New Graph

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

 

 

π = [4, 1, 3, 4, 2, 2]

POSSIBLE RANKS

1 x 16
2 x 8
4 x 4

BASE DETERMINANT 2831/16384, .1727905273

NullSpace of Δ

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

Nullspace of A

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

 

 
1 . Coloring, {}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [48, 9, 39, 16, 12, 4] . FixedPtCheck, [48, 9, 39, 16, 12, 4]

det(A + τ Δ) =   0

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

[4, 1, 3, 4, 2, 2]

+              \ ;      -              \ ;      Δ

See Matrices

 
[y3, y3 - y2, y2, -y3, -y3 - y1, y1]
  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 5 3 vs 5 3 vs 5 2 vs 3 2 vs 3

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

See Matrix
 

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

 

  p = - s 2 + s 3

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

See Matrix
 

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

 

  p = s 2 - s 3


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, 0, 0, x1, x2]
For A+2Δ :   [y1, y3, y3, y2, %1, %1] %1 := -3 y1 - 3 y3 - y2
For A-2Δ :   [-y1 - 3 y2 - 3 y3, y1, y1, y2, y3, y3]

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

 
rank of M is 4 , rank of N is 3

M               N

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

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

ker M, [0, -3 λ2, λ2, 0, -λ1, λ1]
Range M, [x2, x1, 3 x1, x4, x3, x3]

τ= 19 , r'= 1/2

Ranges

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

`

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

 

 
2 . Coloring, {2}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 8` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 8` (` 15 + 2τ - 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , -32` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 16` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 16` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [912, 171, 759, 400, 228, 100] . FixedPtCheck, [912, 171, 759, 400, 228, 100]

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

See Matrix for A+τΔ

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

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

[y2, y3, y1, y4, 0, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s 3 + s 4

 » SYNC'D 1/32 , 0.03125000000

 
3 . Coloring, {3}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 8` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 8` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -32` (` 5 - τ + 3τ 2 + τ 3 ` )` , 16` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 16` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-816, -153, -741, -688, -204, -172] . FixedPtCheck, [816, 153, 741, 688, 204, 172]

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

See Matrix for A+τΔ

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

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

[y3, y4, y1, y2, 0, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = s 3 - s 4

 » SYNC'D 21/256 , 0.08203125000

 
4 . Coloring, {4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `8` (` 5 - τ + 3τ 2 + τ 3 ` )` , -2` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 2` (` 15 - 2τ + 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , 8` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 4` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [688, 129, 687, 304, 172, 228] . FixedPtCheck, [688, 129, 687, 304, 172, 228]

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

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

See Matrix
 

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

 

  p = s 2 - s 4   p' = s 2 - s 3

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

[y1, 0, 0, y2, y3, 0]  

See Matrices
 

 » SYNC'D 1/4 , 0.2500000000

 
5 . Coloring, {5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-16` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -4` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 4` (` 1 + τ ` )`` (` 15 + 2τ - 2τ 3 + τ 4 ` )` , -16` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 8` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 8` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [912, 57, 759, 400, 228, 100] . FixedPtCheck, [912, 57, 759, 400, 228, 100]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 2 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, 3}},   net cycles: 0 .    order:   2

See Matrix
 

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

 

  p = - s 2 + s 3

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

[0, y1, 0, y3, y2, y4]  

See Matrices
 

 » SYNC'D 3/32 , 0.09375000000

 
6 . Coloring, {6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-16` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 4` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -4` (`15 + τ + 18τ 2 - 2τ 3 - τ 4 + τ 5 ` )` , -16` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 8` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 8` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )``]`

For τ=1/2, [-816, -153, -631, -304, -204, -76] . FixedPtCheck, [816, 153, 631, 304, 204, 76]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 2 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, 3}},   net cycles: -1 .    order:   2

See Matrix
 

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

 

  p = s 2 - s 4   p' = s 2 - s 3

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

[0, 0, y1, y2, y3, y4]  

See Matrices
 

 » SYNC'D 5/64 , 0.07812500000

 
7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-8` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 2` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , -2` (` 15 + 2τ - 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , 8` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 4` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-816, -153, -759, -784, -204, -196] . FixedPtCheck, [816, 153, 759, 784, 204, 196]

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

See Matrix for A+τΔ

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

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

[y1, y2, y3, y4, 0, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = s 3 - s 4

 » SYNC'D 1/4 , 0.2500000000

 
8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32` (` 5 - τ + 3τ 2 + τ 3 ` )` , 8` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 8` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 32` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 16` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 16` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-688, -129, -741, -400, -172, -300] . FixedPtCheck, [688, 129, 741, 400, 172, 300]

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

See Matrix for A+τΔ

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

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

[y2, y3, y4, y1, 0, y5]  

See Matrices
 

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

[y1, 0, 0, y2, y3, 0]  

See Matrices
 

 » SYNC'D 17/128 , 0.1328125000

 
9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32` (` - 5 + τ 2 ` )` , 8` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 8` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -32` (` - 5 + τ ` )`` (` - 1 + τ ` )` , -16` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 16` (` - 5 + τ ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-304, -19, -255, -144, -76, -36] . FixedPtCheck, [304, 19, 255, 144, 76, 36]

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

See Matrix for A+τΔ

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

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

[y1, 0, y2, y3, 0, 0]  

See Matrices
 

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

See Matrix
 

[7 y2, 7 y1, 0, 9 y2 + 9 y1 + 9 y3 - 7 y4, 7 y3, 7 y4]

 

  p = s + s 2 - s 4 - s 5

 » SYNC'D 7/128 , 0.05468750000

 
10 . Coloring, {2, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 8` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 8` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 32` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 16` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -16` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-816, -153, -637, -400, -204, -100] . FixedPtCheck, [816, 153, 637, 400, 204, 100]

det(A + τ Δ) =   1` (` 1 + τ ` )`` (` - 1 + τ ` )` 3 ` (` τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 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: {{1, 3}},   net cycles: 0 .    order:   4

[y1, y2, y3, y4, 0, 0]  

See Matrices
 

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

[y5, 0, y4, y2, y3, y1]  

See Matrices
 

 » SYNC'D 269/2048 , 0.1313476562

 
11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 8` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 8` (` 15 - 2τ + 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , 32` (` 5 - τ + 3τ 2 + τ 3 ` )` , 16` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 16` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [400, 75, 687, 688, 100, 516] . FixedPtCheck, [400, 75, 687, 688, 100, 516]

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

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, y3, y4, y2, 0, y5]  

See Matrices
 

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

[y1, 0, 0, y2, y3, 0]  

See Matrices
 

 » SYNC'D 3/16 , 0.1875000000

 
12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 8` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 8` (` 15 + 2τ - 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , -32` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -16` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 16` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [816, 51, 759, 784, 204, 196] . FixedPtCheck, [816, 51, 759, 784, 204, 196]

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

See Matrix for A+τΔ

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

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

[y1, 0, y2, y3, 0, 0]  

See Matrices
 

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

[y2, y1, 0, y4, y5, y3]  

See Matrices
 

 » SYNC'D 95/512 , 0.1855468750

 
13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 8` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 8` (`15 + τ + 18τ 2 - 2τ 3 - τ 4 + τ 5 ` )` , 32` (` 5 - τ + 3τ 2 + τ 3 ` )` , 16` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -16` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [784, 147, 631, 688, 196, 172] . FixedPtCheck, [784, 147, 631, 688, 196, 172]

det(A + τ Δ) =   1` (` 1 + τ ` )`` (` τ ` )`` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 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: {{1, 3, 4}},   net cycles: 0 .    order:   3

[y1, y2, y3, y4, 0, 0]  

See Matrices
 

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

[y1, 0, y2, y4, y5, y3]  

See Matrices
 

 » SYNC'D 479/4096 , 0.1169433594

 
14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-688, -43, -741, -400, -172, -300] . FixedPtCheck, [688, 43, 741, 400, 172, 300]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 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, 3}},   net cycles: 0 .    order:   4

See Matrix
 

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

 

  p = s 3 - s 4

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

[y1, y2, 0, y3, y4, 0]  

See Matrices
 

 » SYNC'D 11/128 , 0.08593750000

 
15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-400, -75, -357, -304, -100, -228] . FixedPtCheck, [400, 75, 357, 304, 100, 228]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 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, 6}},   net cycles: 1 .    order:   2

See Matrix
 

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

 

  p = - s 2 + s 4   p' = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 4, 5}},   net cycles: 0 .    order:   3

[y1, 0, y2, y3, y4, 0]  

See Matrices
 

 » SYNC'D 7/128 , 0.05468750000

 
16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [816, 51, 637, 400, 204, 100] . FixedPtCheck, [816, 51, 637, 400, 204, 100]

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

See Matrix for A+τΔ

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

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

See Matrix
 

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

 

  p = - s 2 + s 3

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

[0, y3, y4, y5, y1, y2]  

See Matrices
 

 » SYNC'D 1/4 , 0.2500000000

 
17 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [400, 75, 741, 784, 100, 588] . FixedPtCheck, [400, 75, 741, 784, 100, 588]

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

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, y1, y2, y4, 0, y3]  

See Matrices
 

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

[y3, 0, 0, y2, y1, 0]  

See Matrices
 

 » SYNC'D 3/8 , 0.3750000000

 
18 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-16 , -4` (` - 1 + τ ` )` 2 , 4` (` 1 + τ ` )`` (` - 3 + τ ` )` , -16 , 8` (` - 1 + τ ` )` , 8` (` - 1 + τ ` )``]`

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

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

See Matrix for A+τΔ

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

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

[y1, 0, y2, y3, 0, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 67/512 , 0.1308593750

 
19 . Coloring, {2, 3, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-16 , 4` (` 1 + τ ` )`` (` - 1 + τ ` )` , -4` (` 3 + τ 2 ` )` , -16 , 8` (` - 1 + τ ` )` , 8` (` - 1 + τ ` )``]`

For τ=1/2, [-16, -3, -13, -16, -4, -4] . FixedPtCheck, [16, 3, 13, 16, 4, 4]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 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: {{1, 3, 4}},   net cycles: 0 .    order:   3

[y1, y2, y3, y4, 0, 0]  

See Matrices
 

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

[y4, 0, y3, y2, y1, y5]  

See Matrices
 

 » SYNC'D 45/512 , 0.08789062500

 
20 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32` (` 5 - τ + 3τ 2 + τ 3 ` )` , 8` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 8` (` 15 + 2τ - 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , 32` (` 1 + τ ` )`` (` - 5 + τ ` )`` (` - 1 + τ ` )` , -16` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 16` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [688, 43, 759, 432, 172, 324] . FixedPtCheck, [688, 43, 759, 432, 172, 324]

det(A + τ Δ) =   1` (` τ ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 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, 3}},   net cycles: 0 .    order:   4

[y4, 0, y2, y3, 0, y1]  

See Matrices
 

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

[y3, y4, 0, y1, y2, 0]  

See Matrices
 

 » SYNC'D 53/256 , 0.2070312500

 
21 . Coloring, {2, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32 , 8` (` - 1 + τ ` )`` (` 1 + τ ` )` , 8` (` 1 + τ ` )`` (` - 3 + τ ` )` , -32 , 16` (` - 1 + τ ` )` , -16` (` 1 + τ ` )``]`

For τ=1/2, [-16, -3, -15, -16, -4, -12] . FixedPtCheck, [16, 3, 15, 16, 4, 12]

det(A + τ Δ) =   1` (` τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 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, 6}},   net cycles: 1 .    order:   2

See Matrix
 

[-7 y3 + 15 y2 - 7 y1, 6 y3, 6 y2, -15 y3 + 27 y2 - 15 y1, 0, 6 y1]

 

  p = s 2 - s 4   p' = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 4, 5}},   net cycles: 0 .    order:   3

[y1, 0, y2, y4, y3, 0]  

See Matrices
 

 » SYNC'D 35/2048 , 0.01708984375

 
22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32` (` 5 - 2τ + τ 2 ` )` , 8` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 8` (`15 - 10τ + 16τ 2 - 6τ 3 + τ 4 ` )` , 32` (` - 1 + τ ` )`` (` - 5 + τ ` )` , -16` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -16` (` - 1 + τ ` )` 2 ` (` - 5 + τ ` )``]`

For τ=1/2, [272, 17, 213, 144, 68, 36] . FixedPtCheck, [272, 17, 213, 144, 68, 36]

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

See Matrix for A+τΔ

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

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

[y3, 0, y1, y2, 0, 0]  

See Matrices
 

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

See Matrix
 

[28 y1 + 25 y3 - 35 y2, -20 y3 + 28 y2, 16 y1, 20 y1 + 27 y3 - 25 y2, 16 y3, 16 y2]

 

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

 » SYNC'D 1715/16384 , 0.1046752930

 
23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 8` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 8` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 32` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -16` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 16` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-400, -25, -741, -784, -100, -588] . FixedPtCheck, [400, 25, 741, 784, 100, 588]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 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, 6}},   net cycles: 0 .    order:   3

[y1, 0, y2, y3, 0, y4]  

See Matrices
 

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

[y1, y2, 0, y3, y4, 0]  

See Matrices
 

 » SYNC'D 27/128 , 0.2109375000

 
24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-32` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 8` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 8` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -32` (` 5 - τ + 3τ 2 + τ 3 ` )` , 16` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -16` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-304, -57, -357, -688, -76, -516] . FixedPtCheck, [304, 57, 357, 688, 76, 516]

det(A + τ Δ) =   1` (` τ ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 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: {{4, 6}},   net cycles: 0 .    order:   4

[y2, y3, y1, y4, 0, y5]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 4, 5}},   net cycles: 0 .    order:   3

[y2, 0, y1, y3, y4, 0]  

See Matrices
 

 » SYNC'D 63/512 , 0.1230468750

 
25 . Coloring, {3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32 , 8` (` - 1 + τ ` )` 2 , 8` (` 3 + τ 2 ` )` , 32 , -16` (` - 1 + τ ` )` , -16` (` - 1 + τ ` )``]`

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

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

See Matrix for A+τΔ

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

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

[y3, 0, y2, y1, 0, 0]  

See Matrices
 

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

[y1, y2, y3, y4, y5, y6]  

See Matrices
 

 » SYNC'D 4007/16384 , 0.2445678711

 
26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [16, 1, 15, 16, 4, 12] . FixedPtCheck, [16, 1, 15, 16, 4, 12]

det(A + τ Δ) =   0

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

[4, 1, 3, 4, 2, 2]

+              \ ;      -              \ ;      Δ

See Matrices

 
[0, -y1, y1, 0, -y2, y2]
  p' = s 4   p' = s 3   p = s 3

         S+              \ ;      S-              \ ;      NM
See Matrices

CmmCk true, true, true


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

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

See Matrix
 

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

 

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

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

See Matrix
 

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

 

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


  « NOT SYNC'D »

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

Range of {ΩΔi}: [0, μ2, -μ2, 0, μ1, -μ1]

 
rank of M is 6 , rank of N is 4

M               N

$ [ [0, 8, 24, 36, 18, 18] , [8, 0, 0, 9, 9, 0] , [24, 0, 0, 27, 9, 18] , [36, 9, 27, 0, 16, 16] , [18, 9, 9, 16, 0, 0] , [18, 0, 18, 16, 0, 0] ] $     $ [ [0, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1] , [1, 0, 0, 1, 1, 1] , [1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 0, 0] , [1, 1, 1, 1, 0, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, -1, 1, 0, -2, 2]

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

τ= 10 , r'= 3/4

Ranges

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

ker N, [0, μ1, -μ1, 0, -μ2, μ2]
Range of N
    [y1, y3, y3, y4, y2, y2]

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

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

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

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

` ` See level-4 partition graph.

`

Right Group
Coloring {4, 5, 6}
Rank4
R,B [3, 1, 1, 6, 4, 4], [5, 4, 4, 1, 2, 3]
π2 [8, 24, 36, 18, 18, 0, 9, 9, 0, 27, 9, 18, 16, 16, 0]
u2 [1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 2)
wpp [1, 2, 2, 1, 2, 2]
π4 [0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0]
u4 [0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0]

 

 
27 . Coloring, {2, 3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-16` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -4` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 4` (` 15 + 2τ - 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , 16` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 8` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 8` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [400, 25, 759, 816, 100, 612] . FixedPtCheck, [400, 25, 759, 816, 100, 612]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 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: {{3, 4, 6}},   net cycles: 1 .    order:   3

[0, 0, y1, y2, 0, y3]  

See Matrices
 

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

[y3, y4, 0, y1, y2, 0]  

See Matrices
 

 » SYNC'D 3/8 , 0.3750000000

 
28 . Coloring, {2, 3, 4, 6}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-304, -57, -375, -784, -76, -588] . FixedPtCheck, [304, 57, 375, 784, 76, 588]

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

See Matrix for A+τΔ

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

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

See Matrix
 

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

 

  p = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 4, 5}},   net cycles: 0 .    order:   3

[y1, 0, y3, y4, y2, 0]  

See Matrices
 

 » SYNC'D 9/32 , 0.2812500000

 
29 . Coloring, {2, 3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-4` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -1` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 1` (` 1 + τ ` )`` (`15 - 10τ + 16τ 2 - 6τ 3 + τ 4 ` )` , 4` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 2` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -2` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [784, 49, 639, 816, 196, 204] . FixedPtCheck, [784, 49, 639, 816, 196, 204]

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

See Matrix for A+τΔ

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

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

[y1, 0, y2, y3, 0, 0]  

See Matrices
 

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

[y1, y2, y3, 0, y4, y5]  

See Matrices
 

 » SYNC'D 3/8 , 0.3750000000

 
30 . Coloring, {2, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32` (` 5 + 2τ + τ 2 ` )` , 8` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 8` (` 15 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -32` (` - 5 + τ ` )`` (` 1 + τ ` )` , -16` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -16` (` - 5 + τ ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [400, 25, 381, 432, 100, 324] . FixedPtCheck, [400, 25, 381, 432, 100, 324]

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

See Matrix for A+τΔ

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

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

See Matrix
 

[-8 y2 + 7 y1, 0, y2, y1, 0, -9 y2 + 8 y1]

 

  p' = - s + s 3   p = - s + s 3

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

[y1, y4, y5, y2, y3, 0]  

See Matrices
 

 » SYNC'D 1/64 , 0.01562500000

 
31 . Coloring, {3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `32` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 8` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 , 8` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -32` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -16` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , -16` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [304, 19, 375, 784, 76, 588] . FixedPtCheck, [304, 19, 375, 784, 76, 588]

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

See Matrix for A+τΔ

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

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

See Matrix
 

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

 

  p = - s 3 + s 4

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

[y1, y3, y2, y4, y5, 0]  

See Matrices
 

 » SYNC'D 39/256 , 0.1523437500

 
32 . Coloring, {2, 3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `8` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 2` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 , -2` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 15 + 3τ - 3τ 2 + τ 3 ` )` , 8` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -4` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 4` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [304, 19, 381, 816, 76, 612] . FixedPtCheck, [304, 19, 381, 816, 76, 612]

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

See Matrix for A+τΔ

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

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

See Matrix
 

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

 

  p = - s 2 + s 3

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

[y1, y2, y3, 0, y4, 0]  

See Matrices
 

 » SYNC'D 1/4 , 0.2500000000

 
SUMMARY
Graph Type
CC
ν(A)
 1
ν(Δ)
 1
π
 [4, 1, 3, 4, 2, 2]
Dbly Stoch
false

 
SANDWICH
Total 1
No .ColoringRank
1 {} 2

 
RT GROUPS
Total 1
No .ColoringRankSolv
1 {4, 5, 6} 4 Not Solvable

 
CC Colorings
Total 1
No .ColoringSandwich,Rank
1 {} true, 2

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
30 0 29 , 29 19 , 24 2 32 32