - 2τ - 12τ 2 + 2τ 3 + τ 4 ` )` , -2` (` - 5 + τ ` )`` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` , -2` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )` , -2` (` - 5 - τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 2` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 2` (` 5 - τ + 12τ 2 - τ 4 + τ 5 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-990, -122, -834, -486, -249, -369, -279, -239] . FixedPtCheck, [990, 122, 834, 486, 249, 369, 279, 239]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   4

[y1, 0, y4, y5, y3, 0, y2, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 8}},   net cycles: 0 .    order:   4

[0, y1, 0, 0, y2, y5, y4, y3]  

See Matrices
 

 » SYNC'D 87/2048 , 0.04248046875

 
28 . Coloring, {6, 8}

R: [3, 3, 1, 1, 7, 4, 5, 2]    B: [6, 8, 8, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` 1 + τ ` )` , 2` (` - 1 + τ ` )` , -2` (` 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )``]`

For τ=1/2, [-3, -1, -3, -1, -1, -1, -1, -1] . FixedPtCheck, [3, 1, 3, 1, 1, 1, 1, 1]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y2, y1, -y2, y1, -y1, y2, -y1, y2]
  p = s + 4s 5

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 2 - 2s 4 + 4s 5   p' = s 3 - 2s 4 + 2s 5   p' = s + 4s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 6 3 vs 7 3 vs 7 2 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {5, 7}},   net cycles: 0 .    order:   2

See Matrix
 

[-y1 + 3 y2, y1, -y1 + 3 y2, y1, y2, 0, y2, 0]

 

  p = - s 2 + s 5   p = - s 2 + s 6   p = - s 2 + s 3   p = - s 2 + s 4

Omega Rank for B :  cycles: {{4, 6, 7}, {2, 5, 8}},   net cycles: 2 .    order:   3

See Matrix
 

[0, y3, 0, y3, y1, y2, y1, y2]

 

  p' = s - s 4   p = s - s 4   p' = - s 2 + s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x3, x4, x2, x1, 4 x3 - 2 x1 - 2 x2, 4 x4 + 4 x2 + 2 x1]
For A+2Δ :   [y1, -y3, y2, y3, -y5, -3 y1 - 3 y2 - y4, y5, y4]
For A-2Δ :   [y4, y5, -y4 - 3 y3 - 3 y1, -y5, y2, y3, -y2, y1]

Range of {ΩΔi}: [-μ2, -μ1, -μ2, -μ1, μ1, μ2, μ1, μ2]

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] ] $     $ [ [0, 1, 2, 1, 1, 1, 1, 1] , [1, 0, 1, 2, 1, 1, 1, 1] , [2, 1, 0, 1, 1, 1, 1, 1] , [1, 2, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 2, 1] , [1, 1, 1, 1, 1, 0, 1, 2] , [1, 1, 1, 1, 2, 1, 0, 1] , [1, 1, 1, 1, 1, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [1, 0, 1, 0, 0, -1, 0, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[1], [1], [3], [2]]
Action of B on ranges, [[4], [4], [2], [3]]
β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

ker N, [μ1, %1, μ1, %1, μ3, μ2, μ3, μ2] %1 := -μ1 - μ3 - μ2
Range of N
    [y2, y1, y3 + y5 - y2, -y1 + y3 + y5, -y4 + y3 + y5, y5, y4, y3]

Partitions

Action of R on partitions, [[6], [7], [1], [4], [1], [6], [4], [7]]
Action of B on partitions, [[3], [8], [2], [2], [5], [8], [5], [3]]

α([{3, 4, 5, 6}, {1, 2, 7, 8}]) = 1/8
α([{2, 3, 6, 7}, {1, 4, 5, 8}]) = 1/8
α([{1, 4, 7, 8}, {2, 3, 5, 6}]) = 1/8
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/8
α([{2, 3, 5, 8}, {1, 4, 6, 7}]) = 1/8
α([{1, 2, 6, 7}, {3, 4, 5, 8}]) = 1/8
α([{1, 2, 5, 8}, {3, 4, 6, 7}]) = 1/8
α([{2, 3, 7, 8}, {1, 4, 5, 6}]) = 1/8

b1 = {1, 2, 5, 8} ` , ` b2 = {2, 3, 5, 8} ` , ` b3 = {2, 3, 6, 7} ` , ` b4 = {1, 4, 5, 8} ` , ` b5 = {2, 3, 7, 8} ` , ` b6 = {1, 4, 5, 6} ` , ` b7 = {1, 4, 6, 7} ` , ` b8 = {3, 4, 5, 6} ` , ` b9 = {1, 2, 7, 8} ` , ` b10 = {1, 4, 7, 8} ` , ` b11 = {3, 4, 7, 8} ` , ` b12 = {1, 2, 5, 6} ` , ` b13 = {2, 3, 5, 6} ` , ` b14 = {1, 2, 6, 7} ` , ` b15 = {3, 4, 5, 8} ` , ` b16 = {3, 4, 6, 7}

Action of R and B on the blocks of the partitions: = [B, 9, 1, 10, 1, 10, 8, E, F, 8, C, B, 9, F, E, C] [2, 2, 6, 5, D, A, 7, A, D, 3, 3, 4, 4, 6, 5, 7]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {6, 8}
Rank2
R,B [3, 3, 1, 1, 7, 4, 5, 2], [6, 8, 8, 6, 2, 7, 4, 5]
π2 [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0]
u2 [1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
29 . Coloring, {7, 8}

R: [3, 3, 1, 1, 7, 7, 4, 2]    B: [6, 8, 8, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-6` (` 1 + τ ` )` 2 ` (` 5 - 4τ + 3τ 2 ` )` , 6` (` - 1 + τ ` )`` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )` , -2` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 2` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -2` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )` , 2` (` 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 2` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -2` (` - 1 + τ ` )`` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )``]`

For τ=1/2, [-270, -74, -258, -102, -47, -93, -105, -83] . FixedPtCheck, [270, 74, 258, 102, 47, 93, 105, 83]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   4

See Matrix
 

[y4, y3, y1, y2, 0, 0, 2 y3, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{4, 6}, {2, 5, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[0, 5 y1 - 3 y2 + 5 y3 - 3 y4, 0, 3 y1, 3 y2, 3 y3, 0, 3 y4]

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 1/16 , 0.06250000000

 
30 . Coloring, {2, 3, 4}

R: [3, 8, 8, 6, 7, 7, 5, 5]    B: [6, 3, 1, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` 5 - τ + 12τ 2 - τ 4 + τ 5 ` )` , -4` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )` 2 , 4` (` - 1 + τ ` )`` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (`5 + τ + 10τ 2 - 2τ 3 + τ 4 + τ 5 ` )`` (` 1 + τ ` )` , 4` (` - 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-95, -279, -141, -239, -801, -203, -753, -315] . FixedPtCheck, [95, 279, 141, 239, 801, 203, 753, 315]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 5 vs 5

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 0, y3, 0, y2, y3, y1, y2 + 2 y3 - y1]

 

  p = - s 3 + s 4   p = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   3

[y4, y5, y3, y2, 0, y1, 0, 0]  

See Matrices
 

 » SYNC'D 3/64 , 0.04687500000

 
31 . Coloring, {2, 3, 5}

R: [3, 8, 8, 1, 2, 7, 5, 5]    B: [6, 3, 1, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 3 , -4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -4` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 , 12` (` - 5 - 3τ - 3τ 2 + 3τ 3 ` )`` (` 1 + τ ` )` 2 , 12` (` - 1 + τ ` )` 3 ` (` 5 + 4τ + 3τ 2 ` )` , -4` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 `]`

For τ=1/2, [-159, -999, -369, -89, -990, -62, -294, -1026] . FixedPtCheck, [159, 999, 369, 89, 990, 62, 294, 1026]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{2, 5, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y4, y2, y3, 0, y1, 0, y4, y5]

 

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 6}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y3, y1, -y2 + y1 + y4, 0, y4, y3, 0]

 

  p' = s 4 - s 5   p = s 4 - s 6

 » SYNC'D 855/65536 , 0.01304626465

 
32 . Coloring, {2, 3, 6}

R: [3, 8, 8, 1, 7, 4, 5, 5]    B: [6, 3, 1, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )`` (` - 5 + 3τ - 7τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -4` (` 5 + 10τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 12` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )` , 12` (` 5 + τ + 7τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` 2 , 12` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + 3τ 2 ` )` , -12` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [123, 147, 129, 121, 381, 61, 301, 207] . FixedPtCheck, [123, 147, 129, 121, 381, 61, 301, 207]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y4, 0, y4 - y1 - y2 + y3 + y5, y1, y2, 0, y3, y5]

 

  p = - s 5 + s 6

Omega Rank for B :  cycles: {{4, 6, 7}},   net cycles: 0 .    order:   6

[y1, y2, y4, y3, 0, y5, y6, 0]  

See Matrices
 

 » SYNC'D 555/8192 , 0.06774902344

 
33 . Coloring, {2, 3, 7}

R: [3, 8, 8, 1, 7, 7, 4, 5]    B: [6, 3, 1, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 + 3τ - 16τ 2 + 4τ 3 - 3τ 4 + τ 5 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (` 5 + 2τ + 19τ 2 + 7τ 4 - 2τ 5 + τ 6 ` )` , -12` (` 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , 12` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + 3τ 2 ` )` , 4` (` - 5 - 2τ - 12τ 2 + 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , 4` (` - 5 + 3τ - 7τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-687, -375, -609, -713, -762, -350, -834, -738] . FixedPtCheck, [687, 375, 609, 713, 762, 350, 834, 738]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 6 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 3, 4, 5, 7, 8}},   net cycles: 1 .    order:   6

[y4, 0, y3, y1, y2, 0, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6}},   net cycles: 0 .    order:   6

See Matrix
 

[y1, -y1 + y5 - y4 + y2 + y3, y5, y4, y2, y3, 0, 0]

 

  p = s 5 - s 6

 » SYNC'D 2665/65536 , 0.04066467285

 
34 . Coloring, {2, 3, 8}

R: [3, 8, 8, 1, 7, 7, 5, 2]    B: [6, 3, 1, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )` 3 ` (` - 5 + τ ` )``]`

For τ=1/2, [-57, -225, -99, -43, -171, -25, -147, -243] . FixedPtCheck, [57, 225, 99, 43, 171, 25, 147, 243]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{5, 7}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[4 y1 + 4 y2 - 5 y4 - y3, y1, y2, 0, y4, 0, 3 y1 + 3 y2 - 4 y4, y3]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{4, 6}},   net cycles: 0 .    order:   6

See Matrix
 

[-y1 + y2 - y3 + y4 + y5, y1, y2, y3, y4, y5, 0, 0]

 

  p = - s 5 + s 6

 » SYNC'D 463/65536 , 0.007064819336

 
35 . Coloring, {2, 4, 5}

R: [3, 8, 1, 6, 2, 7, 5, 5]    B: [6, 3, 8, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 4` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [73, 137, 89, 25, 141, 37, 63, 125] . FixedPtCheck, [73, 137, 89, 25, 141, 37, 63, 125]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 7 4 vs 7

Omega Rank for R :  cycles: {{1, 3}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, y1, y2, 0, 2 y2, 2 y2 - y1, y3, 2 y2 - y3]

 

  p' = - s 3 + s 4   p' = - s 3 + s 6   p' = - s 3 + s 5   p = s 3 - s 4

Omega Rank for B :  cycles: {{1, 4, 6}, {2, 3, 8}},   net cycles: 1 .    order:   3

See Matrix
 

[5 y1 - y4 - y3 - y2, y1, y1, y4, 0, y3, y2, y1]

 

  p' = - s 3 + s 6   p' = - s 2 + s 5   p = - s 2 + s 5

 » SYNC'D 3999/262144 , 0.01525497437

 
36 . Coloring, {2, 4, 6}

R: [3, 8, 1, 6, 7, 4, 5, 5]    B: [6, 3, 8, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 5 + τ 2 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )` , 4` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -4` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [38, 22, 34, 50, 63, 47, 59, 25] . FixedPtCheck, [38, 22, 34, 50, 63, 47, 59, 25]

det(A + τ Δ) =   1` (` 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Omega Rank for R :  cycles: {{1, 3}, {5, 7}, {4, 6}},   net cycles: 2 .    order:   2

See Matrix
 

[y2, 0, y2, y2, 2 y2, y2, 2 y2 - y1, y1]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 6   p = - s 2 + s 7   p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 4, 6, 7}, {2, 3, 8}},   net cycles: 2 .   

See Matrix
 

[y2, 4 y2 - y1 - y3, y1, y2, 0, y2, y2, y3]

 

  p' = s 3 - s 6   p' = s 2 - s 5   p' = s - s 4   p = s - s 7

 » SYNC'D 145/131072 , 0.001106262207

 
37 . Coloring, {2, 4, 7}

R: [3, 8, 1, 6, 7, 7, 4, 5]    B: [6, 3, 8, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 4` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , -4` (` - 1 + τ ` )`` (` 5 + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [89, 25, 73, 137, 63, 125, 141, 37] . FixedPtCheck, [89, 25, 73, 137, 63, 125, 141, 37]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 7 4 vs 7

Omega Rank for R :  cycles: {{1, 3}, {4, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y1, 0, y1, y3, y2, 2 y1 - y2, 2 y1, 2 y1 - y3]

 

  p = - s 3 + s 7   p = - s 3 + s 6   p = - s 3 + s 5   p = - s 3 + s 4

Omega Rank for B :  cycles: {{1, 4, 6}, {2, 3, 8}},   net cycles: 1 .    order:   3

See Matrix
 

[y3, 5 y3 - y1 - y2 - y4, y1, y3, y2, y3, 0, y4]

 

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = s 3 - s 6

 » SYNC'D 3999/262144 , 0.01525497437

 
38 . Coloring, {2, 4, 8}

R: [3, 8, 1, 6, 7, 7, 5, 2]    B: [6, 3, 8, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + 2τ + τ 2 ` )` , -4` (` - 5 + τ 2 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )` , 4` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [34, 50, 38, 22, 59, 25, 63, 47] . FixedPtCheck, [34, 50, 38, 22, 59, 25, 63, 47]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Omega Rank for R :  cycles: {{1, 3}, {5, 7}, {2, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[y1, y1, y1, 0, 2 y1 - y2, y2, 2 y1, y1]

 

  p = - s 2 + s 3   p = - s 2 + s 5   p = - s 2 + s 6   p = - s 2 + s 7   p = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 4, 6}, {2, 3, 5, 8}},   net cycles: 2 .   

See Matrix
 

[y3, y1, y1, -y3 + 4 y1 - y2, y1, y2, 0, y1]

 

  p = - s + s 4   p' = - s + s 4   p' = - s 2 + s 5   p = - s + s 7

 » SYNC'D 145/131072 , 0.001106262207

 
39 . Coloring, {2, 5, 6}

R: [3, 8, 1, 1, 2, 4, 5, 5]    B: [6, 3, 8, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )` , 4` (` 1 + τ ` )` , 4` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )` , 4` (` 1 + τ ` )``]`

For τ=1/2, [3, 3, 3, 1, 3, 1, 1, 3] . FixedPtCheck, [3, 3, 3, 1, 3, 1, 1, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y4, y3, y2, y4 - y2, y1, 0, 0, -y3 + 2 y4 - y1]

 

  p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{4, 6, 7}, {2, 3, 8}},   net cycles: 2 .    order:   3

See Matrix
 

[0, y3, y3, 5 y3 - y1 - y2, 0, y1, y2, y3]

 

  p' = s 2 - s 5   p' = s - s 4   p = s - s 4

 » SYNC'D 525/32768 , 0.01602172852

 
40 . Coloring, {2, 5, 7}

R: [3, 8, 1, 1, 2, 7, 4, 5]    B: [6, 3, 8, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 - τ - τ 2 + τ 3 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 5 - 2τ + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )` , -4` (` - 5 + τ 2 ` )``]`

For τ=1/2, [51, 35, 47, 21, 34, 18, 22, 38] . FixedPtCheck, [51, 35, 47, 21, 34, 18, 22, 38]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ ` )`` (` 1 + τ 2 ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 4 vs 7 2 vs 7

Omega Rank for R :  cycles: {{1, 3}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, y4, y3, y1, y4, 0, -y2 + 5 y4 - y3 - y1, y4]

 

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

Omega Rank for B :  cycles: {{2, 3, 8}, {5, 7}, {4, 6}},   net cycles: 3 .    order:   6

See Matrix
 

[0, y2, y2, 3 y2 - y1, y2, y1, y2, y2]

 

  p' = - s + s 5   p = - s + s 7   p' = - s + s 3   p = - s + s 3   p = - s + s 5

 » SYNC'D 2469/262144 , 0.009418487549

 
41 . Coloring, {2, 5, 8}

R: [3, 8, 1, 1, 2, 7, 5, 2]    B: [6, 3, 8, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )` 3 ` (` - 5 + τ ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 4` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-147, -243, -171, -25, -99, -43, -57, -225] . FixedPtCheck, [147, 243, 171, 25, 99, 43, 57, 225]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-4 y2 + 3 y4 + 3 y1, y3, y2, 0, y4, 0, -y3 - 5 y2 + 4 y4 + 4 y1, y1]

 

  p = s 3 - s 5   p' = s 3 - s 5

Omega Rank for B :  cycles: {{4, 6}},   net cycles: 0 .    order:   6

See Matrix
 

[0, 0, y5, y4, y2, y3, y1, y5 + y4 + y2 - y3 - y1]

 

  p = - s 5 + s 6

 » SYNC'D 463/65536 , 0.007064819336

 
42 . Coloring, {2, 6, 7}

R: [3, 8, 1, 1, 7, 4, 4, 5]    B: [6, 3, 8, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )` , 12` (` 5 + τ + 7τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` 2 , 12` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + 3τ 2 ` )` , -12` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` 2 , 4` (` - 1 + τ ` )`` (` - 5 + 3τ - 7τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 + 10τ 2 + τ 4 ` )``]`

For τ=1/2, [381, 61, 301, 207, 123, 147, 129, 121] . FixedPtCheck, [381, 61, 301, 207, 123, 147, 129, 121]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - y5, 0, y1, y2, y3, 0, y4, y5]

 

  p = - s 5 + s 6

Omega Rank for B :  cycles: {{2, 3, 8}},   net cycles: 0 .    order:   6

[0, y6, y4, 0, y5, y1, y2, y3]  

See Matrices
 

 » SYNC'D 555/8192 , 0.06774902344

 
43 . Coloring, {2, 6, 8}

R: [3, 8, 1, 1, 7, 4, 5, 2]    B: [6, 3, 8, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` - 5 + τ 2 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )` , 4` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [63, 47, 59, 25, 38, 22, 34, 50] . FixedPtCheck, [63, 47, 59, 25, 38, 22, 34, 50]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Omega Rank for R :  cycles: {{1, 3}, {5, 7}, {2, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[2 y2, y2, 2 y2 - y1, y1, y2, 0, y2, y2]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5   p = - s 2 + s 6   p = - s 2 + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 8}, {4, 6, 7}},   net cycles: 2 .   

See Matrix
 

[0, y3, y3, -y1 - y2 + 4 y3, y3, y1, y2, y3]

 

  p = - s + s 4   p = - s + s 7   p' = - s + s 4   p' = - s 2 + s 5

 » SYNC'D 145/131072 , 0.001106262207

 
44 . Coloring, {2, 7, 8}

R: [3, 8, 1, 1, 7, 7, 4, 2]    B: [6, 3, 8, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 12` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )` , 12` (` 5 + τ + 7τ 2 + 3τ 3 ` )` , -12` (` 5 + 4τ + 3τ 2 ` )`` (` - 1 + τ ` )` , 4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 4` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [69, 37, 61, 31, 19, 25, 33, 43] . FixedPtCheck, [69, 37, 61, 31, 19, 25, 33, 43]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[6 y2 - y1 - y3 - y4, y2, y1, y3, 0, 0, y4, y2]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 3, 5, 8}, {4, 6}},   net cycles: 2 .    order:   4

See Matrix
 

[0, y1, 4 y1 - 5 y2 - y3 + 4 y4, y2, y3, 3 y1 - 4 y2 + 3 y4, 0, y4]

 

  p = s - s 5   p' = s - s 5

 » SYNC'D 179/16384 , 0.01092529297

 
45 . Coloring, {3, 4, 5}

R: [3, 3, 8, 6, 2, 7, 5, 5]    B: [6, 8, 1, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 12` (` 1 + τ ` )` 2 ` (` 5 - 4τ + 3τ 2 ` )` , 12` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` 2 , -4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [73, 267, 255, 37, 270, 46, 102, 258] . FixedPtCheck, [73, 267, 255, 37, 270, 46, 102, 258]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 6 vs 6 2 vs 6

Omega Rank for R :  cycles: {{2, 3, 5, 8}},   net cycles: 0 .    order:   4

[0, y5, y4, 0, y3, y2, y1, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 8}, {1, 4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[2 y2, y2, 0, 2 y2, 0, 2 y2 - y1, y1, y2]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5   p = - s 2 + s 6

 » SYNC'D 4447/262144 , 0.01696395874

 
46 . Coloring, {3, 4, 6}

R: [3, 3, 8, 6, 7, 4, 5, 5]    B: [6, 8, 1, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 4` (` 5 - τ + 3τ 2 + τ 3 ` )` , 12` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 12` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )` , 12` (` 5 + τ + 7τ 2 + 3τ 3 ` )` , -12` (` - 1 + τ ` )`` (` 5 + 4τ + 3τ 2 ` )``]`

For τ=1/2, [19, 25, 33, 43, 69, 37, 61, 31] . FixedPtCheck, [19, 25, 33, 43, 69, 37, 61, 31]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{5, 7}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[0, 0, y2, y3, y1, y3, -y2 + 6 y3 - y1 - y4, y4]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 8}, {1, 4, 6, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, y2, 0, 4 y1 - 5 y2 - y3 + 4 y4, 0, y3, y4, 3 y1 - 4 y2 + 3 y4]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 179/16384 , 0.01092529297

 
47 . Coloring, {3, 4, 7}

R: [3, 3, 8, 6, 7, 7, 4, 5]    B: [6, 8, 1, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -4` (` 1 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -12` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` , 12` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 4` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [89, 59, 111, 245, 138, 206, 258, 98] . FixedPtCheck, [89, 59, 111, 245, 138, 206, 258, 98]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 6 vs 6 4 vs 6

Omega Rank for R :  cycles: {{4, 6, 7}},   net cycles: 0 .    order:   6

[0, 0, y1, y2, y3, y4, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 8}, {1, 4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[2 y2 + 2 y4 - y1 - y3, y2 + y4, 0, y1, y2, y3, 0, y4]

 

  p' = - s 2 + s 5   p = - s 2 + s 5

 » SYNC'D 1665/32768 , 0.05081176758

 
48 . Coloring, {3, 4, 8}

R: [3, 3, 8, 6, 7, 7, 5, 2]    B: [6, 8, 1, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` , 4` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` , 4` (` 1 + τ ` )``]`

For τ=1/2, [1, 3, 3, 1, 3, 1, 3, 3] . FixedPtCheck, [1, 3, 3, 1, 3, 1, 3, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 3 vs 6

Omega Rank for R :  cycles: {{5, 7}, {2, 3, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, -y1 + 2 y3 + 2 y2 - y4, y1, 0, y3, y2, y3 + y2, y4]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 5, 8}, {1, 4, 6}},   net cycles: 2 .    order:   3

See Matrix
 

[5 y3 - y1 - y2, y3, 0, y1, y3, y2, 0, y3]

 

  p = s - s 4   p' = s - s 4   p' = s 2 - s 5

 » SYNC'D 525/32768 , 0.01602172852

 
49 . Coloring, {3, 5, 6}

R: [3, 3, 8, 1, 2, 4, 5, 5]    B: [6, 8, 1, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 12` (` 5 - 4τ + 3τ 2 ` )`` (` 1 + τ ` )` 2 , 12` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )` , 4` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -4` (` - 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [102, 258, 270, 46, 255, 37, 73, 267] . FixedPtCheck, [102, 258, 270, 46, 255, 37, 73, 267]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 6 vs 6 2 vs 6

Omega Rank for R :  cycles: {{2, 3, 5, 8}},   net cycles: 0 .    order:   4

[y1, y2, y3, y4, y5, 0, 0, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 8}, {4, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[2 y2 - y1, y2, 0, y1, 0, 2 y2, 2 y2, y2]

 

  p' = s 2 - s 3   p' = - s 3 + s 4   p' = - s 3 + s 5   p = s 2 - s 4

 » SYNC'D 4447/262144 , 0.01696395874

 
50 . Coloring, {3, 5, 7}

R: [3, 3, 8, 1, 2, 7, 4, 5]    B: [6, 8, 1, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (`5 + 2τ + 2τ 2 - 2τ 3 + τ 4 ` )``]`

For τ=1/2, [51, 89, 105, 33, 85, 21, 37, 101] . FixedPtCheck, [51, 89, 105, 33, 85, 21, 37, 101]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 4 vs 7 2 vs 7

Omega Rank for R :  cycles: {{2, 3, 5, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y1, -y1 + y3 + y4, y3 + y4, y2, y3 + y4 - y2, 0, y3, y4]

 

  p = - s 4 + s 7   p = - s 4 + s 5   p = - s 4 + s 6

Omega Rank for B :  cycles: {{5, 7}, {4, 6}, {2, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[y2, y1, 0, -y2 + 2 y1, y1, 2 y1, y1, y1]

 

  p = s 2 - s 3   p' = - s 2 + s 6   p' = - s 2 + s 4   p' = - s 2 + s 5   p' = - s 2 + s 3

 » SYNC'D 285/262144 , 0.001087188721

 
51 . Coloring, {3, 5, 8}

R: [3, 3, 8, 1, 2, 7, 5, 2]    B: [6, 8, 1, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 12` (` - 5 - 3τ - 3τ 2 + 3τ 3 ` )`` (` 1 + τ ` )` 2 , 12` (` - 1 + τ ` )` 3 ` (` 5 + 4τ + 3τ 2 ` )` , -4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -4` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 , 4` (` - 1 + τ ` )` 2 ` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )` 3 ` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-294, -1026, -990, -62, -369, -89, -159, -999] . FixedPtCheck, [294, 1026, 990, 62, 369, 89, 159, 999]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y4, y1, y2, 0, y3, 0, y4, y5]

 

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 6}},   net cycles: -1 .    order:   4

See Matrix
 

[y4, 0, 0, y2, y1, y2 + y1 - y3, y3, y4]

 

  p = s 4 - s 6   p' = s 4 - s 5

 » SYNC'D 855/65536 , 0.01304626465

 
52 . Coloring, {3, 6, 7}

R: [3, 3, 8, 1, 7, 4, 4, 5]    B: [6, 8, 1, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-12` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )` , 12` (` 1 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` - 5 - 2τ - 12τ 2 + 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , 4` (` - 5 + 3τ - 7τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 4` (` - 5 + 3τ - 16τ 2 + 4τ 3 - 3τ 4 + τ 5 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )`` (` 1 + τ ` )` , -4` (` 5 + 2τ + 19τ 2 + 7τ 4 - 2τ 5 + τ 6 ` )``]`

For τ=1/2, [-762, -350, -834, -738, -687, -375, -609, -713] . FixedPtCheck, [762, 350, 834, 738, 687, 375, 609, 713]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 6 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 3, 4, 5, 7, 8}},   net cycles: 1 .    order:   6

[y6, 0, y5, y4, y3, 0, y2, y1]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[-y1 + y5 + y2 - y3 + y4, y1, 0, 0, y5, y2, y3, y4]

 

  p = - s 5 + s 6

 » SYNC'D 2665/65536 , 0.04066467285

 
53 . Coloring, {3, 6, 8}

R: [3, 3, 8, 1, 7, 4, 5, 2]    B: [6, 8, 1, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 4` (` 5 + 4τ + 6τ 2 + τ 4 ` )``]`

For τ=1/2, [63, 125, 141, 37, 89, 25, 73, 137] . FixedPtCheck, [63, 125, 141, 37, 89, 25, 73, 137]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 7 4 vs 7

Omega Rank for R :  cycles: {{5, 7}, {2, 3, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + 2 y2, y1, 2 y2, 2 y2 - y3, y2, 0, y2, y3]

 

  p = - s 3 + s 6   p = - s 3 + s 7   p = - s 3 + s 4   p = - s 3 + s 5

Omega Rank for B :  cycles: {{4, 6, 7}, {2, 5, 8}},   net cycles: 1 .    order:   3

See Matrix
 

[-y1 + 5 y4 - y2 - y3, y4, 0, y1, y4, y2, y3, y4]

 

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

 » SYNC'D 3999/262144 , 0.01525497437

 
54 . Coloring, {3, 7, 8}

R: [3, 3, 8, 1, 7, 7, 4, 2]    B: [6, 8, 1, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-12` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 12` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 4` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [138, 206, 258, 98, 89, 59, 111, 245] . FixedPtCheck, [138, 206, 258, 98, 89, 59, 111, 245]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 6 vs 6 4 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}},   net cycles: 0 .    order:   6

[y4, y3, y1, y2, 0, 0, y6, y5]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + y2, -y3 + 2 y2 - y4, 0, y1, y3, y2, 0, y4]

 

  p = s 2 - s 5   p' = s 2 - s 5

 » SYNC'D 1665/32768 , 0.05081176758

 
55 . Coloring, {4, 5, 6}

R: [3, 3, 1, 6, 2, 4, 5, 5]    B: [6, 8, 8, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` 5 + τ + 7τ 2 + 3τ 3 ` )` , -12` (` 5 + 4τ + 3τ 2 ` )`` (` - 1 + τ ` )` , 12` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` , 12` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 4` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [61, 31, 69, 37, 33, 43, 19, 25] . FixedPtCheck, [61, 31, 69, 37, 33, 43, 19, 25]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[-y1 - y2 + 6 y4 - y3, y1, y2, y4, y3, y4, 0, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 8}, {1, 4, 6, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[-5 y1 + 4 y4 + 4 y3 - y2, y1, 0, y4, 0, y3, y2, -4 y1 + 3 y4 + 3 y3]

 

  p' = s - s 5   p = s - s 5

 » SYNC'D 179/16384 , 0.01092529297

 
56 . Coloring, {4, 5, 7}

R: [3, 3, 1, 6, 2, 7, 4, 5]    B: [6, 8, 8, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 + τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 - τ - τ 2 + τ 3 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )` , -4` (` - 5 + τ 2 ` )` , 4` (` 5 - 2τ + τ 2 ` )` , 4` (` - 5 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [47, 21, 51, 35, 22, 38, 34, 18] . FixedPtCheck, [47, 21, 51, 35, 22, 38, 34, 18]

det(A + τ Δ) =   1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 4 vs 7 2 vs 7

Omega Rank for R :  cycles: {{1, 3}, {4, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y4, y3, y2, y1, -y4 - y3 - y2 + 5 y1, y1, y1, 0]

 

  p' = s 4 - s 6   p' = s 3 - s 5   p = s 3 - s 7

Omega Rank for B :  cycles: {{5, 7}, {1, 4, 6}, {2, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[y1, 3 y1 - y2, 0, y1, y1, y1, y1, y2]

 

  p = - s + s 3   p = - s + s 5   p' = - s + s 3   p' = - s + s 5   p = - s + s 7

 » SYNC'D 2469/262144 , 0.009418487549

 
57 . Coloring, {4, 5, 8}

R: [3, 3, 1, 6, 2, 7, 5, 2]    B: [6, 8, 8, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + 3τ 2 ` )` , -12` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 12` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )` , 12` (` 5 + τ + 7τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` 2 , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -4` (` 5 + 10τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 4` (` - 1 + τ ` )`` (` - 5 + 3τ - 7τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [301, 207, 381, 61, 129, 121, 123, 147] . FixedPtCheck, [301, 207, 381, 61, 129, 121, 123, 147]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 3}},   net cycles: 0 .    order:   6

See Matrix
 

[y3, y2, y1, 0, y3 + y2 - y1 - y5 + y4, y5, y4, 0]

 

  p = - s 5 + s 6

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   6

[y2, 0, 0, y1, y3, y4, y5, y6]  

See Matrices
 

 » SYNC'D 555/8192 , 0.06774902344

 
58 . Coloring, {4, 6, 7}

R: [3, 3, 1, 6, 7, 4, 4, 5]    B: [6, 8, 8, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )` 3 ` (` - 5 + τ ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-171, -25, -147, -243, -57, -225, -99, -43] . FixedPtCheck, [171, 25, 147, 243, 57, 225, 99, 43]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[y3, 0, -4 y3 + 3 y1 + 3 y4, -5 y3 - y2 + 4 y1 + 4 y4, y2, y1, y4, 0]

 

  p = - s 3 + s 5   p' = s 3 - s 5

Omega Rank for B :  cycles: {{2, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y5, y4, 0, 0, y3, y2, y1, y5 + y4 - y3 - y2 + y1]

 

  p = s 5 - s 6

 » SYNC'D 463/65536 , 0.007064819336

 
59 . Coloring, {4, 6, 8}

R: [3, 3, 1, 6, 7, 4, 5, 2]    B: [6, 8, 8, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + 2τ + τ 2 ` )` , -4` (` - 5 + τ 2 ` )` , -4` (` 5 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [59, 25, 63, 47, 34, 50, 38, 22] . FixedPtCheck, [59, 25, 63, 47, 34, 50, 38, 22]

det(A + τ Δ) =   1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Omega Rank for R :  cycles: {{1, 3}, {5, 7}, {4, 6}},   net cycles: 2 .    order:   2

See Matrix
 

[y2, -y2 + 2 y1, 2 y1, y1, y1, y1, y1, 0]

 

  p' = - s 2 + s 4   p' = - s 2 + s 5   p' = - s 2 + s 6   p = s 2 - s 3   p' = - s 2 + s 3

Omega Rank for B :  cycles: {{1, 4, 6, 7}, {2, 5, 8}},   net cycles: 2 .   

See Matrix
 

[y2, y1, 0, y2, 4 y2 - y1 - y3, y2, y2, y3]

 

  p' = s 2 - s 5   p = - s + s 4   p' = - s + s 4   p = - s + s 7

 » SYNC'D 145/131072 , 0.001106262207

 
60 . Coloring, {4, 7, 8}

R: [3, 3, 1, 6, 7, 7, 4, 2]    B: [6, 8, 8, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` , 4` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` , 4` (` 1 + τ ` )` , -4` (` - 1 + τ ` )``]`

For τ=1/2, [3, 1, 3, 3, 1, 3, 3, 1] . FixedPtCheck, [3, 1, 3, 3, 1, 3, 3, 1]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y1, y2, y1 + y2, y4, 0, y3, 2 y1 + 2 y2 - y4 - y3, 0]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 4, 6}, {2, 5, 8}},   net cycles: 2 .    order:   3

See Matrix
 

[y2, 5 y2 - y1 - y3, 0, y2, y1, y2, 0, y3]

 

  p' = s 2 - s 5   p = - s + s 4   p' = - s + s 4

 » SYNC'D 525/32768 , 0.01602172852

 
61 . Coloring, {5, 6, 7}

R: [3, 3, 1, 1, 2, 4, 4, 5]    B: [6, 8, 8, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 4` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 4` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [306, 86, 294, 114, 83, 105, 47, 95] . FixedPtCheck, [306, 86, 294, 114, 83, 105, 47, 95]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y3, y4, 2 y1, y1, 0, 0, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 8}},   net cycles: 1 .    order:   2

See Matrix
 

[0, y1, 0, 0, -5 y1 - y3 + 4 y2, y3, y2, -4 y1 + 3 y2]

 

  p = - s 2 + s 4   p' = - s 2 + s 4

 » SYNC'D 9/256 , 0.03515625000

 
62 . Coloring, {5, 6, 8}

R: [3, 3, 1, 1, 2, 4, 5, 2]    B: [6, 8, 8, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (`5 + τ + 10τ 2 - 2τ 3 + τ 4 + τ 5 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ - τ 2 + τ 3 ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )`` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` 5 - τ + 12τ 2 - τ 4 + τ 5 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [-753, -315, -801, -203, -141, -239, -95, -279] . FixedPtCheck, [753, 315, 801, 203, 141, 239, 95, 279]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 5 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, y1 + y2 - 2 y3, y3, y3, 0, 0, 0]

 

  p' = - s 3 + s 4   p = s 3 - s 4

Omega Rank for B :  cycles: {{4, 6, 7}},   net cycles: 0 .    order:   3

[0, 0, 0, y5, y4, y3, y2, y1]  

See Matrices
 

 » SYNC'D 3/64 , 0.04687500000

 
63 . Coloring, {5, 7, 8}

R: [3, 3, 1, 1, 2, 7, 4, 2]    B: [6, 8, 8, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -4` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 4` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` - 5 + 2τ - 4τ 2 - 2τ 3 + τ 4 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [294, 114, 306, 86, 47, 95, 83, 105] . FixedPtCheck, [294, 114, 306, 86, 47, 95, 83, 105]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, 2 y3, y1, y4, 0, 0, y3, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{5, 7}, {4, 6}},   net cycles: 1 .    order:   2

See Matrix
 

[0, 0, 0, -4 y1 + 3 y2 + 3 y3, -5 y1 + 4 y2 + 4 y3, y1, y2, y3]

 

  p = - s 2 + s 4   p' = - s 2 + s 4

 » SYNC'D 9/256 , 0.03515625000

 
64 . Coloring, {6, 7, 8}

R: [3, 3, 1, 1, 7, 4, 4, 2]    B: [6, 8, 8, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )` 2 , 4` (` - 1 + τ ` )`` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (` 1 + τ ` )`` (`5 + τ + 10τ 2 - 2τ 3 + τ 4 + τ 5 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ - τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` 5 - τ + 12τ 2 - τ 4 + τ 5 ` )``]`

For τ=1/2, [-801, -203, -753, -315, -95, -279, -141, -239] . FixedPtCheck, [801, 203, 753, 315, 95, 279, 141, 239]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 5 vs 5

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y3, y1, y2 - y1 + 2 y3, 0, 0, y3, 0]

 

  p' = s 3 - s 4   p = s 3 - s 5

Omega Rank for B :  cycles: {{2, 5, 8}},   net cycles: 0 .    order:   3

[0, y5, 0, 0, y4, y3, y2, y1]  

See Matrices
 

 » SYNC'D 3/64 , 0.04687500000

 
65 . Coloring, {2, 3, 4, 5}

R: [3, 8, 8, 6, 2, 7, 5, 5]    B: [6, 3, 1, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` - 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 , -2` (` - 1 + τ ` )`` (` 1 + τ ` )` , 2` (` - 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 , 2` (` - 1 + τ ` )` 2 , -2` (` - 1 + τ ` )`` (` 1 + τ ` )` , 2` (` 1 + τ ` )` 2 `]`

For τ=1/2, [1, 9, 3, 1, 9, 1, 3, 9] . FixedPtCheck, [1, 9, 3, 1, 9, 1, 3, 9]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{2, 5, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[0, y1, y2, 0, y3, y2, y4, y5]

 

  p = s 3 - s 6

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: -1 .    order:   3

See Matrix
 

[y5, y4, y3, y2, 0, y1, y4, 0]

 

  p = - s 3 + s 6

 » SYNC'D 1269/32768 , 0.03872680664

 
66 . Coloring, {2, 3, 4, 6}

R: [3, 8, 8, 6, 7, 4, 5, 5]    B: [6, 3, 1, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 5 + 10τ 2 + τ 4 ` )` , 4` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` 1 + τ ` )` , 4` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , 4` (` 5 + 3τ + 16τ 2 + 4τ 3 + 3τ 4 + τ 5 ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [86, 150, 102, 242, 411, 203, 359, 189] . FixedPtCheck, [86, 150, 102, 242, 411, 203, 359, 189]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 6 6 vs 6

Omega Rank for R :  cycles: {{5, 7}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[0, 0, 3 y2 - y1, y2, y1, y2, 3 y2 - y3, y3]

 

  p = - s 3 + s 4   p = - s 3 + s 5   p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 4, 6, 7}},   net cycles: 0 .    order:   4

[y1, y2, y6, y5, 0, y4, y3, 0]  

See Matrices
 

 » SYNC'D 59/4096 , 0.01440429688

 
67 . Coloring, {2, 3, 4, 7}

R: [3, 8, 8, 6, 7, 7, 4, 5]    B: [6, 3, 1, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` - 5 + τ - 10τ 2 - 2τ 3 - τ 4 + τ 5 ` )`` (` - 1 + τ ` )` , 2` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -2` (` 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 2` (` 5 + 2τ + 19τ 2 + 7τ 4 - 2τ 5 + τ 6 ` )` , -2` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , 2` (` 5 - 2τ + 19τ 2 + 7τ 4 + 2τ 5 + τ 6 ` )` , 2` (` 1 + τ ` )`` (`5 + τ + 10τ 2 - 2τ 3 + τ 4 + τ 5 ` )` , 2` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [233, 177, 219, 713, 411, 593, 753, 297] . FixedPtCheck, [233, 177, 219, 713, 411, 593, 753, 297]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 6 vs 6 6 vs 6

Omega Rank for R :  cycles: {{4, 6, 7}},   net cycles: 0 .    order:   6

[0, 0, y1, y2, y3, y4, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   6

[y1, y2, y5, y3, y4, y6, 0, 0]  

See Matrices
 

 » SYNC'D 665/16384 , 0.04058837891

 
68 . Coloring, {2, 3, 4, 8}

R: [3, 8, 8, 6, 7, 7, 5, 2]    B: [6, 3, 1, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -4` (` - 1 + τ ` )`` (`5 - 2τ + 2τ 2 + 2τ 3 + τ 4 ` )` , 4` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 - τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [50, 306, 114, 86, 279, 77, 267, 315] . FixedPtCheck, [50, 306, 114, 86, 279, 77, 267, 315]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 6 6 vs 6

Omega Rank for R :  cycles: {{5, 7}, {2, 8}},   net cycles: 0 .    order:   2

See Matrix
 

[0, y1, y2, 0, y1, y2, y1 + y2, y1 + y2]

 

  p' = s 3 - s 5   p' = s 4 - s 5   p = s 2 - s 6   p' = s 2 - s 5

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   6

[y4, y3, y2, y1, y5, y6, 0, 0]  

See Matrices
 

 » SYNC'D 1409/65536 , 0.02149963379

 
69 . Coloring, {2, 3, 5, 6}

R: [3, 8, 8, 1, 2, 4, 5, 5]    B: [6, 3, 1, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 1` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -1` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -1` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 3` (` 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , -3` (` 5 + 4τ + 3τ 2 ` )`` (` - 1 + τ ` )` 3 , -3` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` - 1 + τ ` )` , 3` (` 1 + τ ` )` 3 ` (` 5 - 4τ + 3τ 2 ` )``]`

For τ=1/2, [75, 387, 153, 49, 381, 31, 103, 405] . FixedPtCheck, [75, 387, 153, 49, 381, 31, 103, 405]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[0, 0, 0, 0, y1, -y1, -y1, y1]
  p = s 2

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 2   p' = s 3   p' = s 4   p' = s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

Omega Rank for R :  cycles: {{2, 5, 8}},   net cycles: 0 .    order:   6

[y2, y1, y4, y5, y6, 0, 0, y3]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6, 7}},   net cycles: 0 .    order:   6

[y2, y1, y3, y4, 0, y5, y6, 0]  

See Matrices
 

 » SYNC'D 15/512 , 0.02929687500

 
70 . Coloring, {2, 3, 5, 7}

R: [3, 8, 8, 1, 2, 7, 4, 5]    B: [6, 3, 1, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , -1` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 3` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , 3` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` 2 , 1` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [99, 267, 141, 85, 254, 46, 98, 306] . FixedPtCheck, [99, 267, 141, 85, 254, 46, 98, 306]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y4, y3, y2, -y4 - y3 + y2, y1, -y4 - y2 - y5, y4 - y2 - y1, y5]
  p = s - 2s 3 + 8s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 7 vs 7 4 vs 7

Omega Rank for R :  cycles: {{2, 5, 8}},   net cycles: 0 .    order:   6

[y1, y2, y3, y4, y5, 0, y6, y7]  

See Matrices
 

Omega Rank for B :  cycles: {{5, 7}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[y2, -y2 - y1 + 3 y3, 3 y3 - y4, y1, y3, y4, y3, 0]

 

  p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 5

 » SYNC'D 135/8192 , 0.01647949219

 
71 . Coloring, {2, 3, 5, 8}

R: [3, 8, 8, 1, 2, 7, 5, 2]    B: [6, 3, 1, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 1` (` 1 + τ ` )` 3 , -1` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -1` (` - 1 + τ ` )` 3 , -1` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -1` (` - 1 + τ ` )` 3 , 1` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 1` (` 1 + τ ` )` 3 `]`

For τ=1/2, [3, 27, 9, 1, 9, 1, 3, 27] . FixedPtCheck, [3, 27, 9, 1, 9, 1, 3, 27]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y2, y1, -y2 - y1 - y3, y3, -y2 - y1 - y3, y3, y2, y1]
  p = s 2 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 3 + 2s 5   p' = s 2 + 2s 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 3 vs 6 3 vs 6 3 vs 6 3 vs 6

Omega Rank for R :  cycles: {{2, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y3, y2, 0, y2, 0, y1, y3]

 

  p = - s 3 + s 6   p = - s 3 + s 4   p = - s 3 + s 5

Omega Rank for B :  cycles: {{4, 6}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, y2, y3, y2, y3, y1, 0]

 

  p' = s 3 - s 5   p = s 3 - s 6   p' = s 4 - s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, x3, x2, x1, 2 x2, -4 x3 + 2 x1]
For A+2Δ :   [-3 y3 - 3 y5 - y4, y3, y2, y1, -3 y3 - y2 - 3 y5, 9 y3 + 9 y5 - y1, y4, y5]
For A-2Δ :   [-3 y3 - 3 y1 - y2, 9 y3 + 9 y1 - y5, -3 y3 - 3 y1 - y4, y3, y4, y1, y2, y5]

Range of {ΩΔi}: [μ2, μ3, %1, μ1, %1, μ1, μ2, μ3] %1 := -μ3 - μ1 - μ2

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $     $ [ [0, 5, 6, 3, 3, 6, 9, 4] , [5, 0, 3, 4, 6, 5, 4, 9] , [6, 3, 0, 5, 9, 4, 3, 6] , [3, 4, 5, 0, 4, 9, 6, 5] , [3, 6, 9, 4, 0, 5, 6, 3] , [6, 5, 4, 9, 5, 0, 3, 4] , [9, 4, 3, 6, 6, 3, 0, 5] , [4, 9, 6, 5, 3, 4, 5, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, 1, 0, -1, 0, -1, 0, 1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x6, x5, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [2], [2], [1]]
Action of B on ranges, [[4], [3], [1], [4]]
β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

ker N, [μ3, %1, μ1, μ2, μ1, μ2, μ3, %1] %1 := -μ3 - μ1 - μ2
Range of N
    [y5, y4, y3, y2, y1, y3 - y2 + y1, -y5 + y3 + y1, -y4 + y3 + y1]

Partitions

Action of R on partitions, [[3], [3], [6], [2], [1], [6]]
Action of B on partitions, [[2], [3], [5], [3], [5], [4]]

α([{4, 5, 7, 8}, {1, 2, 3, 6}]) = 1/9
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/9
α([{2, 3, 6, 7}, {1, 4, 5, 8}]) = 2/9
α([{1, 3, 4, 8}, {2, 5, 6, 7}]) = 1/9
α([{1, 2, 4, 5}, {3, 6, 7, 8}]) = 2/9
α([{1, 5, 6, 8}, {2, 3, 4, 7}]) = 2/9

b1 = {4, 5, 7, 8} ` , ` b2 = {1, 2, 3, 6} ` , ` b3 = {1, 2, 3, 4} ` , ` b4 = {5, 6, 7, 8} ` , ` b5 = {1, 3, 4, 8} ` , ` b6 = {2, 3, 6, 7} ` , ` b7 = {2, 5, 6, 7} ` , ` b8 = {1, 4, 5, 8} ` , ` b9 = {1, 2, 4, 5} ` , ` b10 = {3, 6, 7, 8} ` , ` b11 = {1, 5, 6, 8} ` , ` b12 = {2, 3, 4, 7}

Action of R and B on the blocks of the partitions: = [6, 8, 8, 6, 3, B, 4, C, 1, 2, C, B] [4, 3, 6, 8, 6, 9, 8, A, A, 9, 5, 7]
with invariant measure [1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 3, 5, 8}
Rank2
R,B [3, 8, 8, 1, 2, 7, 5, 2], [6, 3, 1, 6, 7, 4, 4, 5]
π2 [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u2 [5, 6, 3, 3, 6, 9, 4, 3, 4, 6, 5, 4, 9, 5, 9, 4, 3, 6, 4, 9, 6, 5, 5, 6, 3, 3, 4, 5] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
72 . Coloring, {2, 3, 6, 7}

R: [3, 8, 8, 1, 7, 4, 4, 5]    B: [6, 3, 1, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-1` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -1` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 1` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -1` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -1` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 1` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [49, 25, 43, 51, 49, 25, 43, 51] . FixedPtCheck, [49, 25, 43, 51, 49, 25, 43, 51]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y2, y3, -y2 - y3 - y1, y1, y2, y3, -y2 - y3 - y1, y1]
  p' = s 3 + 2s 5   p' = s 2 + 2s 4   p = s 2 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 3 vs 6 3 vs 6 3 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3, 4, 5, 7, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y3, 0, y2, y1, y3, 0, y2, y1]

 

  p = - s + s 4   p' = - s + s 4   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y1, y3, y2, 0, y1, y3, y2, 0]

 

  p' = - s 2 + s 5   p = - s + s 4   p' = - s + s 4


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, x2, x1, x3, 2 x1, -4 x2 + 2 x3]
For A+2Δ :   [-3 y1 - 3 y5 - y3, 9 y1 + 9 y5 - y2, -3 y1 - y4 - 3 y5, y1, y3, y2, y4, y5]
For A-2Δ :   [-3 y1 - y2 - 3 y5, y1, -3 y1 - 3 y5 - y3, 9 y1 + 9 y5 - y4, y2, y5, y3, y4]

Range of {ΩΔi}: [%1, μ1, μ2, μ3, %1, μ1, μ2, μ3] %1 := -μ1 - μ2 - μ3

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] ] $     $ [ [0, 3, 2, 1, 5, 2, 3, 4] , [3, 0, 1, 4, 2, 5, 4, 1] , [2, 1, 0, 3, 3, 4, 5, 2] , [1, 4, 3, 0, 4, 1, 2, 5] , [5, 2, 3, 4, 0, 3, 2, 1] , [2, 5, 4, 1, 3, 0, 1, 4] , [3, 4, 5, 2, 2, 1, 0, 3] , [4, 1, 2, 5, 1, 4, 3, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, -1, 0, 1, 0, -1, 0, 1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x8, x7, x6, x5, x4, x3, x2]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [4], [4], [1]]
Action of B on ranges, [[2], [3], [1], [2]]
β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

ker N, [μ3, μ2, μ1, %1, μ3, μ2, μ1, %1] %1 := -μ3 - μ2 - μ1
Range of N
    [y1 + y5 - y2, y1 - y3 + y5, y1, y1 + y5 - y4, y2, y3, y5, y4]

Partitions

Action of R on partitions, [[3], [1], [2], [1]]
Action of B on partitions, [[4], [1], [1], [2]]

α([{2, 3, 5, 8}, {1, 4, 6, 7}]) = 2/5
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/5
α([{4, 5, 6, 7}, {1, 2, 3, 8}]) = 1/5
α([{1, 3, 4, 6}, {2, 5, 7, 8}]) = 1/5

b1 = {1, 2, 3, 4} ` , ` b2 = {4, 5, 6, 7} ` , ` b3 = {5, 6, 7, 8} ` , ` b4 = {2, 3, 5, 8} ` , ` b5 = {1, 2, 3, 8} ` , ` b6 = {1, 3, 4, 6} ` , ` b7 = {2, 5, 7, 8} ` , ` b8 = {1, 4, 6, 7}

Action of R and B on the blocks of the partitions: = [8, 3, 4, 5, 1, 8, 4, 2] [4, 8, 8, 7, 4, 1, 3, 6]
with invariant measure [1, 1, 1, 2, 1, 1, 1, 2]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 3, 6, 7}
Rank2
R,B [3, 8, 8, 1, 7, 4, 4, 5], [6, 3, 1, 6, 2, 7, 5, 2]
π2 [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
u2 [3, 2, 1, 5, 2, 3, 4, 1, 4, 2, 5, 4, 1, 3, 3, 4, 5, 2, 4, 1, 2, 5, 3, 2, 1, 1, 4, 3] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
73 . Coloring, {2, 3, 6, 8}

R: [3, 8, 8, 1, 7, 4, 5, 2]    B: [6, 3, 1, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 1` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -1` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -1` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 1` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 3` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 3` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [111, 375, 177, 89, 258, 50, 206, 414] . FixedPtCheck, [111, 375, 177, 89, 258, 50, 206, 414]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y1 - y3 - y5, -2 y1 - y3 - y5 - y2, y1, y2, 2 y1 - y4 + y3 + y5, y3, y4, y5]
  p = s - 2s 3 - 8s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{5, 7}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y3 - y4, -y1 - y2 + 3 y3, y1, y2, y3, 0, y3, y4]

 

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 6, 7}},   net cycles: 0 .    order:   6

[y1, y2, y3, y4, y5, y6, y7, 0]  

See Matrices
 

 » SYNC'D 135/8192 , 0.01647949219

 
74 . Coloring, {2, 3, 7, 8}

R: [3, 8, 8, 1, 7, 7, 4, 2]    B: [6, 3, 1, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `3` (` - 1 + τ ` )`` (` - 5 + τ - 7τ 2 + 3τ 3 ` )`` (` 1 + τ ` )` , 3` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )` , -3` (` - 1 + τ ` )`` (` 5 + τ + 7τ 2 + 3τ 3 ` )`` (` 1 + τ ` )` , -3` (` - 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , -1` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 1` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 7τ 2 + τ 3 ` )` , 1` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -1` (` - 5 + 3τ - 7τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [141, 309, 183, 127, 129, 67, 147, 369] . FixedPtCheck, [141, 309, 183, 127, 129, 67, 147, 369]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y3, -2 y3 + y2, y3, y2, y1, y3 + y2 + y1, 2 y3 - 2 y2 - y1, -y3 - y2 - y1]
  p' = s 5   p = s 4

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

Omega Rank for R :  cycles: {{2, 8}},   net cycles: 0 .    order:   6

[y1, y2, y3, y4, 0, 0, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{4, 6}},   net cycles: 0 .    order:   6

[y1, y2, y3, y4, y5, y6, 0, 0]  

See Matrices
 

 » SYNC'D 77/1024 , 0.07519531250

 
75 . Coloring, {2, 4, 5, 6}

R: [3, 8, 1, 6, 2, 4, 5, 5]    B: [6, 3, 8, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `6` (` 5 + 3τ 2 ` )` , 6` (` 5 + 4τ + 3τ 2 ` )` , 2` (` 5 + 2τ + τ 2 ` )` , 2` (` 5 - 2τ + τ 2 ` )` , 2` (` 5 + τ 2 ` )`` (` 1 + τ ` )` , -2` (` - 5 + τ - τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [46, 62, 50, 34, 63, 37, 25, 59] . FixedPtCheck, [46, 62, 50, 34, 63, 37, 25, 59]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-2 y4 - y2 - y1 - y3, -y4 - y2 - y1, y3, y4, y2, 2 y4 + y2 + y1 - y5, y1, y5]
  p = s 2 + s 3 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 3 vs 7 4 vs 7

Omega Rank for R :  cycles: {{2, 5, 8}, {1, 3}, {4, 6}},   net cycles: 3 .    order:   6

See Matrix
 

[y2, 4 y2 - y1 - y3, y2, y2, y1, y2, 0, y3]

 

  p' = s 3 - s 6   p' = s - s 4   p = s - s 7   p' = s 2 - s 5

Omega Rank for B :  cycles: {{1, 4, 6, 7}, {2, 3, 8}},   net cycles: 2 .   

See Matrix
 

[y4, y3, y3, y2, 0, -y4 - y2 - y1 + 5 y3, y1, y3]

 

  p = - s + s 5   p' = - s + s 5   p' = - s 2 + s 6

 » SYNC'D 19/2048 , 0.009277343750

 
76 . Coloring, {2, 4, 5, 7}

R: [3, 8, 1, 6, 2, 7, 4, 5]    B: [6, 3, 8, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1 , 1 , 1 , 1 , 1 , 1 , 1 , 1`]`

For τ=1/2, [1, 1, 1, 1, 1, 1, 1, 1] . FixedPtCheck, [1, 1, 1, 1, 1, 1, 1, 1]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[0, 0, 0, 0, 0, 0, 0, 0]
  p' = s 5   p = s

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s   p' = s 2   p' = s 4   p' = s 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
0 vs 6 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R :  cycles: {{2, 5, 8}, {1, 3}, {4, 6, 7}},   net cycles: 3 .    order:   6

See Matrix
 

[y1, y1, y1, y1, y1, y1, y1, y1]

 

  p' = - s 4 + s 5   p' = 1 - s 4   p' = s - s 4   p' = - s 4 + s 6   p' = s 2 - s 4   p' = - s 4 + s 7   p' = s 3 - s 4

Omega Rank for B :  cycles: {{5, 7}, {1, 4, 6}, {2, 3, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[y1, y1, y1, y1, y1, y1, y1, y1]

 

  p' = - 1 + s   p' = - 1 + s 2   p' = - 1 + s 3   p' = - 1 + s 5   p' = - 1 + s 6   p' = - 1 + s 7   p' = - 1 + s 4


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x2, x1, x4, x3, x6, x5]
For A+2Δ :   [y1, -y1 - y6 - y7 - y5 - y3 - y4 - y2, y6, y7, y5, y3, y4, y2]
For A-2Δ :   [y7, y5, y6, y4, y2, y3, y1, -y1 - y6 - y7 - y5 - y3 - y4 - y2]

Range of {ΩΔi}: [0, 0, 0, 0, 0, 0, 0, 0]

 
rank of M is 8 , rank of N is 8

M              \ ;   N

$ [ [0, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 0] ] $     $ [ [0, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, 0, 0, 0, 0, 0, 0, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x4, x2, x3, x7, x5, x6, x8]

τ= 8 , r'= 7/8

Ranges

Action of R on ranges, [[1]]
Action of B on ranges, [[1]]
β({1, 2, 3, 4, 5, 6, 7, 8}) = 1/1

ker N, [0, 0, 0, 0, 0, 0, 0, 0]
Range of N
    [y8, y7, y5, y6, y4, y2, y3, y1]

Partitions
α([{8}, {1}, {2}, {5}, {3}, {4}, {6}, {7}]) = 1/1

b1 = {8} ` , ` b2 = {1} ` , ` b3 = {2} ` , ` b4 = {5} ` , ` b5 = {3} ` , ` b6 = {4} ` , ` b7 = {6} ` , ` b8 = {7}

Action of R and B on the blocks of the partitions: = [3, 5, 4, 1, 2, 8, 6, 7] [5, 6, 1, 8, 3, 7, 2, 4]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-8 partition graph.

`

Right Group
Coloring {2, 4, 5, 7}
Rank8
R,B [3, 8, 1, 6, 2, 7, 4, 5], [6, 3, 8, 1, 7, 4, 5, 2]
π2 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
u2 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 2)
wpp [1, 1, 1, 1, 1, 1, 1, 1]
π8 [1]
u8 [1]

 

 
77 . Coloring, {2, 4, 5, 8}

R: [3, 8, 1, 6, 2, 7, 5, 2]    B: [6, 3, 8, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `3` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 3` (` 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )` , 1` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -1` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -1` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 1` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [206, 414, 258, 50, 177, 89, 111, 375] . FixedPtCheck, [206, 414, 258, 50, 177, 89, 111, 375]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y3 + 2 y5 + y2 + y4, y2, y3, y4, y5, -y1 - 2 y5 - y2 - y4, -y5 - y2 - y4, y1]
  p = s - 2s 3 - 8s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 3}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 3 y1 - y3, y1, 0, 3 y1 - y2 - y4, y2, y3, y4]

 

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   6

[y1, 0, y2, y3, y4, y5, y6, y7]  

See Matrices
 

 » SYNC'D 135/8192 , 0.01647949219

 
78 . Coloring, {2, 4, 6, 7}

R: [3, 8, 1, 6, 7, 4, 4, 5]    B: [6, 3, 8, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 1` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 3` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 3` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` 2 , 1` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 1` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , -1` (` - 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -1` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [258, 50, 206, 414, 111, 375, 177, 89] . FixedPtCheck, [258, 50, 206, 414, 111, 375, 177, 89]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y4 - y3 + y1, -y1 - y3 - y5, y4, y5, y3, y2, y1, y3 - y1 - y2]
  p = s - 2s 3 - 8s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 3}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 0, y1, 3 y1 - y3, y3, y4, y2, 3 y1 - y4 - y2]

 

  p = - s 4 + s 5   p = - s 4 + s 7   p = - s 4 + s 6

Omega Rank for B :  cycles: {{2, 3, 8}},   net cycles: 0 .    order:   6

[y1, y2, y7, 0, y6, y5, y3, y4]  

See Matrices
 

 » SYNC'D 135/8192 , 0.01647949219

 
79 . Coloring, {2, 4, 6, 8}

R: [3, 8, 1, 6, 7, 4, 5, 2]    B: [6, 3, 8, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1 , 1 , 1 , 1 , 1 , 1 , 1 , 1`]`

For τ=1/2, [1, 1, 1, 1, 1, 1, 1, 1] . FixedPtCheck, [1, 1, 1, 1, 1, 1, 1, 1]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[0, 0, 0, 0, 0, 0, 0, 0]
  p' = s   p' = s 2   p' = s 3   p' = s 4   p' = s 5   p = s

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
0 vs 6 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R :  cycles: {{1, 3}, {5, 7}, {4, 6}, {2, 8}},   net cycles: 4 .    order:   2

See Matrix
 

[y1, y1, y1, y1, y1, y1, y1, y1]

 

  p' = - s 4 + s 7   p' = s 2 - s 4   p' = s 3 - s 4   p' = - s 4 + s 5   p' = - s 4 + s 6   p' = 1 - s 4   p' = s - s 4

Omega Rank for B :  cycles: {{2, 3, 5, 8}, {1, 4, 6, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, y1, y1, y1, y1, y1, y1, y1]

 

  p' = - 1 + s 3   p' = - 1 + s 5   p' = - 1 + s 6   p' = - 1 + s 7   p' = - 1 + s 4   p' = - 1 + s   p' = - 1 + s 2


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x3, x2, x5, x6, x4]
For A+2Δ :   [y7, y5, y6, y4, y3, y2, y1, -y7 - y5 - y6 - y4 - y3 - y2 - y1]
For A-2Δ :   [y1, -y7 - y5 - y6 - y4 - y3 - y2 - y1, y4, y2, y3, y6, y5, y7]

Range of {ΩΔi}: [0, 0, 0, 0, 0, 0, 0, 0]

 
rank of M is 8 , rank of N is 8

M              \ ;   N

$ [ [0, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 0] ] $     $ [ [0, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, 0, 0, 0, 0, 0, 0, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 8 , r'= 7/8

Ranges

Action of R on ranges, [[1]]
Action of B on ranges, [[1]]
β({1, 2, 3, 4, 5, 6, 7, 8}) = 1/1

ker N, [0, 0, 0, 0, 0, 0, 0, 0]
Range of N
    [y1, y2, y3, y4, y5, y6, y7, y8]

Partitions
α([{8}, {1}, {2}, {5}, {3}, {4}, {6}, {7}]) = 1/1

b1 = {8} ` , ` b2 = {1} ` , ` b3 = {2} ` , ` b4 = {5} ` , ` b5 = {3} ` , ` b6 = {4} ` , ` b7 = {6} ` , ` b8 = {7}

Action of R and B on the blocks of the partitions: = [3, 5, 1, 8, 2, 7, 6, 4] [5, 6, 4, 1, 3, 8, 2, 7]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-8 partition graph.

`

Right Group
Coloring {2, 4, 6, 8}
Rank8
R,B [3, 8, 1, 6, 7, 4, 5, 2], [6, 3, 8, 1, 2, 7, 4, 5]
π2 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
u2 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 4)
wpp [1, 1, 1, 1, 1, 1, 1, 1]
π8 [1]
u8 [1]

 

 
80 . Coloring, {2, 4, 7, 8}

R: [3, 8, 1, 6, 7, 7, 4, 2]    B: [6, 3, 8, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 5 + 2τ + τ 2 ` )` , 2` (` 5 - 2τ + τ 2 ` )` , 6` (` 5 + 3τ 2 ` )` , 6` (` 5 + 4τ + 3τ 2 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 2` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , -2` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [50, 34, 46, 62, 25, 59, 63, 37] . FixedPtCheck, [50, 34, 46, 62, 25, 59, 63, 37]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y1 - y3 - y5, y2 + y3 + y5, y1, y2, -2 y2 - y3 - y5 - y4, y3, y4, y5]
  p = s 2 + s 3 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 3 vs 7 4 vs 7

Omega Rank for R :  cycles: {{1, 3}, {2, 8}, {4, 6, 7}},   net cycles: 3 .    order:   6

See Matrix
 

[y3, y3, y3, 4 y3 - y1 - y2, 0, y1, y2, y3]

 

  p = - s + s 4   p' = - s + s 4   p' = - s 2 + s 5   p = - s + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 8}, {1, 4, 6}},   net cycles: 2 .   

See Matrix
 

[y3, y1, 5 y3 - y1 - y2 - y4, y3, y2, y3, 0, y4]

 

  p' = - s 2 + s 6   p' = - s + s 5   p = - s + s 5

 » SYNC'D 19/2048 , 0.009277343750

 
81 . Coloring, {2, 5, 6, 7}

R: [3, 8, 1, 1, 2, 4, 4, 5]    B: [6, 3, 8, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 1` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` 5 + 3τ + 7τ 2 + τ 3 ` )` , -1` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -1` (` - 5 + 3τ - 7τ 2 + τ 3 ` )` , 1` (` - 1 + τ ` )`` (` - 5 + τ ` )`` (` 1 + τ ` )` , -1` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -1` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [75, 43, 67, 33, 41, 27, 17, 49] . FixedPtCheck, [75, 43, 67, 33, 41, 27, 17, 49]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y1, y2, -y1 + 5 y2 - y3, y3, y2, 0, 0, y2]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6

Omega Rank for B :  cycles: {{5, 7}, {2, 3, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, y3, y3, 0, 5 y3 - y1 - y2, y1, y2, y3]

 

  p = - s 2 + s 4   p = - s 2 + s 6   p' = s 2 - s 4

 » SYNC'D 1125/32768 , 0.03433227539

 
82 . Coloring, {2, 5, 6, 8}

R: [3, 8, 1, 1, 2, 4, 5, 2]    B: [6, 3, 8, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 4` (` 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (`5 - 2τ + 2τ 2 + 2τ 3 + τ 4 ` )` , 4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [267, 315, 279, 77, 114, 86, 50, 306] . FixedPtCheck, [267, 315, 279, 77, 114, 86, 50, 306]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 8}},   net cycles: 0 .    order:   2

See Matrix
 

[y2 + y1, y2 + y1, y2, y1, y1, 0, 0, y2]

 

  p = - s 2 + s 5   p = - s 2 + s 6   p = - s 2 + s 3   p = - s 2 + s 4

Omega Rank for B :  cycles: {{4, 6, 7}},   net cycles: 0 .    order:   6

[0, 0, y2, y1, y5, y6, y4, y3]  

See Matrices
 

 » SYNC'D 1409/65536 , 0.02149963379

 
83 . Coloring, {2, 5, 7, 8}

R: [3, 8, 1, 1, 2, 7, 4, 2]    B: [6, 3, 8, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 1 + τ ` )` , 1` (` 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , -1` (` - 1 + τ ` )` , -1` (` - 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )``]`

For τ=1/2, [3, 3, 3, 1, 1, 1, 1, 3] . FixedPtCheck, [3, 3, 3, 1, 1, 1, 1, 3]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[4 y1 + 4 y2 - y4 - 5 y3, 3 y1 + 3 y2 - 4 y3, y1, y2, 0, 0, y4, y3]

 

  p = s 3 - s 5   p' = s 3 - s 5

Omega Rank for B :  cycles: {{5, 7}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[0, 0, -5 y1 - y2 + 4 y3 + 4 y4, y1, y2, -4 y1 + 3 y3 + 3 y4, y3, y4]

 

  p' = s 3 - s 5   p = s 3 - s 5

 » SYNC'D 99/16384 , 0.006042480469

 
84 . Coloring, {2, 6, 7, 8}

R: [3, 8, 1, 1, 7, 4, 4, 2]    B: [6, 3, 8, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` 1 + τ ` )` , 4` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , 4` (` 5 + 3τ + 16τ 2 + 4τ 3 + 3τ 4 + τ 5 ` )` , -4` (` - 1 + τ ` )`` (` 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -4` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 5 + 10τ 2 + τ 4 ` )``]`

For τ=1/2, [411, 203, 359, 189, 86, 150, 102, 242] . FixedPtCheck, [411, 203, 359, 189, 86, 150, 102, 242]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 3}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y3 - y2, y3, 3 y3 - y1, y1, 0, 0, y2, y3]

 

  p' = - s 3 + s 5   p' = - s 3 + s 4   p = s 3 - s 4

Omega Rank for B :  cycles: {{2, 3, 5, 8}},   net cycles: 0 .    order:   4

[0, y1, y2, 0, y3, y4, y5, y6]  

See Matrices
 

 » SYNC'D 59/4096 , 0.01440429688

 
85 . Coloring, {3, 4, 5, 6}

R: [3, 3, 8, 6, 2, 4, 5, 5]    B: [6, 8, 1, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )``]`

For τ=1/2, [1, 3, 3, 1, 3, 1, 1, 3] . FixedPtCheck, [1, 3, 3, 1, 3, 1, 1, 3]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y2, y3, -y2 - y3 - y1, y1, -y2 - y3 - y1, y1, y2, y3]
  p = s 2 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 2 - 2s 4   p' = s 3 - 2s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 3 vs 6 3 vs 6 2 vs 6 2 vs 6

Omega Rank for R :  cycles: {{2, 3, 5, 8}, {4, 6}},   net cycles: 2 .    order:   4

See Matrix
 

[0, y1, -y1 + 3 y2, y2, -y1 + 3 y2, y2, 0, y1]

 

  p' = - s + s 5   p' = - s 2 + s 4   p' = - s + s 3   p = s - s 3

Omega Rank for B :  cycles: {{1, 4, 6, 7}, {2, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, y2, 0, -y1 + 3 y2, 0, -y1 + 3 y2, y1, y2]

 

  p = - s + s 3   p' = - s + s 3   p = - s + s 5   p' = - s + s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, x1, x3, x2, -2 x3, -4 x1 - 2 x2]
For A+2Δ :   [9 y1 + 9 y2 - y5, -3 y1 - 3 y2 - y3, y1, -3 y1 - 3 y2 - y4, y2, y4, y5, y3]
For A-2Δ :   [y1, y4, y5, y3, 9 y1 + 9 y2 - y5, -3 y1 - 3 y2 - y3, y2, -3 y1 - 3 y2 - y4]

Range of {ΩΔi}: [μ2, μ1, %1, μ3, %1, μ3, μ2, μ1] %1 := -μ3 - μ2 - μ1

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $     $ [ [0, 3, 4, 5, 5, 4, 9, 6] , [3, 0, 5, 6, 4, 3, 6, 9] , [4, 5, 0, 3, 9, 6, 5, 4] , [5, 6, 3, 0, 6, 9, 4, 3] , [5, 4, 9, 6, 0, 3, 4, 5] , [4, 3, 6, 9, 3, 0, 5, 6] , [9, 6, 5, 4, 4, 5, 0, 3] , [6, 9, 4, 3, 5, 6, 3, 0] ] $

Check is ΩΔN zero? true, πΔ= [-1, 0, 1, 0, 1, 0, -1, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [3], [2], [4]]
Action of B on ranges, [[4], [2], [1], [1]]
β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

ker N, [μ2, μ3, %1, μ1, %1, μ1, μ2, μ3] %1 := -μ1 - μ2 - μ3
Range of N
    [y1 + y3 - y4, y1 + y3 - y5, y1 - y2 + y3, y1, y2, y3, y4, y5]

Partitions

Action of R on partitions, [[6], [5], [1], [6], [1], [2]]
Action of B on partitions, [[4], [1], [5], [3], [1], [4]]

α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 2/9
α([{3, 6, 7, 8}, {1, 2, 4, 5}]) = 1/9
α([{2, 3, 4, 7}, {1, 5, 6, 8}]) = 1/9
α([{1, 3, 4, 8}, {2, 5, 6, 7}]) = 2/9
α([{5, 6, 7, 8}, {1, 2, 3, 4}]) = 1/9
α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 2/9

b1 = {1, 2, 3, 6} ` , ` b2 = {3, 4, 7, 8} ` , ` b3 = {1, 2, 5, 6} ` , ` b4 = {3, 6, 7, 8} ` , ` b5 = {1, 2, 4, 5} ` , ` b6 = {2, 3, 4, 7} ` , ` b7 = {1, 5, 6, 8} ` , ` b8 = {1, 3, 4, 8} ` , ` b9 = {2, 5, 6, 7} ` , ` b10 = {5, 6, 7, 8} ` , ` b11 = {1, 2, 3, 4} ` , ` b12 = {4, 5, 7, 8}

Action of R and B on the blocks of the partitions: = [5, 1, C, B, A, 3, 2, 1, C, 2, 3, 4] [8, 9, 8, 3, 2, A, B, 6, 7, 3, 2, 9]
with invariant measure [2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {3, 4, 5, 6}
Rank2
R,B [3, 3, 8, 6, 2, 4, 5, 5], [6, 8, 1, 1, 7, 7, 4, 2]
π2 [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u2 [3, 4, 5, 5, 4, 9, 6, 5, 6, 4, 3, 6, 9, 3, 9, 6, 5, 4, 6, 9, 4, 3, 3, 4, 5, 5, 6, 3] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
86 . Coloring, {3, 4, 5, 7}

R: [3, 3, 8, 6, 2, 7, 4, 5]    B: [6, 8, 1, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` - 1 + τ ` )`` (` 5 + τ 2 ` )` , 2` (` 5 + τ + τ 2 + τ 3 ` )` , 2` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -2` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 6` (` 5 + 3τ 2 ` )` , 6` (` 5 - 4τ + 3τ 2 ` )` , 2` (` 5 - 2τ + τ 2 ` )` , 2` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [21, 47, 51, 33, 46, 30, 34, 50] . FixedPtCheck, [21, 47, 51, 33, 46, 30, 34, 50]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y1 - 2 y5 - y4 - y3, -y4 - y3 - y2, y1, y2, y4, y5 + y4 + y3, y3, y5]
  p = s 2 - s 3 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 4 vs 7 3 vs 7

Omega Rank for R :  cycles: {{2, 3, 5, 8}, {4, 6, 7}},   net cycles: 2 .   

See Matrix
 

[0, y4, y3, y2, y1, y2, y2, -y4 - y3 - y1 + 5 y2]

 

  p' = - s 2 + s 6   p' = - s + s 5   p = - s + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 8}, {1, 4, 6}},   net cycles: 3 .    order:   6

See Matrix
 

[-y3 + 4 y2 - y1, y2, 0, y3, y2, y1, y2, y2]

 

  p' = s - s 4   p = s - s 7   p' = s 2 - s 5   p' = s 3 - s 6

 » SYNC'D 19/2048 , 0.009277343750

 
87 . Coloring, {3, 4, 5, 8}

R: [3, 3, 8, 6, 2, 7, 5, 2]    B: [6, 8, 1, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-3` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` - 1 + τ ` )` , 3` (` 1 + τ ` )` 3 ` (` 5 - 4τ + 3τ 2 ` )` , 3` (` 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , -3` (` 5 + 4τ + 3τ 2 ` )`` (` - 1 + τ ` )` 3 , -1` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -1` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 1` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 1` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [103, 405, 381, 31, 153, 49, 75, 387] . FixedPtCheck, [103, 405, 381, 31, 153, 49, 75, 387]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y1, -y1, -y1, y1, 0, 0, 0, 0]
  p = s 2

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 2   p' = s 4   p' = s 5   p' = s 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}},   net cycles: 0 .    order:   6

[0, y6, y5, 0, y3, y2, y4, y1]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   6

[y1, 0, 0, y5, y6, y4, y2, y3]  

See Matrices
 

 » SYNC'D 15/512 , 0.02929687500

 
88 . Coloring, {3, 4, 6, 7}

R: [3, 3, 8, 6, 7, 4, 4, 5]    B: [6, 8, 1, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 7τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -1` (` - 5 + 3τ - 7τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 3` (` - 5 + τ - 7τ 2 + 3τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 3` (` 1 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , -3` (` 1 + τ ` )`` (` 5 + τ + 7τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` , -3` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [129, 67, 147, 369, 141, 309, 183, 127] . FixedPtCheck, [129, 67, 147, 369, 141, 309, 183, 127]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y1 - 3 y2 - y3, y1, -y1 + y2 - y3, -y1, -y2, y3, y2, 2 y2 + y3]
  p = s 4

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 4   p' = s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

Omega Rank for R :  cycles: {{4, 6}},   net cycles: 0 .    order:   6

[0, 0, y2, y1, y3, y4, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 8}},   net cycles: 0 .    order:   6

[y6, y5, 0, 0, y4, y3, y2, y1]  

See Matrices
 

 » SYNC'D 77/1024 , 0.07519531250

 
89 . Coloring, {3, 4, 6, 8}

R: [3, 3, 8, 6, 7, 4, 5, 2]    B: [6, 8, 1, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 2` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , -2` (` - 5 + τ - τ 2 + τ 3 ` )` , 2` (` 5 + 2τ + τ 2 ` )` , 2` (` 5 - 2τ + τ 2 ` )` , 6` (` 5 + 3τ 2 ` )` , 6` (` 5 + 4τ + 3τ 2 ` )``]`

For τ=1/2, [25, 59, 63, 37, 50, 34, 46, 62] . FixedPtCheck, [25, 59, 63, 37, 50, 34, 46, 62]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y2, y3 - y5 - y1, -y2 - y3 - y5, y1, -y3 + y5 - y4, y3, y4, y5]
  p = s 2 + s 3 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 6 7 vs 7 7 vs 7 3 vs 7 4 vs 7

Omega Rank for R :  cycles: {{5, 7}, {4, 6}, {2, 3, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[0, -y1 + 4 y3 - y2, y1, y3, y3, y3, y3, y2]

 

  p' = s 2 - s 5   p = - s + s 4   p' = - s + s 4   p = - s + s 7

Omega Rank for B :  cycles: {{1, 4, 6, 7}, {2, 5, 8}},   net cycles: 2 .   

See Matrix
 

[y2, y4, 0, -y2 + 5 y4 - y1 - y3, y4, y1, y3, y4]

 

  p = s - s 5   p' = - s + s 5   p' = - s 2 + s 6

 » SYNC'D 19/2048 , 0.009277343750

 
90 . Coloring, {3, 4, 7, 8}

R: [3, 3, 8, 6, 7, 7, 4, 2]    B: [6, 8, 1, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-1` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 1` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -1` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -1` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 1` (` 5 - τ + 3τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -1` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [25, 43, 51, 49, 25, 43, 51, 49] . FixedPtCheck, [25, 43, 51, 49, 25, 43, 51, 49]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y1 - y2 - y3, y1, y2, y3, -y1 - y2 - y3, y1, y2, y3]
  p' = s 2 + 2s 4   p' = s 3 + 2s 5   p = s 2 - 4s 6

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 3 vs 6 3 vs 6 3 vs 6 3 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}, {4, 6, 7}},   net cycles: 2 .    order:   3

See Matrix
 

[0, y1, y2, y3, 0, y1, y2, y3]

 

  p = s - s 4   p' = s 2 - s 5   p' = - s + s 4

Omega Rank for B :  cycles: {{2, 5, 8}, {1, 4, 6}},   net cycles: 2 .    order:   3

See Matrix
 

[y1, y2, 0, y3, y1, y2, 0, y3]

 

  p' = - s + s 4   p = - s + s 4   p' = s 2 - s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, x3, x2, x1, 2 x2, -4 x3 + 2 x1]
For A+2Δ :   [9 y5 + 9 y1 - y3, -3 y5 - y2 - 3 y1, y5, y4, y3, y2, y1, -3 y5 - 3 y1 - y4]
For A-2Δ :   [y1, -3 y1 - 3 y2 - y3, 9 y1 + 9 y2 - y5, -3 y1 - 3 y2 - y4, y2, y3, y5, y4]

Range of {ΩΔi}: [μ3, μ2, %1, μ1, μ3, μ2, %1, μ1] %1 := -μ3 - μ2 - μ1

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] ] $     $ [ [0, 1, 4, 3, 5, 4, 1, 2] , [1, 0, 3, 2, 4, 5, 2, 3] , [4, 3, 0, 1, 1, 2, 5, 4] , [3, 2, 1, 0, 2, 3, 4, 5] , [5, 4, 1, 2, 0, 1, 4, 3] , [4, 5, 2, 3, 1, 0, 3, 2] , [1, 2, 5, 4, 4, 3, 0, 1] , [2, 3, 4, 5, 3, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [-1, 0, 1, 0, -1, 0, 1, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [3], [4], [2]]
Action of B on ranges, [[2], [4], [1], [1]]
β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

ker N, [%1, μ1, μ2, μ3, %1, μ1, μ2, μ3] %1 := -μ1 - μ2 - μ3
Range of N
    [y1 - y2 + y5, y1 - y3 + y5, y1 - y4 + y5, y1, y2, y3, y4, y5]

Partitions

Action of R on partitions, [[4], [1], [2], [2]]
Action of B on partitions, [[2], [3], [4], [2]]

α([{3, 5, 6, 8}, {1, 2, 4, 7}]) = 1/5
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 2/5
α([{2, 3, 4, 5}, {1, 6, 7, 8}]) = 1/5
α([{5, 6, 7, 8}, {1, 2, 3, 4}]) = 1/5

b1 = {3, 5, 6, 8} ` , ` b2 = {1, 2, 4, 7} ` , ` b3 = {1, 2, 7, 8} ` , ` b4 = {3, 4, 5, 6} ` , ` b5 = {2, 3, 4, 5} ` , ` b6 = {1, 6, 7, 8} ` , ` b7 = {5, 6, 7, 8} ` , ` b8 = {1, 2, 3, 4}

Action of R and B on the blocks of the partitions: = [8, 7, 1, 2, 3, 4, 4, 3] [3, 4, 5, 6, 7, 8, 3, 4]
with invariant measure [1, 1, 2, 2, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {3, 4, 7, 8}
Rank2
R,B [3, 3, 8, 6, 7, 7, 4, 2], [6, 8, 1, 1, 2, 4, 5, 5]
π2 [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
u2 [1, 4, 3, 5, 4, 1, 2, 3, 2, 4, 5, 2, 3, 1, 1, 2, 5, 4, 2, 3, 4, 5, 1, 4, 3, 3, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
91 . Coloring, {3, 5, 6, 7}

R: [3, 3, 8, 1, 2, 4, 4, 5]    B: [6, 8, 1, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + 10τ 2 + τ 4 ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` - 5 + 3τ - 16τ 2 + 4τ 3 - 3τ 4 + τ 5 ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 4` (` - 5 - 3τ - 10τ 2 + 2τ 3 - τ 4 + τ 5 ` )``]`

For τ=1/2, [-150, -242, -294, -102, -229, -63, -73, -281] . FixedPtCheck, [150, 242, 294, 102, 229, 63, 73, 281]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 6 vs 6 3 vs 6

Omega Rank for R :  cycles: {{2, 3, 5, 8}},   net cycles: 0 .    order:   4

[y2, y1, y5, y6, y4, 0, 0, y3]  

See Matrices
 

Omega Rank for B :  cycles: {{5, 7}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y2 - y1, y2, 0, 0, y3, 3 y2 - y3, y1, y2]

 

  p = - s 3 + s 4   p = - s 3 + s 5   p = - s 3 + s 6

 » SYNC'D 59/4096 , 0.01440429688

 
92 . Coloring, {3, 5, 6, 8}

R: [3, 3, 8, 1, 2, 4, 5, 2]    B: [6, 8, 1, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` - 1 + τ ` )`` (` 1 + τ ` )` , 2` (` 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 , 2` (` - 1 + τ ` )` 2 , -2` (` - 1 + τ ` )`` (` 1 + τ ` )` , 2` (` - 1 + τ ` )` 2 , 2` (` - 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 `]`

For τ=1/2, [3, 9, 9, 1, 3, 1, 1, 9] . FixedPtCheck, [3, 9, 9, 1, 3, 1, 1, 9]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y2, y1, y3, y4, y4, 0, 0, y5]

 

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 6, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y5, 0, 0, y1, y2, y3, y4, y5]

 

  p = - s 3 + s 6

 » SYNC'D 1269/32768 , 0.03872680664

 
93 . Coloring, {3, 5, 7, 8}

R: [3, 3, 8, 1, 2, 7, 4, 2]    B: [6, 8, 1, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )` 2 ` (` - 5 - τ + τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )` 2 ` (` - 5 - 3τ - τ 2 + τ 3 ` )` , -4` (` 1 + τ ` )`` (`5 + 2τ + 2τ 2 - 2τ 3 + τ 4 ` )``]`

For τ=1/2, [-114, -294, -306, -50, -89, -41, -53, -303] . FixedPtCheck, [114, 294, 306, 50, 89, 41, 53, 303]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 6 vs 6 2 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}},   net cycles: 0 .    order:   6

[y1, y2, y3, y4, 0, 0, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{5, 7}, {4, 6}},   net cycles: 0 .    order:   2

See Matrix
 

[y2, 0, 0, y1, y2 + y1, y2 + y1, y1, y2]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5   p = - s 2 + s 6

 » SYNC'D 1409/65536 , 0.02149963379

 
94 . Coloring, {3, 6, 7, 8}

R: [3, 3, 8, 1, 7, 4, 4, 2]    B: [6, 8, 1, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` - 1 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` 1 + τ ` )` , 2` (` 5 - 2τ + 19τ 2 + 7τ 4 + 2τ 5 + τ 6 ` )` , 2` (`5 + τ + 10τ 2 - 2τ 3 + τ 4 + τ 5 ` )`` (` 1 + τ ` )` , 2` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 2` (` - 5 + τ - 10τ 2 - 2τ 3 - τ 4 + τ 5 ` )`` (` - 1 + τ ` )` , 2` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , -2` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 2` (` 5 + 2τ + 19τ 2 + 7τ 4 - 2τ 5 + τ 6 ` )``]`

For τ=1/2, [411, 593, 753, 297, 233, 177, 219, 713] . FixedPtCheck, [411, 593, 753, 297, 233, 177, 219, 713]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 6 vs 6 6 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}},   net cycles: 0 .    order:   6

[y1, y2, y3, y4, 0, 0, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 5, 8}},   net cycles: 0 .    order:   6

[y1, y3, 0, 0, y2, y4, y6, y5]  

See Matrices
 

 » SYNC'D 665/16384 , 0.04058837891

 
95 . Coloring, {4, 5, 6, 7}

R: [3, 3, 1, 6, 2, 4, 4, 5]    B: [6, 8, 8, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , 1` (` 1 + τ ` )` , -1` (` - 1 + τ ` )` , -1` (` - 1 + τ ` )``]`

For τ=1/2, [3, 1, 3, 3, 1, 3, 1, 1] . FixedPtCheck, [3, 1, 3, 3, 1, 3, 1, 1]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, y2, 4 y1 + 4 y2 - y3 - 5 y4, 3 y1 + 3 y2 - 4 y4, y3, y4, 0, 0]

 

  p' = s 3 - s 5   p = s 3 - s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 3 y1 + 3 y4 - 4 y3, 0, 0, 4 y1 - y2 + 4 y4 - 5 y3, y2, y4, y3]

 

  p' = - s 3 + s 5   p = - s 3 + s 5

 » SYNC'D 99/16384 , 0.006042480469

 
96 . Coloring, {4, 5, 6, 8}

R: [3, 3, 1, 6, 2, 4, 5, 2]    B: [6, 8, 8, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 + 3τ + 16τ 2 + 4τ 3 + 3τ 4 + τ 5 ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , 4` (`5 - 3τ + 10τ 2 + 2τ 3 + τ 4 + τ 5 ` )` , -4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 5 + 10τ 2 + τ 4 ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [359, 189, 411, 203, 102, 242, 86, 150] . FixedPtCheck, [359, 189, 411, 203, 102, 242, 86, 150]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[y3, -y3 + 3 y1, y2, y1, -y2 + 3 y1, y1, 0, 0]

 

  p = - s 3 + s 4   p = - s 3 + s 6   p = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 4, 6, 7}},   net cycles: 0 .    order:   4

[y1, 0, 0, y2, y3, y4, y5, y6]  

See Matrices
 

 » SYNC'D 59/4096 , 0.01440429688

 
97 . Coloring, {4, 5, 7, 8}

R: [3, 3, 1, 6, 2, 7, 4, 2]    B: [6, 8, 8, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 5 + 3τ + 7τ 2 + τ 3 ` )` , -1` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 1` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 1` (` 5 - τ + 3τ 2 + τ 3 ` )` , -1` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -1` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -1` (` - 5 + 3τ - 7τ 2 + τ 3 ` )` , 1` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ ` )``]`

For τ=1/2, [67, 33, 75, 43, 17, 49, 41, 27] . FixedPtCheck, [67, 33, 75, 43, 17, 49, 41, 27]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y3, y2, -y3 - y2 + 5 y1, y1, 0, y1, y1, 0]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6

Omega Rank for B :  cycles: {{5, 7}, {1, 4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, 0, 0, y2, y1, y2, y3, 5 y2 - y1 - y3]

 

  p' = - s 2 + s 4   p = - s 2 + s 6   p = - s 2 + s 4

 » SYNC'D 1125/32768 , 0.03433227539

 
98 . Coloring, {4, 6, 7, 8}

R: [3, 3, 1, 6, 7, 4, 4, 2]    B: [6, 8, 8, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -4` (` - 1 + τ ` )`` (`5 - 2τ + 2τ 2 + 2τ 3 + τ 4 ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )` , 4` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [279, 77, 267, 315, 50, 306, 114, 86] . FixedPtCheck, [279, 77, 267, 315, 50, 306, 114, 86]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6}},   net cycles: 0 .    order:   2

See Matrix
 

[y1, y2 - y1, y2, y2, 0, y1, y2 - y1, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 5   p = - s 2 + s 6   p = - s 2 + s 4

Omega Rank for B :  cycles: {{2, 5, 8}},   net cycles: 0 .    order:   6

[y6, y5, 0, 0, y4, y3, y2, y1]  

See Matrices
 

 » SYNC'D 1409/65536 , 0.02149963379

 
99 . Coloring, {5, 6, 7, 8}

R: [3, 3, 1, 1, 2, 4, 4, 2]    B: [6, 8, 8, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1` (` 1 + τ ` )` 2 , -1` (` - 1 + τ ` )`` (` 1 + τ ` )` , 1` (` 1 + τ ` )` 2 , -1` (` - 1 + τ ` )`` (` 1 + τ ` )` , 1` (` - 1 + τ ` )` 2 , -1` (` - 1 + τ ` )`` (` 1 + τ ` )` , 1` (` - 1 + τ ` )` 2 , -1` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [9, 3, 9, 3, 1, 3, 1, 3] . FixedPtCheck, [9, 3, 9, 3, 1, 3, 1, 3]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y2, -y1, -y2, -y1, y1, y2, y1, y2]
  p' = s 3   p' = s 5   p' = s 4   p = s 3

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 6 2 vs 6 2 vs 6 2 vs 4 2 vs 4

Omega Rank for R :  cycles: {{1, 3}},   net cycles: -1 .    order:   2

See Matrix
 

[y1, y2, y1, y2, 0, 0, 0, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   2

See Matrix
 

[0, 0, 0, 0, y1, y2, y1, y2]

 

  p = - s 2 + s 4   p = - s 2 + s 3


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, 0, x1, x3, x2, x4]
For A+2Δ :   [y6, y5, y4, y3, y2, y1, -3 y5 - 3 y3 - y2, -3 y6 - 3 y4 - y1]
For A-2Δ :   [-y5 - 3 y3 - 3 y1, y6, y5, -y6 - 3 y4 - 3 y2, y4, y3, y2, y1]

Range of {ΩΔi}: [-μ1, -μ2, -μ1, -μ2, μ2, μ1, μ2, μ1]

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] ] $     $ [ [0, 1, 2, 1, 1, 1, 1, 1] , [1, 0, 1, 2, 1, 1, 1, 1] , [2, 1, 0, 1, 1, 1, 1, 1] , [1, 2, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 2, 1] , [1, 1, 1, 1, 1, 0, 1, 2] , [1, 1, 1, 1, 2, 1, 0, 1] , [1, 1, 1, 1, 1, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [1, 1, 1, 1, -1, -1, -1, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x6, x7, x8, x3, x4, x5, x1, x2]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[1], [1], [2], [2]]
Action of B on ranges, [[4], [4], [3], [3]]
β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

ker N, [μ2, μ3, μ2, μ3, μ1, %1, μ1, %1] %1 := -μ2 - μ3 - μ1
Range of N
    [y2 + y5 - y1, -y4 + y2 + y5, y1, y4, -y3 + y2 + y5, y2, y3, y5]

Partitions

Action of R on partitions, [[3], [2], [2], [3]]
Action of B on partitions, [[1], [4], [4], [1]]

α([{2, 3, 5, 6}, {1, 4, 7, 8}]) = 1/4
α([{3, 4, 5, 8}, {1, 2, 6, 7}]) = 1/4
α([{3, 4, 6, 7}, {1, 2, 5, 8}]) = 1/4
α([{2, 3, 7, 8}, {1, 4, 5, 6}]) = 1/4

b1 = {3, 4, 6, 7} ` , ` b2 = {1, 2, 5, 8} ` , ` b3 = {2, 3, 7, 8} ` , ` b4 = {1, 4, 5, 6} ` , ` b5 = {2, 3, 5, 6} ` , ` b6 = {1, 4, 7, 8} ` , ` b7 = {3, 4, 5, 8} ` , ` b8 = {1, 2, 6, 7}

Action of R and B on the blocks of the partitions: = [8, 7, 2, 1, 2, 1, 8, 7] [4, 3, 5, 6, 6, 5, 3, 4]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {5, 6, 7, 8}
Rank2
R,B [3, 3, 1, 1, 2, 4, 4, 2], [6, 8, 8, 6, 7, 7, 5, 5]
π2 [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0]
u2 [1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
100 . Coloring, {2, 3, 4, 5, 6}

R: [3, 8, 8, 6, 2, 4, 5, 5]    B: [6, 3, 1, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` - 1 + τ ` )` 2 ` (` 5 + 4τ + 3τ 2 ` )` , 12` (` 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -4` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [62, 414, 150, 98, 411, 89, 125, 423] . FixedPtCheck, [62, 414, 150, 98, 411, 89, 125, 423]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 2 vs 6 6 vs 6

Omega Rank for R :  cycles: {{4, 6}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, -y1 + 2 y2, y1, y2, 2 y2, y2, 0, 2 y2]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 6   p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 4, 6, 7}},   net cycles: 0 .    order:   4

[y2, y3, y1, y5, 0, y4, y6, 0]  

See Matrices
 

 » SYNC'D 4447/262144 , 0.01696395874

 
101 . Coloring, {2, 3, 4, 5, 7}

R: [3, 8, 8, 6, 2, 7, 4, 5]    B: [6, 3, 1, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , 4` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [37, 141, 63, 85, 137, 73, 89, 153] . FixedPtCheck, [37, 141, 63, 85, 137, 73, 89, 153]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 4 vs 7 3 vs 7

Omega Rank for R :  cycles: {{4, 6, 7}, {2, 5, 8}},   net cycles: 1 .    order:   3

See Matrix
 

[0, -y1 + 5 y3 - y2 - y4, y1, y3, y2, y3, y3, y4]

 

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 7}, {1, 4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[2 y3, 2 y3 - y2, -y1 + 2 y3, y1, y3, y2, y3, 0]

 

  p = - s 3 + s 4   p = - s 3 + s 5   p = - s 3 + s 6   p = - s 3 + s 7

 » SYNC'D 3999/262144 , 0.01525497437

 
102 . Coloring, {2, 3, 4, 5, 8}

R: [3, 8, 8, 6, 2, 7, 5, 2]    B: [6, 3, 1, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` - 1 + τ ` )` 2 ` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )` , 12` (` 1 + τ ` )` 3 ` (` 5 - 4τ + 3τ 2 ` )` , -4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 , -4` (` 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , 4` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 4` (` 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [74, 810, 258, 38, 279, 47, 105, 801] . FixedPtCheck, [74, 810, 258, 38, 279, 47, 105, 801]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{2, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[0, y3 - y2 + y4, y1, 0, y3, y1, y2, y4]

 

  p = - s 4 + s 6   p = - s 4 + s 5

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, y4, y2, y4, y5, y3, 0]

 

  p = - s 3 + s 6

 » SYNC'D 855/65536 , 0.01304626465

 
103 . Coloring, {2, 3, 4, 6, 7}

R: [3, 8, 8, 6, 7, 4, 4, 5]    B: [6, 3, 1, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 + 2τ - 12τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , 4` (` 5 + 3τ + 7τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -12` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` - 1 + τ ` )` , 12` (` 1 + 3τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` - 5 - 3τ - 10τ 2 + 2τ 3 - τ 4 + τ 5 ` )`` (` - 1 + τ ` )` , 4` (` 5 - 2τ + 19τ 2 + 7τ 4 + 2τ 5 + τ 6 ` )` , -4` (` - 1 + τ ` )`` (` 5 + 3τ + 16τ 2 + 4τ 3 + 3τ 4 + τ 5 ` )` , -4` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [230, 134, 206, 714, 281, 593, 359, 255] . FixedPtCheck, [230, 134, 206, 714, 281, 593, 359, 255]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{4, 6}},   net cycles: 0 .    order:   6

See Matrix
 

[0, 0, y5, y4, y3, y2, y1, y5 + y4 + y3 - y2 - y1]

 

  p = s 5 - s 6

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7}},   net cycles: 1 .    order:   6

[y1, y2, y3, 0, y4, y5, y6, 0]  

See Matrices
 

 » SYNC'D 2665/65536 , 0.04066467285

 
104 . Coloring, {2, 3, 4, 6, 8}

R: [3, 8, 8, 6, 7, 4, 5, 2]    B: [6, 3, 1, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (`5 - 2τ + 2τ 2 + 2τ 3 + τ 4 ` )` , 4` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [41, 177, 75, 89, 141, 77, 125, 189] . FixedPtCheck, [41, 177, 75, 89, 141, 77, 125, 189]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R :  cycles: {{5, 7}, {4, 6}, {2, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[0, -y1 + 2 y2, y1, y2, y2, y2, y2, 2 y2]

 

  p' = s 2 - s 4   p' = s 3 - s 4   p' = - s 4 + s 5   p' = - s 4 + s 6   p = s 2 - s 5

Omega Rank for B :  cycles: {{1, 4, 6, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y3 + y2, y3 + y2 - y4, y1, y3 + y2 - y1, y3, y2, y4, 0 ]

 

  p = - s 4 + s 6   p = - s 4 + s 5   p = - s 4 + s 7

 » SYNC'D 285/262144 , 0.001087188721

 
105 . Coloring, {2, 3, 4, 7, 8}

R: [3, 8, 8, 6, 7, 7, 4, 2]    B: [6, 3, 1, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -12` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 12` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -4` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [98, 258, 138, 254, 141, 215, 267, 297] . FixedPtCheck, [98, 258, 138, 254, 141, 215, 267, 297]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 6 vs 6

Omega Rank for R :  cycles: {{4, 6, 7}, {2, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, y4, y2, y3, 0, -y3 - y1 + 2 y4 + 2 y2, y1, y4 + y2]

 

  p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   6

[y6, y5, y4, y3, y2, y1, 0, 0]  

See Matrices
 

 » SYNC'D 1665/32768 , 0.05081176758

 
106 . Coloring, {2, 3, 5, 6, 7}

R: [3, 8, 8, 1, 2, 4, 4, 5]    B: [6, 3, 1, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -4` (` 1 + τ ` )`` (` 5 + 10τ 2 + τ 4 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 3τ + 7τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 12` (` 1 + 3τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -12` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 12` (` - 1 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 12` (` 1 + τ ` )` 2 ` (` - 5 + τ - 7τ 2 + 3τ 3 ` )``]`

For τ=1/2, [-147, -363, -201, -129, -343, -69, -103, -423] . FixedPtCheck, [147, 363, 201, 129, 343, 69, 103, 423]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 6 vs 6 5 vs 6

Omega Rank for R :  cycles: {{2, 5, 8}},   net cycles: 0 .    order:   6

[y4, y1, y2, y3, y5, 0, 0, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y2, -y2 + y1 + y4 + y3 - y5, y1, 0, y4, y3, y5, 0]

 

  p = - s 5 + s 6

 » SYNC'D 555/8192 , 0.06774902344

 
107 . Coloring, {2, 3, 5, 6, 8}

R: [3, 8, 8, 1, 2, 4, 5, 2]    B: [6, 3, 1, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )` 2 ` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )` , -4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 12` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )`` (` - 1 + τ ` )` 2 , 12` (` 1 + τ ` )` 3 ` (` 5 - 4τ + 3τ 2 ` )``]`

For τ=1/2, [105, 801, 279, 47, 258, 38, 74, 810] . FixedPtCheck, [105, 801, 279, 47, 258, 38, 74, 810]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{2, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y4, y3, y2, y1, y1, 0, 0, -y4 + y3 + y2]

 

  p' = s 4 - s 5   p = s 4 - s 6

Omega Rank for B :  cycles: {{4, 6, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, y3, y2, y3, y4, y5, 0]

 

  p = - s 3 + s 6

 » SYNC'D 855/65536 , 0.01304626465

 
108 . Coloring, {2, 3, 5, 7, 8}

R: [3, 8, 8, 1, 2, 7, 4, 2]    B: [6, 3, 1, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` 5 + τ ` )`` (` - 1 + τ ` )` 3 , 4` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-27, -147, -57, -17, -43, -11, -19, -153] . FixedPtCheck, [27, 147, 57, 17, 43, 11, 19, 153]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{2, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y5, y4, y3, y2, 0, 0, y1, -y5 + y4 + y3 + y2 - y1]

 

  p = s 5 - s 6

Omega Rank for B :  cycles: {{5, 7}, {4, 6}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 0, 4 y1 + 4 y2 - y3 - 5 y4, y2, 3 y1 + 3 y2 - 4 y4, y3, y4, 0]

 

  p' = s 3 - s 5   p = s 3 - s 5

 » SYNC'D 463/65536 , 0.007064819336

 
109 . Coloring, {2, 3, 6, 7, 8}

R: [3, 8, 8, 1, 7, 4, 4, 2]    B: [6, 3, 1, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 - 3τ - 10τ 2 + 2τ 3 - τ 4 + τ 5 ` )`` (` - 1 + τ ` )` , 4` (` 5 - 2τ + 19τ 2 + 7τ 4 + 2τ 5 + τ 6 ` )` , -4` (` 5 + 3τ + 16τ 2 + 4τ 3 + 3τ 4 + τ 5 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 4` (` - 5 + 2τ - 12τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , 4` (` 5 + 3τ + 7τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -12` (` - 1 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 12` (` 1 + 3τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [281, 593, 359, 255, 230, 134, 206, 714] . FixedPtCheck, [281, 593, 359, 255, 230, 134, 206, 714]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{2, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 + y3 + y2 - y5 - y4, y1, y3, y2, 0, 0, y5, y4]

 

  p = - s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7}},   net cycles: 1 .    order:   6

[y1, y5, y6, 0, y4, y2, y3, 0]  

See Matrices
 

 » SYNC'D 2665/65536 , 0.04066467285

 
110 . Coloring, {2, 4, 5, 6, 7}

R: [3, 8, 1, 6, 2, 4, 4, 5]    B: [6, 3, 8, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 + τ ` )`` (` 1 + τ ` )` , 4` (` - 5 + τ 2 ` )` , -4` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` 1 + τ ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )` , -4` (` 1 + τ ` )`` (` 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` - 5 - τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [-54, -38, -50, -66, -37, -63, -25, -41] . FixedPtCheck, [54, 38, 50, 66, 37, 63, 25, 41]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R :  cycles: {{1, 3}, {4, 6}, {2, 5, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[y1, y1, y1, y2, y1, 3 y1 - y2, 0, y1]

 

  p' = - s + s 5   p' = - s 2 + s 6   p' = - s 2 + s 4   p' = - s + s 3   p = s - s 3

Omega Rank for B :  cycles: {{5, 7}, {2, 3, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y4, y3, y3, 0, y1, y2, -y4 - y1 - y2 + 5 y3, y3]

 

  p' = s 3 - s 5   p = - s 3 + s 5   p = - s 3 + s 7

 » SYNC'D 2469/262144 , 0.009418487549

 
111 . Coloring, {2, 4, 5, 6, 8}

R: [3, 8, 1, 6, 2, 4, 5, 2]    B: [6, 3, 8, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (`5 - 2τ + 2τ 2 + 2τ 3 + τ 4 ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )` , 4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [125, 189, 141, 77, 75, 89, 41, 177] . FixedPtCheck, [125, 189, 141, 77, 75, 89, 41, 177]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R :  cycles: {{1, 3}, {4, 6}, {2, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[y2, 2 y2, y2, y2, 2 y2 - y1, y2, 0, y1]

 

  p = s 2 - s 3   p' = - s 2 + s 3   p' = - s 2 + s 4   p' = - s 2 + s 5   p' = - s 2 + s 6

Omega Rank for B :  cycles: {{1, 4, 6, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y3 - y4, 0, -y1 + y3, y1, -y2 + y3, y2, y3, y4]

 

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

 » SYNC'D 285/262144 , 0.001087188721

 
112 . Coloring, {2, 4, 5, 7, 8}

R: [3, 8, 1, 6, 2, 7, 4, 2]    B: [6, 3, 8, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 5 + 2τ + τ 2 ` )` , -4` (` 5 + τ ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )`` (` - 5 + τ ` )` , 4` (` - 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` - 5 - τ + τ 2 + τ 3 ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )` , -4` (` 1 + τ ` )`` (` 5 + τ 2 ` )``]`

For τ=1/2, [-50, -66, -54, -38, -25, -41, -37, -63] . FixedPtCheck, [50, 66, 54, 38, 25, 41, 37, 63]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R :  cycles: {{1, 3}, {4, 6, 7}, {2, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[y1, 3 y1 - y2, y1, y1, 0, y1, y1, y2]

 

  p' = s - s 3   p' = s 2 - s 4   p' = - s 3 + s 5   p' = - s 4 + s 6   p = s - s 5

Omega Rank for B :  cycles: {{1, 4, 6}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, 0, y1, y2, -y1 + 5 y2 - y3 - y4, y2, y3, y4]

 

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

 » SYNC'D 2469/262144 , 0.009418487549

 
113 . Coloring, {2, 4, 6, 7, 8}

R: [3, 8, 1, 6, 7, 4, 4, 2]    B: [6, 3, 8, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (`5 - 2τ + 2τ 2 + 2τ 3 + τ 4 ` )` , 4` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` 5 + τ 2 ` )` , 4` (` - 1 + τ ` )`` (` - 5 - τ + τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [141, 77, 125, 189, 41, 177, 75, 89] . FixedPtCheck, [141, 77, 125, 189, 41, 177, 75, 89]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R :  cycles: {{1, 3}, {4, 6}, {2, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[y2, y2, y2, 2 y2, 0, 2 y2 - y1, y1, y2]

 

  p' = s 4 - s 6   p = s 2 - s 7   p' = s 5 - s 6   p' = s 3 - s 6   p' = s 2 - s 6

Omega Rank for B :  cycles: {{2, 3, 5, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y2, -y2 + y3 + y4, y1, 0, y3 + y4, -y1 + y3 + y4, y3, y4]

 

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

 » SYNC'D 285/262144 , 0.001087188721

 
114 . Coloring, {2, 5, 6, 7, 8}

R: [3, 8, 1, 1, 2, 4, 4, 2]    B: [6, 3, 8, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 4` (` 1 + τ ` )`` (` - 5 - 2τ - 4τ 2 + 2τ 3 + τ 4 ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 - τ + τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -4` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-333, -285, -321, -123, -86, -114, -50, -294] . FixedPtCheck, [333, 285, 321, 123, 86, 114, 50, 294]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 4 vs 5

Omega Rank for R :  cycles: {{1, 3}, {2, 8}},   net cycles: 1 .    order:   2

See Matrix
 

[y1, 3 y1 - 4 y3, 4 y1 - y2 - 5 y3, y2, 0, 0, 0, y3]

 

  p' = s 2 - s 4   p = s 2 - s 4

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 0, y1, 0, y2, 2 y1, y3, y4]

 

  p = - s 3 + s 5

 » SYNC'D 9/256 , 0.03515625000

 
115 . Coloring, {3, 4, 5, 6, 7}

R: [3, 3, 8, 6, 2, 4, 4, 5]    B: [6, 8, 1, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )`` (` - 5 + τ ` )`` (` - 1 + τ ` )` , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 12` (` - 5 + τ - 7τ 2 + 3τ 3 ` )` , -12` (` 1 + τ ` )`` (` 5 - 4τ + 3τ 2 ` )` , 12` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` , 12` (` - 5 - 3τ - 3τ 2 + 3τ 3 ` )``]`

For τ=1/2, [-27, -49, -57, -51, -47, -45, -23, -55] . FixedPtCheck, [27, 49, 57, 51, 47, 45, 23, 55]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{2, 3, 5, 8}, {4, 6}},   net cycles: 2 .    order:   4

See Matrix
 

[0, 4 y2 - 5 y1 + 4 y4 - y3, y2, y1, y4, 3 y2 - 4 y1 + 3 y4, 0, y3]

 

  p' = - s + s 5   p = - s + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[6 y4 - y1 - y2 - y3, y4, 0, 0, y1, y2, y3, y4]

 

  p' = - s 3 + s 5   p = - s 3 + s 5

 » SYNC'D 179/16384 , 0.01092529297

 
116 . Coloring, {3, 4, 5, 6, 8}

R: [3, 3, 8, 6, 2, 4, 5, 2]    B: [6, 8, 1, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 4` (` 1 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , -4` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 12` (` - 1 + τ ` )` 2 ` (` 5 + 4τ + 3τ 2 ` )` , 12` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [125, 423, 411, 89, 150, 98, 62, 414] . FixedPtCheck, [125, 423, 411, 89, 150, 98, 62, 414]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 2 vs 6 6 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}, {4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[0, 2 y1, 2 y1, y1, 2 y1 - y2, y1, 0, y2]

 

  p = s 2 - s 6   p' = s 2 - s 5   p' = s 4 - s 5   p' = s 3 - s 5

Omega Rank for B :  cycles: {{1, 4, 6, 7}},   net cycles: 0 .    order:   4

[y1, 0, 0, y2, y3, y4, y5, y6]  

See Matrices
 

 » SYNC'D 4447/262144 , 0.01696395874

 
117 . Coloring, {3, 4, 5, 7, 8}

R: [3, 3, 8, 6, 2, 7, 4, 2]    B: [6, 8, 1, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 1 + τ ` )` , -4` (` 1 + τ ` )` , -4` (` 1 + τ ` )` , 4` (` - 1 + τ ` )` , 4` (` - 1 + τ ` )` , 4` (` - 1 + τ ` )` , 4` (` - 1 + τ ` )` , -4` (` 1 + τ ` )``]`

For τ=1/2, [-1, -3, -3, -1, -1, -1, -1, -3] . FixedPtCheck, [1, 3, 3, 1, 1, 1, 1, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 6 4 vs 6

Omega Rank for R :  cycles: {{4, 6, 7}, {2, 3, 8}},   net cycles: 2 .    order:   3

See Matrix
 

[0, -y2 + 5 y3 - y1, y2, y3, 0, y3, y3, y1]

 

  p' = - s + s 4   p = - s + s 4   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 4, 6}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 - y2 + 2 y3 + 2 y4, 0, 0, y1, y3 + y4, y2, y3, y4]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

 » SYNC'D 525/32768 , 0.01602172852

 
118 . Coloring, {3, 4, 6, 7, 8}

R: [3, 3, 8, 6, 7, 4, 4, 2]    B: [6, 8, 1, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )` , -4` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -12` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 12` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )``]`

For τ=1/2, [141, 215, 267, 297, 98, 258, 138, 254] . FixedPtCheck, [141, 215, 267, 297, 98, 258, 138, 254]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 4 vs 6 6 vs 6

Omega Rank for R :  cycles: {{2, 3, 8}, {4, 6}},   net cycles: 1 .    order:   6

See Matrix
 

[0, y1, -y1 + 2 y3 + 2 y4 - y2, y3 + y4, 0, y3, y4, y2]

 

  p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 5, 8}},   net cycles: 0 .    order:   6

[y6, y5, 0, 0, y3, y4, y2, y1]  

See Matrices
 

 » SYNC'D 1665/32768 , 0.05081176758

 
119 . Coloring, {3, 5, 6, 7, 8}

R: [3, 3, 8, 1, 2, 4, 4, 2]    B: [6, 8, 1, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ - 12τ 2 + τ 4 + τ 5 ` )` , 4` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 4` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , -4` (` - 1 + τ ` )`` (` - 5 + τ - 10τ 2 - 2τ 3 - τ 4 + τ 5 ` )` , 4` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 - τ + τ 2 + τ 3 ` )` , -4` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 , 4` (` 1 + τ ` )`` (` - 5 - 3τ - 10τ 2 + 2τ 3 - τ 4 + τ 5 ` )``]`

For τ=1/2, [-333, -807, -855, -159, -233, -123, -89, -843] . FixedPtCheck, [333, 807, 855, 159, 233, 123, 89, 843]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 5 vs 5 3 vs 5

Omega Rank for R :  cycles: {{2, 3, 8}},   net cycles: 0 .    order:   3

[y1, y2, y3, y4, 0, 0, 0, y5]  

See Matrices
 

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, 0, 0, y2, y3, y2 + y3 - 2 y1, y1]

 

  p' = s 3 - s 4   p = s 3 - s 5

 » SYNC'D 3/64 , 0.04687500000

 
120 . Coloring, {4, 5, 6, 7, 8}

R: [3, 3, 1, 6, 2, 4, 4, 2]    B: [6, 8, 8, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` - 5 - 2τ - 4τ 2 + 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , -4` (` - 5 - τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )` , 4` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -4` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 4` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-321, -123, -333, -285, -50, -294, -86, -114] . FixedPtCheck, [321, 123, 333, 285, 50, 294, 86, 114]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 4 vs 5

Omega Rank for R :  cycles: {{1, 3}, {4, 6}},   net cycles: 1 .    order:   2

See Matrix
 

[-y1 + 4 y3 - 5 y2, y1, y3, 3 y3 - 4 y2, 0, y2, 0, 0]

 

  p' = s 2 - s 4   p = - s 2 + s 4

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, 0, 0, y2, y3, y4, 2 y1]

 

  p = s 3 - s 5

 » SYNC'D 9/256 , 0.03515625000

 
121 . Coloring, {2, 3, 4, 5, 6, 7}

R: [3, 8, 8, 6, 2, 4, 4, 5]    B: [6, 3, 1, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` - 5 - 3τ - 10τ 2 + 2τ 3 - τ 4 + τ 5 ` )` , -4` (` 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 4` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 4` (` 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-102, -294, -150, -258, -281, -219, -125, -333] . FixedPtCheck, [102, 294, 150, 258, 281, 219, 125, 333]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{4, 6}, {2, 5, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, 3 y2, -3 y2 + 5 y1 - 3 y4 + 5 y3 - 3 y5, 3 y1, 3 y4, 3 y3, 0, 3 y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y4, y5, y6, 0, y1, y2, y3, 0]  

See Matrices
 

 » SYNC'D 3891/65536 , 0.05937194824

 
122 . Coloring, {2, 3, 4, 5, 6, 8}

R: [3, 8, 8, 6, 2, 4, 5, 2]    B: [6, 3, 1, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` - 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 , -2` (` 1 + τ ` )`` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` 2 , -2` (` 1 + τ ` )`` (` - 1 + τ ` )` , 2` (` - 1 + τ ` )` 2 , 2` (` - 1 + τ ` )` 2 , 2` (` 1 + τ ` )` 2 `]`

For τ=1/2, [1, 9, 3, 1, 3, 1, 1, 9] . FixedPtCheck, [1, 9, 3, 1, 3, 1, 1, 9]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y1, y2, -y1 - y2 - y3, y3, -y1 - y2 - y3, y3, y1, y2]
  p' = s + 4s 4 - 8s 5   p' = s 2 - 2s 3 + 4s 4 - 4s 5   p = s + 4s 4 - 8s 5

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 2 vs 6 2 vs 6

Omega Rank for R :  cycles: {{4, 6}, {2, 8}},   net cycles: 0 .    order:   2

See Matrix
 

[0, -y1 + 3 y2, y1, y2, y1, y2, 0, -y1 + 3 y2]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5   p = - s 2 + s 6

Omega Rank for B :  cycles: {{1, 4, 6, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1 + y2, 0, y1, y2, y1, y2, y1 + y2, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5   p = - s 2 + s 6


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x2, x3, 4 x1 - 2 x3, 4 x2 - 8 x1 + 4 x3, -8 x2 - 4 x3]
For A+2Δ :   [-y4, -y3, -y1, -y2, y1, y2, y4, y3]
For A-2Δ :   [-y2, -y4, y3, -y1, -y3, y1, y2, y4]

Range of {ΩΔi}: [μ3, μ2, μ1, %1, μ1, %1, μ3, μ2] %1 := -μ3 - μ2 - μ1

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $     $ [ [0, 5, 4, 4, 3, 3, 7, 2] , [5, 0, 2, 4, 5, 3, 2, 7] , [4, 2, 0, 2, 7, 5, 3, 5] , [4, 4, 2, 0, 5, 7, 3, 3] , [3, 5, 7, 5, 0, 2, 4, 2] , [3, 3, 5, 7, 2, 0, 4, 4] , [7, 2, 3, 3, 4, 4, 0, 5] , [2, 7, 5, 3, 2, 4, 5, 0] ] $

Check is ΩΔN zero? true, πΔ= [-1, 1, 0, 0, 0, 0, -1, 1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x6, x4, x5, x2, x3, x1, x8, x7]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [2], [2], [4]]
Action of B on ranges, [[4], [3], [1], [1]]
β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

ker N, [μ1, %1, μ3, μ2, μ3, μ2, μ1, %1] %1 := -μ3 - μ2 - μ1
Range of N
    [y4, y5, y3, y3 + y1 - y2, y1, y2, -y4 + y3 + y1, -y5 + y3 + y1]

Partitions

Action of R on partitions, [[2], [2], [1], [6], [1], [6]]
Action of B on partitions, [[4], [3], [5], [6], [6], [5]]

α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 1/7
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/7
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/14
α([{5, 6, 7, 8}, {1, 2, 3, 4}]) = 1/14
α([{1, 3, 4, 8}, {2, 5, 6, 7}]) = 3/14
α([{2, 3, 4, 7}, {1, 5, 6, 8}]) = 5/14

b1 = {1, 2, 3, 6} ` , ` b2 = {1, 4, 5, 8} ` , ` b3 = {3, 4, 7, 8} ` , ` b4 = {1, 3, 4, 8} ` , ` b5 = {1, 2, 5, 6} ` , ` b6 = {2, 5, 6, 7} ` , ` b7 = {2, 3, 6, 7} ` , ` b8 = {2, 3, 4, 7} ` , ` b9 = {1, 5, 6, 8} ` , ` b10 = {5, 6, 7, 8} ` , ` b11 = {4, 5, 7, 8} ` , ` b12 = {1, 2, 3, 4}

Action of R and B on the blocks of the partitions: = [2, 7, 1, 1, B, B, 2, 9, 8, 8, 7, 9] [C, 3, 6, 8, 4, 9, 5, 6, 4, 9, A, 8]
with invariant measure [2, 2, 1, 3, 1, 3, 2, 5, 5, 1, 2, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 3, 4, 5, 6, 8}
Rank2
R,B [3, 8, 8, 6, 2, 4, 5, 2], [6, 3, 1, 1, 7, 7, 4, 5]
π2 [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u2 [5, 4, 4, 3, 3, 7, 2, 2, 4, 5, 3, 2, 7, 2, 7, 5, 3, 5, 5, 7, 3, 3, 2, 4, 2, 4, 4, 5] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
123 . Coloring, {2, 3, 4, 5, 7, 8}

R: [3, 8, 8, 6, 2, 7, 4, 2]    B: [6, 3, 1, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` 5 + τ ` )`` (` - 1 + τ ` )` 2 , -4` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ ` )` , 4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 4` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 4` (` 5 - τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-22, -150, -54, -34, -47, -31, -35, -153] . FixedPtCheck, [22, 150, 54, 34, 47, 31, 35, 153]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 8 vs 8 8 vs 8 3 vs 6 5 vs 6

Omega Rank for R :  cycles: {{4, 6, 7}, {2, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[0, -y1 + 5 y2 - y3, y1, y2, 0, y2, y2, y3]

 

  p = - s 2 + s 6   p = - s 2 + s 4   p' = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 4, 6}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-3 y1 - 3 y2 + 5 y3 - 3 y4 + 5 y5, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 2641/131072 , 0.02014923096

 
124 . Coloring, {2, 3, 4, 6, 7, 8}

R: [3, 8, 8, 6, 7, 4, 4, 2]    B: [6, 3, 1, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 2` (` 5 - τ + 3τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 2` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 2` (` 5 - τ + 3τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 2` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [19, 43, 25, 51, 19, 43, 25, 51] . FixedPtCheck, [19, 43, 25, 51, 19, 43, 25, 51]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[y1, -y3 - y2 - y1, y3, y2, y1, -y3 - y2 - y1, y3, y2]
  p' = s + 4s 4   p = s + 4s 4

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s 2 + 4s 5
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 6 4 vs 8 4 vs 8 2 vs 6 3 vs 6

Omega Rank for R :  cycles: {{4, 6}, {2, 8}},   net cycles: 0 .    order:   2

See Matrix
 

[0, y2, y1, y2 + y1, 0, y2, y1, y2 + y1]

 

  p' = s 4 - s 5   p' = s 3 - s 5   p' = s 2 - s 5   p = s 2 - s 6

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y3, y2, y1, 0, y3, y2, y1, 0]

 

  p = - s + s 4   p' = - s + s 4   p' = - s 2 + s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x3, x2, x1, 4 x3, 4 x2, 4 x1]
For A+2Δ :   [-y1, y4, y2, y3, y1, -y4, -y2, -y3]
For A-2Δ :   [-y3, -y4, -y1, -y2, y3, y4, y1, y2]

Range of {ΩΔi}: [%1, μ1, μ2, μ3, %1, μ1, μ2, μ3] %1 := -μ3 - μ1 - μ2

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] ] $     $ [ [0, 3, 2, 2, 4, 1, 2, 2] , [3, 0, 1, 2, 1, 4, 3, 2] , [2, 1, 0, 1, 2, 3, 4, 3] , [2, 2, 1, 0, 2, 2, 3, 4] , [4, 1, 2, 2, 0, 3, 2, 2] , [1, 4, 3, 2, 3, 0, 1, 2] , [2, 3, 4, 3, 2, 1, 0, 1] , [2, 2, 3, 4, 2, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [-1, 0, 0, 1, -1, 0, 0, 1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x5, x6, x7, x8, x1, x2, x3, x4]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[3], [4], [4], [2]]
Action of B on ranges, [[2], [3], [1], [1]]
β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

ker N, [μ2, %1, μ1, μ3, μ2, %1, μ1, μ3] %1 := -μ2 - μ1 - μ3
Range of N
    [y5 - y4 + y1, y5 + y1 - y3, y5 + y1 - y2, y5, y4, y3, y2, y1]

Partitions

Action of R on partitions, [[1], [4], [3], [3], [1]]
Action of B on partitions, [[2], [5], [1], [2], [1]]

α([{2, 3, 4, 5}, {1, 6, 7, 8}]) = 3/8
α([{2, 5, 7, 8}, {1, 3, 4, 6}]) = 1/4
α([{4, 5, 6, 7}, {1, 2, 3, 8}]) = 1/8
α([{2, 3, 5, 8}, {1, 4, 6, 7}]) = 1/8
α([{5, 6, 7, 8}, {1, 2, 3, 4}]) = 1/8

b1 = {2, 3, 5, 8} ` , ` b2 = {2, 3, 4, 5} ` , ` b3 = {1, 6, 7, 8} ` , ` b4 = {4, 5, 6, 7} ` , ` b5 = {1, 2, 3, 8} ` , ` b6 = {1, 4, 6, 7} ` , ` b7 = {5, 6, 7, 8} ` , ` b8 = {2, 5, 7, 8} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {1, 3, 4, 6}

Action of R and B on the blocks of the partitions: = [5, 3, 2, 4, 5, 4, 2, 1, 3, 6] [8, 8, A, 3, 2, A, 3, 7, 2, 9]
with invariant measure [1, 3, 3, 1, 1, 1, 1, 2, 1, 2]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 3, 4, 6, 7, 8}
Rank2
R,B [3, 8, 8, 6, 7, 4, 4, 2], [6, 3, 1, 1, 2, 7, 5, 5]
π2 [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
u2 [3, 2, 2, 4, 1, 2, 2, 1, 2, 1, 4, 3, 2, 1, 2, 3, 4, 3, 2, 2, 3, 4, 3, 2, 2, 1, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
125 . Coloring, {2, 3, 5, 6, 7, 8}

R: [3, 8, 8, 1, 2, 4, 4, 2]    B: [6, 3, 1, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 - 3τ - τ 2 + τ 3 ` )` , 2` (` 1 + τ ` )`` (` - 5 - τ - 12τ 2 + τ 4 + τ 5 ` )` , -2` (` 1 + τ ` )`` (` - 5 - 2τ - 4τ 2 + 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , -2` (` 1 + τ ` )`` (` 5 - τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -2` (` - 5 + 2τ - 12τ 2 - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , 2` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 3 , -6` (` - 1 + τ ` )` 2 ` (` 5 - 3τ + 3τ 2 + 3τ 3 ` )` , 6` (` 1 + τ ` )` 2 ` (` - 5 + τ - 7τ 2 + 3τ 3 ` )``]`

For τ=1/2, [-159, -807, -321, -105, -230, -66, -74, -846] . FixedPtCheck, [159, 807, 321, 105, 230, 66, 74, 846]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

Omega Rank for R :  cycles: {{2, 8}},   net cycles: 0 .    order:   4

[y2, y1, y4, y3, 0, 0, 0, y5]  

See Matrices
 

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

[y1, 0, y2, 0, y3, y4, y5, 0]  

See Matrices
 

 » SYNC'D 87/2048 , 0.04248046875

 
126 . Coloring, {2, 4, 5, 6, 7, 8}

R: [3, 8, 1, 6, 2, 4, 4, 2]    B: [6, 3, 8, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , 2` (` 1 + τ ` )` , -2` (` - 1 + τ ` )` , 2` (` 1 + τ ` )` , -2` (` - 1 + τ ` )` , 2` (` 1 + τ ` )``]`

For τ=1/2, [3, 3, 3, 3, 1, 3, 1, 3] . FixedPtCheck, [3, 3, 3, 3, 1, 3, 1, 3]

det(A + τ Δ) =   0

Delta Range :  [-y1 - y3 - y5, -y6 - y2 - y4, y1, y6, y2, y3, y4, y5]

[1, 1, 1, 1, 1, 1, 1, 1]

+              \ ;          -              \ ;          Δ

See Matrices

 
[-y1, -y2, -y1, -y2, y2, y1, y2, y1]
  p = s - 4s 5

           S+              \ ;          S-              \ ;          NM
See Matrices

CmmCk true, true, true

  p' = s - 4s 5   p' = s 3 - 2s 5   p' = s 2 - 2s 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 6 3 vs 7 3 vs 7 2 vs 6 3 vs 6

Omega Rank for R :  cycles: {{1, 3}, {4, 6}, {2, 8}},   net cycles: 3 .    order:   2

See Matrix
 

[y1, 3 y1 - y2, y1, 3 y1 - y2, 0, y2, 0, y2]

 

  p = - s + s 5   p' = s 3 - s 5   p = - s + s 3   p' = s - s 5

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, y1, 0, y2, y3, y2, y3]

 

  p = - s 3 + s 5   p = - s 3 + s 4   p = - s 3 + s 6


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x4, x3, x2, x1, -4 x4 - 2 x2, -4 x3 - 2 x1]
For A+2Δ :   [y5, y4, -y5, y3, y2, y1, -3 y4 - 3 y3 - y2, -y1]
For A-2Δ :   [-y1, -3 y4 - 3 y3 - y2, y1, y2, y3, y5, y4, -y5]

Range of {ΩΔi}: [-μ1, μ2, -μ1, μ2, -μ2, μ1, -μ2, μ1]

 
rank of M is 8 , rank of N is 5

M              \ ;   N

$ [ [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] ] $     $ [ [0, 1, 2, 1, 1, 1, 1, 1] , [1, 0, 1, 2, 1, 1, 1, 1] , [2, 1, 0, 1, 1, 1, 1, 1] , [1, 2, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 1, 2, 1] , [1, 1, 1, 1, 1, 0, 1, 2] , [1, 1, 1, 1, 2, 1, 0, 1] , [1, 1, 1, 1, 1, 2, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, 1, 0, 1, -1, 0, -1, 0]

ker M, [0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8]

τ= 32 , r'= 1/2

Ranges

Action of R on ranges, [[1], [4], [2], [2]]
Action of B on ranges, [[4], [1], [3], [3]]
β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

ker N, [%1, μ1, %1, μ1, μ2, μ3, μ2, μ3] %1 := -μ1 - μ2 - μ3
Range of N
    [y3 + y5 - y1, -y2 + y3 + y5, y1, y2, -y4 + y3 + y5, y3, y4, y5]

Partitions

Action of R on partitions, [[6], [5], [2], [4], [2], [6], [5], [4]]
Action of B on partitions, [[7], [8], [1], [3], [3], [8], [7], [1]]

α([{2, 3, 5, 6}, {1, 4, 7, 8}]) = 1/8
α([{3, 4, 6, 7}, {1, 2, 5, 8}]) = 1/8
α([{1, 4, 5, 6}, {2, 3, 7, 8}]) = 1/8
α([{1, 2, 6, 7}, {3, 4, 5, 8}]) = 1/8
α([{2, 3, 5, 8}, {1, 4, 6, 7}]) = 1/8
α([{2, 3, 6, 7}, {1, 4, 5, 8}]) = 1/8
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 1/8
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/8

b1 = {2, 3, 5, 6} ` , ` b2 = {1, 4, 7, 8} ` , ` b3 = {3, 4, 7, 8} ` , ` b4 = {3, 4, 6, 7} ` , ` b5 = {1, 2, 5, 8} ` , ` b6 = {2, 3, 5, 8} ` , ` b7 = {2, 3, 6, 7} ` , ` b8 = {1, 2, 7, 8} ` , ` b9 = {1, 4, 5, 6} ` , ` b10 = {2, 3, 7, 8} ` , ` b11 = {3, 4, 5, 6} ` , ` b12 = {1, 2, 6, 7} ` , ` b13 = {3, 4, 5, 8} ` , ` b14 = {1, 4, 5, 8} ` , ` b15 = {1, 4, 6, 7} ` , ` b16 = {1, 2, 5, 6}

Action of R and B on the blocks of the partitions: = [E, 7, C, F, 6, 5, E, 6, 4, 5, F, D, C, 7, 4, D] [8, B, 1, 10, 3, A, 10, B, 2, 1, 8, 9, A, 3, 9, 2]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 4, 5, 6, 7, 8}
Rank2
R,B [3, 8, 1, 6, 2, 4, 4, 2], [6, 3, 8, 1, 7, 7, 5, 5]
π2 [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0]
u2 [1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1] (dim 1)
wpp [4, 4, 4, 4, 4, 4, 4, 4]

 

 
127 . Coloring, {3, 4, 5, 6, 7, 8}

R: [3, 3, 8, 6, 2, 4, 4, 2]    B: [6, 8, 1, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-2` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 - τ + τ 2 + τ 3 ` )` , 2` (` 1 + τ ` )`` (` - 5 - 2τ - 4τ 2 + 2τ 3 + τ 4 ` )` , 2` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -2` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 2` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -6` (` 5 + 4τ + 3τ 2 ` )`` (` - 1 + τ ` )` 2 , 6` (` - 5 - 3τ - 3τ 2 + 3τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-123, -321, -333, -159, -98, -150, -62, -330] . FixedPtCheck, [123, 321, 333, 159, 98, 150, 62, 330]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

Omega Rank for R :  cycles: {{2, 3, 8}, {4, 6}},   net cycles: 2 .    order:   6

See Matrix
 

[0, 3 y4, 3 y3, 3 y2, 0, 3 y1, 0, -3 y4 - 3 y3 + 5 y2 + 5 y1]

 

  p = s + s 2 - s 4 - s 5

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[2 y4, 0, 0, 0, y1, y2, y3, y4]

 

  p = - s 3 + s 5

 » SYNC'D 1/16 , 0.06250000000

 
128 . Coloring, {2, 3, 4, 5, 6, 7, 8}

R: [3, 8, 8, 6, 2, 4, 4, 2]    B: [6, 3, 1, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `4` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 4` (` 1 + τ ` )`` (` - 5 - 2τ - 4τ 2 + 2τ 3 + τ 4 ` )` , -4` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 - τ + τ 2 + τ 3 ` )` , 4` (` 5 - τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 4` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )` , 4` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-57, -321, -123, -105, -95, -93, -47, -333] . FixedPtCheck, [57, 321, 123, 105, 95, 93, 47, 333]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 6 7 vs 7 7 vs 7 3 vs 5 4 vs 5

Omega Rank for R :  cycles: {{4, 6}, {2, 8}},   net cycles: 1 .    order:   2

See Matrix
 

[0, -y1 - 5 y2 + 4 y3, y1, -4 y2 + 3 y3, 0, y2, 0, y3]

 

  p = - s 2 + s 4   p' = - s 2 + s 4

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 + y3 + y4 - y2, 0, y1, 0, y3, y4, y2, 0]

 

  p = s 4 - s 5

 » SYNC'D 5/512 , 0.009765625000

 
SUMMARY
Graph Type
CC
ν(A)
 2
ν(Δ)
 2
π
 [1, 1, 1, 1, 1, 1, 1, 1]
Dbly Stoch
true

 
SANDWICH
Total 18
No .ColoringRank
1 {2, 6} 2
2 {2, 3, 6, 7} 2
3 {5, 6, 7, 8} 2
4 {} 4
5 {4, 8} 2
6 {2, 3, 4, 5, 6, 8} 2
7 {2, 3, 5, 8} 2
8 {4, 6} 2
9 {3, 4, 7, 8} 2
10 {3, 7} 2
11 {2, 8} 2
12 {2, 4, 5, 6, 7, 8} 2
13 {5, 7} 2
14 {2, 4} 2
15 {6, 8} 2
16 {3, 5} 2
17 {2, 3, 4, 6, 7, 8} 2
18 {3, 4, 5, 6} 2

 
RT GROUPS
Total 2
No .ColoringRankSolv
1 {2, 4, 6, 8} 8 ["group", Not Solvable]
2 {2, 4, 5, 7} 8 ["group", Not Solvable]

 
CC Colorings
Total 2
No .ColoringSandwich,Rank
1 {5, 6, 7, 8} true, 2
2 {} true, 4

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
96 0 108 , 108 24 , 32 20 128 128