New Graph

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

 

 

π = [3, 4, 6, 5, 2, 4]

POSSIBLE RANKS

1 x 24
2 x 12
3 x 8
4 x 6

BASE DETERMINANT 55/256, .2148437500

NullSpace of Δ

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

Nullspace of A

[{5, 6},{3}]

 

 
1 . Coloring, {}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-12` (` - 1 + τ ` )` 2 ` (` 3 + 2τ + τ 2 ` )` , 24` (` - 1 + τ ` )`` (` 2 + τ + τ 2 ` )` , -24` (` 3 + τ 2 ` )` , 12` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 24` (` - 1 + τ ` )`` (` 1 + τ ` )` , 48` (` - 1 + τ ` )``]`

For τ=1/2, [-17, -44, -104, -111, -24, -32] . FixedPtCheck, [17, 44, 104, 111, 24, 32]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, y1, y2, y3, y4, 0]  

See Matrices
 

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

[y4, y3, y2, 0, 0, y1]  

See Matrices
 

 » SYNC'D 1/16 , 0.06250000000

 
2 . Coloring, {2}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 24` (` - 1 + τ ` )`` (` 2 + τ + τ 2 ` )` , -24` (` 3 + τ 2 ` )` , -12` (` 5 + 2τ 2 + τ 4 ` )` , 24` (` 1 + τ ` )`` (` - 1 + τ ` )` , 48` (` - 1 + τ ` )``]`

For τ=1/2, [-39, -44, -104, -89, -24, -32] . FixedPtCheck, [39, 44, 104, 89, 24, 32]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, y2, y3, y4, y5, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 3, 4, 6}},   net cycles: 0 .    order:   4

[y3, y4, y5, y1, 0, y2]  

See Matrices
 

 » SYNC'D 1/16 , 0.06250000000

 
3 . Coloring, {3}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [15, 48, 52, 49, 12, 16] . FixedPtCheck, [15, 48, 52, 49, 12, 16]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, y1, y2, y3, y4, 0]  

See Matrices
 

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

[y4, 0, y3, y2, 0, y1]  

See Matrices
 

 » SYNC'D 5/8 , 0.6250000000

 
4 . Coloring, {4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 3 + τ ` )` , -24` (` 4 - τ + 3τ 2 + τ 3 + τ 4 ` )` , 24` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 2 + τ + τ 2 ` )` , 24` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 24` (` - 2 - τ - 2τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 48` (` - 2 - τ - 2τ 2 + τ 3 ` )``]`

For τ=1/2, [-35, -71, -77, -111, -69, -92] . FixedPtCheck, [35, 71, 77, 111, 69, 92]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{2, 4, 5, 6}},   net cycles: 0 .    order:   4

[0, y5, y4, y3, y2, y1]  

See Matrices
 

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

[y1, y2, y3, 0, 0, y4]  

See Matrices
 

 » SYNC'D 45/256 , 0.1757812500

 
5 . Coloring, {5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-13, -16, -52, -51, -12, -16] . FixedPtCheck, [13, 16, 52, 51, 12, 16]

det(A + τ Δ) =   0

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

[3, 4, 6, 5, 2, 4]

+              \ ;      -              \ ;      Δ

See Matrices

 
[-y3 - y1, y3, -y3 - y2, y1, y2, y3]
  p = s 3 - 2s 4

         S+              \ ;      S-              \ ;      NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 4 3 vs 5 3 vs 5 3 vs 4 2 vs 4

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

See Matrix
 

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

 

  p = - s 3 + s 4

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

See Matrix
 

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

 

  p = s - s 3   p' = s - s 3


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, 0, x1, -2 x1]
For A+2Δ :   [y1, 9 y1 + 9 y3 - y2, -4 y1 - 3 y3, y1, y3, y2]
For A-2Δ :   [%1, y3, y2, %1, 12 y3 + y2 + 12 y1, y1] %1 := -3 y3 - y2 - 3 y1

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

 
rank of M is 6 , rank of N is 3

M               N

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

Check is ΩΔN zero? true, πΔ= [-1, -4, 2, 5, 2, -4]

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

τ= 18 , r'= 1/2

Ranges

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

ker N, [μ1 + μ3 - μ2, -μ1 - μ3, μ1, μ2, μ3, -μ1 - μ3]
Range of N
    [y1, y2, -y1 + y2 + y3, y1, -y1 + y2 + y3, y3]

Partitions

Action of R on partitions, [[2], [2]]
Action of B on partitions, [[2], [1]]

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

b1 = {3, 5, 6} ` , ` b2 = {1, 2, 4} ` , ` b3 = {2, 3, 5} ` , ` b4 = {1, 4, 6}

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

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

` ` See level-2 partition graph.

`

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

 

 
6 . Coloring, {6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-12` (` - 3 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 24` (` - 2 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -24` (` 1 + τ ` )`` (` - 3 + τ ` )` , 12` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 24` (` - 1 + τ ` )` 2 , -48` (` - 1 + τ ` )``]`

For τ=1/2, [11, 36, 120, 117, 8, 32] . FixedPtCheck, [11, 36, 120, 117, 8, 32]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, y2, y3, y1, 0, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 5, 6}},   net cycles: 0 .    order:   3

[y1, y2, 0, 0, y4, y3]  

See Matrices
 

 » SYNC'D 5/32 , 0.1562500000

 
7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [39, 48, 52, 25, 12, 16] . FixedPtCheck, [39, 48, 52, 25, 12, 16]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, y2, y4, 0, y3, 0]  

See Matrices
 

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

[y2, 0, y1, y3, 0, y4]  

See Matrices
 

 » SYNC'D 5/8 , 0.6250000000

 
8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` 3 - τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 24` (` 4 - τ + 3τ 2 + τ 3 + τ 4 ` )` , 24` (` 6 - τ + τ 2 + τ 3 + τ 4 ` )` , 24` (` 5 + 2τ 2 + τ 4 ` )` , 24` (` 1 + τ ` )` 2 ` (` 2 - τ + τ 2 ` )` , 48` (` 1 + τ ` )`` (` 2 - τ + τ 2 ` )``]`

For τ=1/2, [69, 71, 95, 89, 63, 84] . FixedPtCheck, [69, 71, 95, 89, 63, 84]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, y2, y3, y4, y5, y6]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 3, 4}},   net cycles: 0 .    order:   3

[y1, y4, y3, y2, 0, y5]  

See Matrices
 

 » SYNC'D 33/256 , 0.1289062500

 
9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-21, -16, -52, -43, -12, -16] . FixedPtCheck, [21, 16, 52, 43, 12, 16]

det(A + τ Δ) =   0

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

[3, 4, 6, 5, 2, 4]

+              \ ;      -              \ ;      Δ

See Matrices

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

         S+              \ ;      S-              \ ;      NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 4 3 vs 5 3 vs 5 3 vs 4 2 vs 4

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

See Matrix
 

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

 

  p = s 3 - s 4

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

See Matrix
 

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

 

  p = - s + s 3   p' = - s + s 3


  « NOT SYNC'D »

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

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

 
rank of M is 6 , rank of N is 3

M               N

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

Check is ΩΔN zero? true, πΔ= [3, -4, 2, 1, 2, -4]

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

τ= 18 , r'= 1/2

Ranges

Action of R on ranges, [[4], [1], [2], [4]]
Action of B on ranges, [[3], [3], [4], [3]]
β({1, 3}) = 1/12
β({1, 5}) = 1/6
β({2, 6}) = 1/3
β({3, 4}) = 5/12

ker N, [-μ1 - μ3, μ3, -μ3 - μ2, μ1, μ2, μ3]
Range of N
    [y1, y1 + y2 - y3, y2, y1, y2, y3]

Partitions

Action of R on partitions, [[2], [2]]
Action of B on partitions, [[2], [1]]

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

b1 = {3, 5, 6} ` , ` b2 = {1, 2, 4} ` , ` b3 = {2, 3, 5} ` , ` b4 = {1, 4, 6}

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

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

` ` See level-2 partition graph.

`

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

 

 
10 . Coloring, {2, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` - 1 + τ ` )`` (` - 3 - τ - τ 2 + τ 3 ` )` , 24` (` 1 + τ ` )`` (` - 2 + τ ` )`` (` - 1 + τ ` )` , -24` (` 1 + τ ` )`` (` - 3 + τ ` )` , -12` (` 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 24` (` - 1 + τ ` )` 2 , -48` (` - 1 + τ ` )``]`

For τ=1/2, [29, 36, 120, 99, 8, 32] . FixedPtCheck, [29, 36, 120, 99, 8, 32]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y3, y4, y2, y1, 0, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 5, 6}},   net cycles: 0 .    order:   3

[y4, y5, 0, y1, y2, y3]  

See Matrices
 

 » SYNC'D 5/32 , 0.1562500000

 
11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )` , 24` (` - 4 - τ + τ 3 ` )`` (` 1 + τ ` )` , 24` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 2 + τ + τ 2 ` )` , 24` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -24` (` 1 + τ ` )` 2 ` (` 2 - τ + τ 2 ` )` , -48` (` 1 + τ ` )`` (` 2 - τ + τ 2 ` )``]`

For τ=1/2, [-6, -15, -11, -14, -9, -12] . FixedPtCheck, [6, 15, 11, 14, 9, 12]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{2, 4, 5, 6}},   net cycles: 0 .    order:   4

[0, y1, y2, y5, y3, y4]  

See Matrices
 

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

[y3, 0, y4, y2, 0, y1]  

See Matrices
 

 » SYNC'D 45/256 , 0.1757812500

 
12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-12` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 24` (` 1 + τ ` )`` (` 2 - τ + τ 2 ` )` , 24` (` 3 + τ 2 ` )` , 12` (` 5 + 2τ 2 + τ 4 ` )` , -24` (` - 1 + τ ` )`` (` 1 + τ ` )` , -48` (` - 1 + τ ` )``]`

For τ=1/2, [39, 84, 104, 89, 24, 32] . FixedPtCheck, [39, 84, 104, 89, 24, 32]

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

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

[y5, y4, y3, y2, y1, 0]  

See Matrices
 

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

[y4, y5, y2, y3, 0, y1]  

See Matrices
 

 » SYNC'D 3/16 , 0.1875000000

 
13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [13, 48, 60, 51, 4, 16] . FixedPtCheck, [13, 48, 60, 51, 4, 16]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, y3, y2, y1, 0, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 5, 6}},   net cycles: 0 .    order:   3

[y2, 0, 0, y1, y3, y4]  

See Matrices
 

 » SYNC'D 5/8 , 0.6250000000

 
14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-24` (` 3 - τ + 5τ 2 + τ 3 ` )` , 24` (` 4 + τ + 2τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -24` (` 6 - τ + τ 2 + τ 3 + τ 4 ` )` , -24` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 24` (` - 2 - τ - 2τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 48` (` - 2 - τ - 2τ 2 + τ 3 ` )``]`

For τ=1/2, [-62, -41, -95, -102, -69, -92] . FixedPtCheck, [62, 41, 95, 102, 69, 92]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y5, 0, y1, y2, y3, y4]  

See Matrices
 

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

[y1, y2, y3, 0, 0, y4]  

See Matrices
 

 » SYNC'D 189/512 , 0.3691406250

 
15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` - 1 + τ ` )` 2 ` (` 3 + 2τ + τ 2 ` )` , -24` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 4 - τ + τ 2 ` )` , 24` (` 1 + τ ` )`` (` 6 - 3τ + τ 3 ` )` , 24` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 24` (` - 1 + τ ` )`` (` - 2 - τ - 2τ 2 + τ 3 ` )` , -48` (` - 2 - τ - 2τ 2 + τ 3 ` )``]`

For τ=1/2, [17, 45, 111, 117, 23, 92] . FixedPtCheck, [17, 45, 111, 117, 23, 92]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, y4, y1, y2, 0, y3]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 5, 6}},   net cycles: 0 .    order:   3

[y4, y5, y3, 0, y2, y1]  

See Matrices
 

 » SYNC'D 343/1024 , 0.3349609375

 
16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-7, -16, -60, -57, -4, -16] . FixedPtCheck, [7, 16, 60, 57, 4, 16]

det(A + τ Δ) =   0

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

[3, 4, 6, 5, 2, 4]

+              \ ;      -              \ ;      Δ

See Matrices

 
[-y2 - y1, y2, -y3 - y2, y1, y3, y2]
  p = s 3 - 2s 4

         S+              \ ;      S-              \ ;      NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 4 3 vs 5 3 vs 5 2 vs 3 2 vs 4

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

See Matrix
 

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

 

  p = s 2 - s 3

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

See Matrix
 

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

 

  p = s - s 3   p' = s - s 3


  « NOT SYNC'D »

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

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

 
rank of M is 6 , rank of N is 3

M               N

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

Check is ΩΔN zero? true, πΔ= [-1, -4, 6, 5, -2, -4]

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

τ= 18 , r'= 1/2

Ranges

Action of R on ranges, [[4], [1], [4], [4]]
Action of B on ranges, [[3], [3], [2], [3]]
β({1, 3}) = 1/12
β({1, 5}) = 1/6
β({2, 6}) = 1/3
β({3, 4}) = 5/12

ker N, [μ2, μ3, -μ3 - μ1, -μ2 - μ3, μ1, μ3]
Range of N
    [y3, y3 + y2 - y1, y2, y3, y2, y1]

Partitions

Action of R on partitions, [[2], [2]]
Action of B on partitions, [[2], [1]]

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

b1 = {3, 5, 6} ` , ` b2 = {1, 2, 4} ` , ` b3 = {2, 3, 5} ` , ` b4 = {1, 4, 6}

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

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

` ` See level-2 partition graph.

`

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

 

 
17 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` , 24` (` 1 + τ ` )`` (` - 4 - τ + τ 3 ` )` , -24` (` 6 - τ + τ 2 + τ 3 + τ 4 ` )` , 24` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 24` (` 1 + τ ` )` 2 ` (` 2 + τ ` )`` (` - 1 + τ ` )` , 48` (` 1 + τ ` )`` (` 2 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-18, -21, -19, -10, -9, -12] . FixedPtCheck, [18, 21, 19, 10, 9, 12]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y5, y3, y4, 0, y1, y2]  

See Matrices
 

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

[y1, 0, y4, y2, 0, y3]  

See Matrices
 

 » SYNC'D 81/1024 , 0.07910156250

 
18 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` 1 + τ ` )` 2 ` (` 3 - 2τ + τ 2 ` )` , 24` (` 1 + τ ` )`` (` 2 - τ + τ 2 ` )` , 24` (` 3 + τ 2 ` )` , -12` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -24` (` 1 + τ ` )`` (` - 1 + τ ` )` , -48` (` - 1 + τ ` )``]`

For τ=1/2, [81, 84, 104, 47, 24, 32] . FixedPtCheck, [81, 84, 104, 47, 24, 32]

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

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

[y1, y2, y4, 0, y3, 0]  

See Matrices
 

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

[0, y1, y2, y3, 0, y4]  

See Matrices
 

 » SYNC'D 3/16 , 0.1875000000

 
19 . Coloring, {2, 3, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [37, 48, 60, 27, 4, 16] . FixedPtCheck, [37, 48, 60, 27, 4, 16]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, y2, y3, 0, 0, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 5, 6}},   net cycles: 0 .    order:   3

[y1, 0, 0, y2, y3, y4]  

See Matrices
 

 » SYNC'D 5/8 , 0.6250000000

 
20 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-24` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 24` (` 4 + τ + 2τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 24` (` - 6 + τ - 3τ 2 - τ 3 + τ 4 ` )` , -24` (` 5 - τ + 3τ 2 + τ 3 ` )` , -24` (` 2 - τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -48` (` 2 - τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-78, -41, -101, -86, -63, -84] . FixedPtCheck, [78, 41, 101, 86, 63, 84]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y2, y4, y3, y5]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 3, 4}},   net cycles: 0 .    order:   3

[0, y3, y4, y1, 0, y2]  

See Matrices
 

 » SYNC'D 85/1024 , 0.08300781250

 
21 . Coloring, {2, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 24` (` 4 - τ + τ 2 ` )`` (` - 1 + τ ` )` , 24` (` - 2 + τ ` )`` (` 3 + τ 2 ` )` , 24` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 24` (` - 1 + τ ` )`` (` 2 - τ + τ 2 ` )` , -48` (` 2 - τ + τ 2 ` )``]`

For τ=1/2, [-13, -15, -39, -33, -7, -28] . FixedPtCheck, [13, 15, 39, 33, 7, 28]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, y2, y5, y3, 0, y4]  

See Matrices
 

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

See Matrix
 

[2 y3, 2 y2, -15 y3 - 2 y2 + 13 y1, 2 y1, 2 y3 + 2 y2 - 2 y1, -13 y3 - 2 y2 + 11 y1]

 

  p' = - 1 + s 3   p' = - s + s 4   p' = - s 2 + s 5

 » SYNC'D 3/128 , 0.02343750000

 
22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-15, -16, -60, -49, -4, -16] . FixedPtCheck, [15, 16, 60, 49, 4, 16]

det(A + τ Δ) =   0

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

[3, 4, 6, 5, 2, 4]

+              \ ;      -              \ ;      Δ

See Matrices

 
[-y3 - y1, y3, -y3 - y2, y1, y2, y3]
  p = s 3 - 2s 4

         S+              \ ;      S-              \ ;      NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
3 vs 4 3 vs 5 3 vs 5 2 vs 3 2 vs 4

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

See Matrix
 

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

 

  p = - s 2 + s 3

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

See Matrix
 

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

 

  p = s - s 3   p' = - s + s 3


  « NOT SYNC'D »

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

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

 
rank of M is 6 , rank of N is 3

M               N

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

Check is ΩΔN zero? true, πΔ= [3, -4, 6, 1, -2, -4]

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

τ= 18 , r'= 1/2

Ranges

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

ker N, [-μ1 - μ3, μ3, -μ3 - μ2, μ1, μ2, μ3]
Range of N
    [y1, y1 + y2 - y3, y2, y1, y2, y3]

Partitions

Action of R on partitions, [[2], [2]]
Action of B on partitions, [[2], [1]]

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

b1 = {3, 5, 6} ` , ` b2 = {1, 2, 4} ` , ` b3 = {2, 3, 5} ` , ` b4 = {1, 4, 6}

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

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

` ` See level-2 partition graph.

`

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

 

 
23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` 1 + τ ` )`` (` 3 - τ + τ 2 + τ 3 ` )` , 24` (` 1 + τ ` )`` (` 4 - τ + τ 3 ` )` , 24` (` 6 - τ + τ 2 + τ 3 + τ 4 ` )` , 24` (` 5 + 2τ 2 + τ 4 ` )` , 24` (` 1 + τ ` )` 2 ` (` 2 - τ + τ 2 ` )` , 48` (` 1 + τ ` )`` (` 2 - τ + τ 2 ` )``]`

For τ=1/2, [69, 87, 95, 89, 63, 84] . FixedPtCheck, [69, 87, 95, 89, 63, 84]

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

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

[y4, y1, y2, y3, y6, y5]  

See Matrices
 

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

[y5, y4, y1, y2, 0, y3]  

See Matrices
 

 » SYNC'D 9/128 , 0.07031250000

 
24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 24` (` 4 - 3τ + τ 2 ` )`` (` 1 + τ ` )` , 24` (` 6 - 3τ + τ 3 ` )` , 24` (` 5 - 2τ + τ 2 ` )` , -24` (` - 1 + τ ` )`` (` 2 - τ + τ 2 ` )` , 48` (` 2 - τ + τ 2 ` )``]`

For τ=1/2, [10, 33, 37, 34, 7, 28] . FixedPtCheck, [10, 33, 37, 34, 7, 28]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, y3, y4, y2, 0, y1]  

See Matrices
 

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

See Matrix
 

[13 y1 + 13 y2 - 11 y3 - 11 y4, 0, 11 y1, 11 y2, 11 y3, 11 y4]

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 37/1024 , 0.03613281250

 
25 . Coloring, {3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-12` (` - 1 + τ ` )`` (` - 3 - τ - τ 2 + τ 3 ` )` , 24` (` - 2 - τ - 2τ 2 + τ 3 ` )` , 24` (` 1 + τ ` )`` (` - 3 + τ ` )` , 12` (` 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , -24` (` - 1 + τ ` )` 2 , 48` (` - 1 + τ ` )``]`

For τ=1/2, [-29, -92, -120, -99, -8, -32] . FixedPtCheck, [29, 92, 120, 99, 8, 32]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y4, y1, y2, y3, 0, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 5, 6}},   net cycles: 0 .    order:   4

[y1, y4, 0, y3, y2, y5]  

See Matrices
 

 » SYNC'D 15/32 , 0.4687500000

 
26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-24` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 24` (` - 1 + τ ` )`` (` - 4 - τ + τ 3 ` )` , -24` (` - 2 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -24` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 24` (` - 1 + τ ` )`` (` - 2 - τ - 2τ 2 + τ 3 ` )` , -48` (` - 2 - τ - 2τ 2 + τ 3 ` )``]`

For τ=1/2, [26, 35, 117, 114, 23, 92] . FixedPtCheck, [26, 35, 117, 114, 23, 92]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y4, y3, 0, y2]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 5, 6}},   net cycles: 0 .    order:   4

[y3, y4, y5, 0, y1, y2]  

See Matrices
 

 » SYNC'D 105/512 , 0.2050781250

 
27 . Coloring, {2, 3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` 1 + τ ` )` 2 ` (` - 3 + τ 2 ` )` , -24` (` 1 + τ ` )`` (` 4 - τ + τ 3 ` )` , 24` (` - 6 + τ - 3τ 2 - τ 3 + τ 4 ` )` , 24` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 24` (` 1 + τ ` )` 2 ` (` 2 + τ ` )`` (` - 1 + τ ` )` , 48` (` 1 + τ ` )`` (` 2 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-99, -87, -101, -47, -45, -60] . FixedPtCheck, [99, 87, 101, 47, 45, 60]

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

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

[y3, y1, y2, 0, y5, y4]  

See Matrices
 

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

See Matrix
 

[0, 2 y3, 2 y2, 2 y1, 0, 3 y3]

 

  p = - s 2 + s 4

 » SYNC'D 65/1024 , 0.06347656250

 
28 . Coloring, {2, 3, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` 3 + τ 2 ` )` , 24` (` 1 + τ ` )`` (` 4 - 3τ + τ 2 ` )` , -24` (` - 2 + τ ` )`` (` 3 + τ 2 ` )` , 24` (` - 1 + τ ` )`` (` - 5 + τ ` )` , 24` (` - 1 + τ ` )` 2 ` (` 2 + τ ` )` , -48` (` - 1 + τ ` )`` (` 2 + τ ` )``]`

For τ=1/2, [26, 33, 39, 18, 5, 20] . FixedPtCheck, [26, 33, 39, 18, 5, 20]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, y3, y4, 0, 0, y2]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 21/128 , 0.1640625000

 
29 . Coloring, {2, 3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `12` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 24` (` - 2 - τ - 2τ 2 + τ 3 ` )` , 24` (` 1 + τ ` )`` (` - 3 + τ ` )` , -12` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -24` (` - 1 + τ ` )` 2 , 48` (` - 1 + τ ` )``]`

For τ=1/2, [-75, -92, -120, -53, -8, -32] . FixedPtCheck, [75, 92, 120, 53, 8, 32]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, y2, y3, 0, 0, 0]  

See Matrices
 

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

[0, y1, 0, y2, y3, y4]  

See Matrices
 

 » SYNC'D 15/32 , 0.4687500000

 
30 . Coloring, {2, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-24` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 24` (` - 4 - τ + τ 3 ` )`` (` - 1 + τ ` )` , 24` (` 6 + 3τ + τ 2 - 3τ 3 + τ 4 ` )` , -24` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -24` (` - 1 + τ ` )`` (` 2 - τ + τ 2 ` )`` (` 1 + τ ` )` , 48` (` 2 - τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [6, 5, 17, 14, 3, 12] . FixedPtCheck, [6, 5, 17, 14, 3, 12]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y3, y2, 0, y4]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 3, 4}},   net cycles: 0 .    order:   3

[0, y1, y2, y3, y4, y5]  

See Matrices
 

 » SYNC'D 289/1024 , 0.2822265625

 
31 . Coloring, {3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 24` (` - 4 + τ - 2τ 2 + τ 3 ` )` , 24` (` - 2 + τ ` )`` (` 3 + τ 2 ` )` , 24` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 24` (` - 1 + τ ` )`` (` 2 - τ + τ 2 ` )` , -48` (` 2 - τ + τ 2 ` )``]`

For τ=1/2, [-13, -31, -39, -33, -7, -28] . FixedPtCheck, [13, 31, 39, 33, 7, 28]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{2, 3, 4, 6}},   net cycles: 0 .    order:   4

[y4, y3, y2, y1, 0, y5]  

See Matrices
 

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

See Matrix
 

[10 y1, -23 y1 + 39 y2 - 23 y3 - 10 y4, -11 y1 + 23 y2 - 11 y3, 10 y2, 10 y3, 10 y4]

 

  p' = - 1 + s 4   p' = - s + s 5

 » SYNC'D 3/256 , 0.01171875000

 
32 . Coloring, {2, 3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `24` (` 1 + τ ` )` 2 ` (` 3 - 2τ + τ 2 ` )` , -24` (` 1 + τ ` )`` (` - 4 + τ - 2τ 2 + τ 3 ` )` , 24` (` 6 + 3τ + τ 2 - 3τ 3 + τ 4 ` )` , 24` (` - 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )` , 24` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 2 + τ ` )` , -48` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 2 + τ ` )``]`

For τ=1/2, [81, 93, 119, 53, 15, 60] . FixedPtCheck, [81, 93, 119, 53, 15, 60]

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

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

[y4, y3, y2, 0, 0, y1]  

See Matrices
 

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

[0, y1, y5, y3, y4, y2]  

See Matrices
 

 » SYNC'D 13/64 , 0.2031250000

 
SUMMARY
Graph Type
NOT CC
ν(A)
 1
ν(Δ)
 2
π
 [3, 4, 6, 5, 2, 4]
Dbly Stoch
false

 
SANDWICH
Total 4
No .ColoringRank
1 {2, 5} 2
2 {5} 2
3 {2, 5, 6} 2
4 {5, 6} 2

 
RT GROUPS
Total 0
No .ColoringRankSolv

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
28 0 24 , 23 28 , 23 4 32 32