New Graph

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

π = [1, 2, 2, 3, 2, 2]

POSSIBLE RANKS

1 x 12
2 x 6
3 x 4

BASE DETERMINANT 231/2048, .1127929688

NullSpace of Δ

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

Nullspace of A

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

1 . Coloring, {}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-1` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 2` (` 3 + τ ` )`` (` 1 + τ ` )` , -2` (` - 1 + τ ` )`` (` 3 + τ ` )` , 1` (` 9 - 2τ + τ ² ` )`` (` 1 + τ ` )` , 2` (` 3 + τ ² ` )` , -2` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [13, 84, 28, 99, 52, 60] . FixedPtCheck, [13, 84, 28, 99, 52, 60]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₃, -y₂, y₂, y₃, -y₁, y₁]

p = s ³ + 2s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 4	3 vs 5	3 vs 5	1 vs 4	3 vs 5

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

See Matrix

[0, 2 y₁, 0, 2 y₁, y₁, y₁]

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

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

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

« NOT SYNC'D »

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

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

rank of M is 5 , rank of N is 3

M N

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

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

ker M, [0, 0, 0, 0, -λ₁, λ₁]
Range M, [x₄, x₁, x₂, x₃, x₅, x₅]

τ= 12 , r'= 2/3

Ranges

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

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {}

Rank 3

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

π₂ [0, 2, 0, 1, 1, 0, 4, 2, 2, 2, 2, 2, 3, 3, 0]

u₂ [1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)

wpp [2, 2, 2, 2, 2, 2]

π₃ [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 1, 0, 0]

u₃ [0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

Right Group
Coloring	{}
Rank	3
R,B	[2, 4, 4, 2, 6, 5], [3, 6, 5, 3, 1, 4]
π₂	[0, 2, 0, 1, 1, 0, 4, 2, 2, 2, 2, 2, 3, 3, 0]
u₂	[1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)
wpp	[2, 2, 2, 2, 2, 2]
π₃	[0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 1, 0, 0]
u₃	[0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

2 . Coloring, {2}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-3` (` 3 - τ + 5τ ² + τ ³ ` )`` (` - 1 + τ ` )` , -6` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )` , 6` (` - 1 + τ ` )` ² ` (` 3 + τ ` )` , 3` (` - 1 + τ ` )`` (` - 9 + 4τ + τ ² ` )`` (` 1 + τ ` )` , 6` (` 3 - τ + 5τ ² + τ ³ ` )` , 6` (` 1 + τ ` )`` (` 3 + τ ² ` )``]`

For τ=1/2, [31, 84, 28, 81, 124, 156] . FixedPtCheck, [31, 84, 28, 81, 124, 156]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	4 vs 5	5 vs 5	3 vs 4	4 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 1/4, 3/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1/10, 3/10, 0, 0, 0] , [0, 3/10, 9/10, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [0, 11/8, 3/8, 1/3, -2] , [3/4, 27/8, 29/8, 1, -26/3] , [-1/4, -11/8, -5/8, -1, 10/3] , [0, 11/8, 3/8, 1/3, -2] , [0, -3/8, 1/8, 1, -2/3] , [0, -35/8, -31/8, -5/3, 10] ] $ x $ [ [3/2, 1, 3, 7/2, 2, 1] , [3/2, 5/4, 15/4, 9/4, 5/2, 3/4] , [15/8, 15/16, 45/16, 39/16, 3, 15/16] , [9/4, 69/64, 207/64, 135/64, 75/32, 63/64] , [225/128, 279/256, 837/256, 603/256, 171/64, 219/256] ] $

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

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

See Matrix

[0, y₃, 0, y₃ + y₁ - y₂, y₁, y₂]

p = - s ³ + s ⁴

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

[y₁, 0, y₃, y₂, y₄, 0]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0] ] $ = $ [ [0, 7/36, -5/36, 1/36] , [1/4, -5/36, 1/36, -1/18] , [0, 1/36, 7/36, -5/36] , [0, 7/36, -5/36, 1/36] , [0, -5/36, 1/36, 7/36] , [1/4, -5/36, 1/36, -1/18] ] $ x $ [ [2, 0, 4, 4, 2, 0] , [2, 0, 6, 0, 4, 0] , [4, 0, 2, 0, 6, 0] , [6, 0, 4, 0, 2, 0] ] $

» SYNC'D 1/4 , 0.2500000000

3 . Coloring, {3}

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

` See graph

` ` See pair graph

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

For τ=1/2, [3, 12, 4, 13, 12, 12] . FixedPtCheck, [3, 12, 4, 13, 12, 12]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[y₁, y₂, -y₂, -y₁ - y₂, y₂, 0]

p' = s ² + 2s ³ p = s ² - 4s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 4	2 vs 4	2 vs 4	2 vs 4	2 vs 4

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

See Matrix

[0, y₁, 0, y₂, y₁, y₂]

p = - s + s ³ p' = - s + s ³

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

See Matrix

[y₂, 0, y₁, y₁, 0, y₂]

p = - s ² + s ⁴ p = - s ² + s ³

« NOT SYNC'D »

Nullspace of {Ω&Deltaⁱ} :
[0, x₁, x₂, -4 x₁ + 2 x₂]
For A+2Δ : [-3 y₃ - 4 y₁, y₄, -3 y₄ - y₃ - 3 y₂, y₃, y₂, y₁]
For A-2Δ : [y₄, y₃, y₂, -3 y₄ - 4 y₁, 9 y₄ - y₃ - 3 y₂ + 12 y₁, y₁]

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

rank of M is 6 , rank of N is 2

M N

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

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

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

τ= 18 , r'= 1/2

Ranges

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

ker N, [-μ₁ - μ₃, μ₁, -μ₄ - μ₂, μ₃, μ₄, μ₂]
Range of N
[y₁, y₁, y₂, y₁, y₂, y₂]

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

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

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

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

` ` See level-2 partition graph.

Right Group

Coloring {3}

Rank 2

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

π₂ [0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0]

u₂ [0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0] (dim 1)

wpp [3, 3, 3, 3, 3, 3]

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

4 . Coloring, {4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `2` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -12` (` - 1 + τ ` )` , 4` (` 3 + τ ² ` )` , 2` (` 9 - 2τ + τ ² ` )` , -4` (` - 3 + τ ` )` , 4` (` 3 - 2τ + τ ² ` )``]`

For τ=1/2, [5, 12, 26, 33, 20, 18] . FixedPtCheck, [5, 12, 26, 33, 20, 18]

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

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₁, -y₂, y₂, y₁, y₃, -y₃]

p = s + 2s ³ + 4s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 4	4 vs 6	4 vs 6	2 vs 5	3 vs 6

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

See Matrix

[0, y₁, -y₁ + 2 y₂, 2 y₂, y₂, y₂]

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

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

See Matrix

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

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

« NOT SYNC'D »

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

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

rank of M is 6 , rank of N is 3

M N

$ [ [0, 4, 1, 0, 2, 3] , [4, 0, 0, 6, 6, 4] , [1, 0, 0, 9, 4, 6] , [0, 6, 9, 0, 8, 7] , [2, 6, 4, 8, 0, 0] , [3, 4, 6, 7, 0, 0] ] $ $ [ [0, 1, 1, 0, 1, 1] , [1, 0, 0, 1, 1, 1] , [1, 0, 0, 1, 1, 1] , [0, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 0, 0] , [1, 1, 1, 1, 0, 0] ] $

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

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

τ= 12 , r'= 2/3

Ranges

Action of R on ranges, [[5], [4], [4], [7], [6], [7], [6]]
Action of B on ranges, [[3], [7], [6], [2], [5], [1], [4]]
β({1, 2, 5}) = 1/10
β({1, 2, 6}) = 1/10
β({1, 3, 6}) = 1/20
β({2, 4, 5}) = 1/5
β({2, 4, 6}) = 1/10
β({3, 4, 5}) = 1/5
β({3, 4, 6}) = 1/4

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {4}

Rank 3

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

π₂ [4, 1, 0, 2, 3, 0, 6, 6, 4, 9, 4, 6, 8, 7, 0]

u₂ [1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)

wpp [2, 2, 2, 2, 2, 2]

π₃ [0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 4, 2, 0, 4, 5, 0, 0]

u₃ [0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

Right Group
Coloring	{4}
Rank	3
R,B	[2, 4, 4, 3, 6, 5], [3, 6, 5, 2, 1, 4]
π₂	[4, 1, 0, 2, 3, 0, 6, 6, 4, 9, 4, 6, 8, 7, 0]
u₂	[1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)
wpp	[2, 2, 2, 2, 2, 2]
π₃	[0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 4, 2, 0, 4, 5, 0, 0]
u₃	[0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

5 . Coloring, {5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-3, -12, -4, -13, -4, -4] . FixedPtCheck, [3, 12, 4, 13, 4, 4]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[y₁, -y₂, y₂, -y₁ - y₂, 0, y₂]

p = s ² - 4s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s ² - 2s ³

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 4	3 vs 5	3 vs 5	3 vs 4	2 vs 4

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

See Matrix

[y₁ - y₂ + y₃, y₁, 0, y₂, y₃, 0]

p = - s ³ + s ⁴

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

See Matrix

[0, 0, y₁, y₂, y₂, y₁]

p = s - s ³ p' = s - s ³

« NOT SYNC'D »

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

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

rank of M is 6 , rank of N is 3

M N

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

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

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

τ= 18 , r'= 1/2

Ranges

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

Sandwich

Coloring {5}

Rank 2

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

π₂ [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0]

u₂ [3, 2, 0, 3, 1, 1, 3, 0, 2, 2, 1, 3, 3, 1, 2] (dim 1)

wpp [3, 3, 3, 3, 3, 3]

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

6 . Coloring, {6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-3` (` - 1 + τ ` )` ³ ` (` 3 + τ ` )` , 6` (` 1 + τ ` )`` (` 3 + τ ² ` )` , -6` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 3` (` 1 + τ ` )`` (` 9 - τ - τ ² + τ ³ ` )` , 6` (` - 1 + τ ` )` ² ` (` 3 + τ ` )` , 6` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [7, 156, 52, 201, 28, 60] . FixedPtCheck, [7, 156, 52, 201, 28, 60]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	3 vs 4	4 vs 4	2 vs 3	4 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 1/4, 3/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [99/100, 0, 0, -9/100, 3/100, 3/100] , [0, 1/10, 3/10, 0, 0, 0] , [0, 3/10, 9/10, 0, 0, 0] , [-9/100, 0, 0, 19/100, 27/100, 27/100] , [3/100, 0, 0, 27/100, 91/100, -9/100] , [3/100, 0, 0, 27/100, -9/100, 91/100] ] $ =
$ [ [-1/16, -11/16, -1/6, 1] , [15/16, 5/16, 1/2, -5/3] , [-3/16, 7/16, -1/2, 1/3] , [-1/16, -11/16, -1/6, 1] , [1/16, 3/16, 5/6, -1] , [-3/16, 7/16, -1/2, 1/3] ] $ x $ [ [3/2, 1, 3, 3/2, 3, 2] , [9/4, 3/4, 9/4, 3/2, 15/4, 3/2] , [45/16, 15/16, 45/16, 9/8, 45/16, 3/2] , [135/64, 63/64, 189/64, 21/16, 207/64, 45/32] ] $

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

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

See Matrix

[0, y₂, 0, y₁, 0, -y₂ + y₁]

p = s ² - s ³

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

[y₄, 0, y₃, 0, y₂, y₁]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 1, 0] ] $ x $ [ [1, 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, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, -5/36, 1/36, 7/36] , [1/2, -5/36, 1/36, -11/36] , [0, 7/36, -5/36, 1/36] , [0, -5/36, 1/36, 7/36] , [0, 1/36, 7/36, -5/36] , [0, 7/36, -5/36, 1/36] ] $ x $ [ [2, 0, 4, 0, 4, 2] , [4, 0, 2, 0, 6, 0] , [6, 0, 4, 0, 2, 0] , [2, 0, 6, 0, 4, 0] ] $

» SYNC'D 1/4 , 0.2500000000

7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `3` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 6` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -6` (` 3 + τ ` )`` (` - 1 + τ ` )` ² , -3` (` - 1 + τ ` )`` (` - 9 - τ + τ ² + τ ³ ` )` , -6` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 6` (` 1 + τ ` )` ² ` (` - 3 + τ ` )``]`

For τ=1/2, [-39, -84, -28, -73, -156, -180] . FixedPtCheck, [39, 84, 28, 73, 156, 180]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	4 vs 4	4 vs 4	3 vs 3	3 vs 3

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 1/4, 3/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [19/20, 0, 0, -1/20, 3/20, 3/20] , [0, 1/10, 3/10, 0, 0, 0] , [0, 3/10, 9/10, 0, 0, 0] , [-1/20, 0, 0, 19/20, 3/20, 3/20] , [3/20, 0, 0, 3/20, 11/20, -9/20] , [3/20, 0, 0, 3/20, -9/20, 11/20] ] $ =
$ [ [1/4, 3/2, -1/3, -4/3] , [9/4, 15/2, -1/3, -28/3] , [-3/4, -5/2, 2/3, 8/3] , [1/4, 3/2, -1/3, -4/3] , [-3/4, -11/2, -1/3, 20/3] , [-3/4, -5/2, 2/3, 8/3] ] $ x $ [ [3/2, 1, 3, 9/2, 1, 1] , [3/4, 3/2, 9/2, 15/4, 1, 1/2] , [3/4, 9/8, 27/8, 39/8, 5/4, 5/8] , [15/16, 45/32, 135/32, 123/32, 1, 19/32] ] $

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

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

[0, y₃, 0, 0, y₂, y₁]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] ] $ x $ [ [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, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [1/4, -1/12, -1/12] , [0, 1/6, -1/12] , [0, -1/12, 1/6] , [1/4, -1/12, -1/12] , [0, 1/6, -1/12] , [0, -1/12, 1/6] ] $ x $ [ [0, 4, 0, 0, 4, 4] , [0, 0, 0, 0, 4, 8] , [0, 0, 0, 0, 8, 4] ] $

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

[y₁, 0, y₂, y₃, 0, 0]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ] $ = $ [ [0, 1/6, -1/12] , [0, -1/12, 1/6] , [0, -1/12, 1/6] , [0, 1/6, -1/12] , [1/2, -1/12, -1/3] , [0, -1/12, 1/6] ] $ x $ [ [2, 0, 4, 6, 0, 0] , [0, 0, 8, 4, 0, 0] , [0, 0, 4, 8, 0, 0] ] $

» SYNC'D 1/4 , 0.2500000000

8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `2` (` 3 + τ ` )`` (` - 1 + τ ` )` , 12` (` - 1 + τ ` )` , 4` (` - 3 + τ ² ` )` , 2` (` - 9 + 4τ + τ ² ` )` , -4` (` 3 + τ ` )` , 4` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-7, -12, -22, -27, -28, -30] . FixedPtCheck, [7, 12, 22, 27, 28, 30]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	6 vs 6	6 vs 6	3 vs 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [-77/3096, 4879/3096, 1241/1161, -1969/1161, -2048/1161, 1072/1161] , [-247/1548, -400/387, -502/1161, -64/1161, 2944/1161, -896/1161] , [-7/1548, 20/387, 2618/1161, 2480/1161, -2624/1161, -2432/1161] , [-677/3096, 679/3096, -1111/1161, -505/1161, -1280/1161, 2992/1161] , [631/3096, 1579/3096, 425/1161, 1835/1161, 1504/1161, -4496/1161] , [2479/3096, -1997/3096, -1495/1161, -2509/1161, 1120/1161, 2800/1161] ] $ x $ [ [3/2, 5/2, 3/2, 7/2, 2, 1] , [3/2, 3, 2, 3, 11/8, 9/8] , [33/32, 21/8, 15/8, 115/32, 57/32, 35/32] , [171/128, 189/64, 107/64, 417/128, 215/128, 141/128] , [645/512, 711/256, 465/256, 1771/512, 783/512, 593/512] , [2349/2048, 2979/1024, 1853/1024, 6975/2048, 3383/2048, 2205/2048] ] $

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

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

See Matrix

[0, y₁, 5 y₁ - 6 y₂ + 5 y₃, y₂, y₃, 6 y₁ - 7 y₂ + 6 y₃]

p = - s ² + s ⁴ p' = - s ² + s ⁴

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

See Matrix

[5 y₂, 7 y₂ + 7 y₁ - 5 y₃ + 7 y₄, 5 y₁, 5 y₃, 5 y₄, 0]

p = - s - s ² + s ⁴ + s ⁵

» SYNC'D 5/256 , 0.01953125000

9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `3` (` 3 - τ + 5τ ² + τ ³ ` )`` (` 1 + τ ` )` , 6` (` 3 + τ ² ` )`` (` 1 + τ ` )` , -6` (` 3 + τ ² ` )`` (` - 1 + τ ` )` , -3` (` 9 + 7τ + 7τ ² + τ ³ ` )`` (` - 1 + τ ` )` , 6` (` 3 - τ + 5τ ² + τ ³ ` )` , -6` (` - 3 - τ - 5τ ² + τ ³ ` )``]`

For τ=1/2, [93, 156, 52, 115, 124, 148] . FixedPtCheck, [93, 156, 52, 115, 124, 148]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	4 vs 5	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 1/4, 3/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1/10, 3/10, 0, 0, 0] , [0, 3/10, 9/10, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [0, -5/8, 1/24, -4, 14/3] , [3/4, 5/12, 1/4, 8/3, -4] , [-1/4, 17/12, 1/4, 8/3, -4] , [0, -5/8, 1/24, -4, 14/3] , [0, -35/24, 7/8, -4/3, 2] , [0, 7/8, -35/24, 4, -10/3] ] $ x $ [ [1/2, 1, 3, 7/2, 2, 2] , [1/2, 1, 3, 3, 11/4, 7/4] , [11/16, 7/8, 21/8, 45/16, 43/16, 37/16] , [43/64, 7/8, 21/8, 195/64, 163/64, 143/64] , [163/256, 119/128, 357/128, 765/256, 647/256, 545/256] ] $

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

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

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

See Matrix

[0, 0, y₁, y₂, y₃, -y₁ + y₂ + y₃]

p = - s + s ² - s ³ + s ⁴

» SYNC'D 3/32 , 0.09375000000

10 . Coloring, {2, 6}

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

` See graph

` ` See pair graph

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

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

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₁, y₂, -y₂, y₁, -y₂, y₂]

p = s ³

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s ³

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 4	2 vs 4	2 vs 4	1 vs 3	2 vs 4

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

See Matrix

[0, y₁, 0, y₁, 0, y₁]

p = - s + s ³ p = - s + s ²

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

See Matrix

[-y₂ + y₁, 0, y₁, y₂, y₁, 0]

p = - s ² + s ³ p = - s ² + s ⁴

« NOT SYNC'D »

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

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

rank of M is 6 , rank of N is 3

M N

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

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

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

τ= 12 , r'= 2/3

Ranges

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

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {2, 6}

Rank 3

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

π₂ [0, 1, 0, 1, 0, 0, 2, 0, 2, 1, 2, 0, 1, 2, 0]

u₂ [1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1] (dim 1)

wpp [2, 2, 2, 2, 2, 2]

π₃ [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0]

u₃ [1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1]

Right Group
Coloring	{2, 6}
Rank	3
R,B	[2, 6, 4, 2, 6, 4], [3, 4, 5, 3, 1, 5]
π₂	[0, 1, 0, 1, 0, 0, 2, 0, 2, 1, 2, 0, 1, 2, 0]
u₂	[1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1] (dim 1)
wpp	[2, 2, 2, 2, 2, 2]
π₃	[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0]
u₃	[1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1]

11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-3, -4, -6, -7, -12, -10] . FixedPtCheck, [3, 4, 6, 7, 12, 10]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	4 vs 5	5 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [819/820, 0, 0, -9/820, 27/820, 3/820] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-9/820, 0, 0, 739/820, 243/820, 27/820] , [27/820, 0, 0, 243/820, 91/820, -81/820] , [3/820, 0, 0, 27/820, -81/820, 811/820] ] $ =
$ [ [-181/3172, 72/61, 98/793, -1904/2379, -288/793] , [-9/3172, -19/61, -258/793, -1048/793, 4864/2379] , [3/244, 1/61, 86/61, 24/61, -320/183] , [-725/3172, 8/61, -694/793, 2608/2379, -32/793] , [2695/3172, -31/61, -986/793, -1688/2379, 1344/793] , [3/244, 1/61, 86/61, 24/61, -320/183] ] $ x $ [ [3/2, 5/2, 3/2, 7/2, 1, 2] , [3/4, 3, 2, 13/4, 7/8, 17/8] , [21/32, 21/8, 11/8, 123/32, 33/32, 79/32] , [99/128, 195/64, 93/64, 453/128, 123/128, 285/128] , [369/512, 729/256, 375/256, 1803/512, 471/512, 1293/512] ] $

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

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

See Matrix

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

p = s ⁴ - s ⁵

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

[y₁, y₂, y₃, y₄, 0, y₅]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, 1/2, 1/36, -11/36, -5/36] , [0, 0, 1/36, -11/36, 13/36] , [0, 0, 13/36, 1/36, -11/36] , [0, 0, -11/36, 13/36, 1/36] , [1/2, -1/4, -11/36, -5/36, 5/18] , [0, 0, 13/36, 1/36, -11/36] ] $ x $ [ [2, 3, 1, 4, 0, 2] , [0, 4, 2, 3, 0, 3] , [0, 3, 0, 5, 0, 4] , [0, 5, 0, 4, 0, 3] , [0, 4, 0, 3, 0, 5] ] $

» SYNC'D 27/256 , 0.1054687500

12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `3` (` 3 + τ ` )`` (` 1 + τ ` )` ² ` (` - 1 + τ ` )` , -6` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 6` (` 3 + τ ² ` )`` (` - 1 + τ ` )` , -3` (`9 - 4τ + 6τ ² + 4τ ³ + τ ⁴ ` )` , 6` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -6` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-63, -156, -52, -145, -84, -60] . FixedPtCheck, [63, 156, 52, 145, 84, 60]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	4 vs 4	4 vs 4	4 vs 4	3 vs 3

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 1/4, 3/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [91/100, 0, 0, -9/100, 27/100, 3/100] , [0, 1/10, 3/10, 0, 0, 0] , [0, 3/10, 9/10, 0, 0, 0] , [-9/100, 0, 0, 91/100, 27/100, 3/100] , [27/100, 0, 0, 27/100, 19/100, -9/100] , [3/100, 0, 0, 3/100, -9/100, 99/100] ] $ =
$ [ [-3/4, -5/2, 2/3, 8/3] , [-3/4, -11/2, -1/3, 20/3] , [1/4, 3/2, -1/3, -4/3] , [-3/4, -5/2, 2/3, 8/3] , [9/4, 15/2, -1/3, -28/3] , [1/4, 3/2, -1/3, -4/3] ] $ x $ [ [1/2, 1, 3, 7/2, 1, 3] , [1/4, 1, 3, 19/4, 3/2, 3/2] , [3/8, 5/4, 15/4, 29/8, 9/8, 15/8] , [9/32, 1, 3, 145/32, 45/32, 57/32] ] $

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

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

[y₁, y₂, 0, y₃, y₄, 0]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 1, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0] ] $ = $ [ [0, 0, 1/6, -1/12] , [0, 0, -1/12, 1/6] , [1/4, -1/8, -1/12, 1/24] , [0, 0, 1/6, -1/12] , [0, 1/4, -1/12, -1/12] , [1/4, -1/8, -1/12, 1/24] ] $ x $ [ [2, 4, 0, 2, 4, 0] , [4, 4, 0, 4, 0, 0] , [0, 8, 0, 4, 0, 0] , [0, 4, 0, 8, 0, 0] ] $

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

[0, 0, y₃, y₂, 0, y₁]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, -1/12, 1/6] , [1/4, -1/12, -1/12] , [0, 1/6, -1/12] , [0, -1/12, 1/6] , [1/4, -1/12, -1/12] , [0, 1/6, -1/12] ] $ x $ [ [0, 0, 4, 4, 0, 4] , [0, 0, 4, 8, 0, 0] , [0, 0, 8, 4, 0, 0] ] $

» SYNC'D 1/4 , 0.2500000000

13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `3` (` - 1 + τ ` )` ² ` (` 1 + τ ` )`` (` - 3 + τ ` )` , -6` (` 3 + τ ² ` )`` (` 1 + τ ` )` , 6` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , -3` (`9 + 4τ + 6τ ² - 4τ ³ + τ ⁴ ` )` , -6` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 6` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-15, -156, -52, -193, -60, -84] . FixedPtCheck, [15, 156, 52, 193, 60, 84]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	3 vs 4	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 1/4, 3/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1/10, 3/10, 0, 0, 0] , [0, 3/10, 9/10, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [0, -3/8, 9/8, 4, -14/3] , [3/4, -3/4, -5/4, -8/3, 4] , [-1/4, 1/4, -5/4, -8/3, 4] , [0, -3/8, 9/8, 4, -14/3] , [0, 1/8, 5/8, -4, 10/3] , [0, 9/8, -3/8, 4/3, -2] ] $ x $ [ [3/2, 1, 3, 5/2, 2, 2] , [3/2, 1, 3, 3, 9/4, 5/4] , [27/16, 9/8, 27/8, 45/16, 27/16, 21/16] , [81/64, 9/8, 27/8, 201/64, 117/64, 81/64] , [351/256, 141/128, 423/128, 801/256, 459/256, 333/256] ] $

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

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

See Matrix

[0, y₁ + y₂ - y₃, 0, y₁, y₂, y₃]

p = - s ³ + s ⁴

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

See Matrix

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

p = s ⁴ - s ⁵

» SYNC'D 3/32 , 0.09375000000

14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` 1 + τ ` )` ² ` (` - 3 + τ ` )`` (` - 1 + τ ` )` , -36` (` 3 + 2τ + 3τ ² ` )`` (` - 1 + τ ` )` , 12` (` 3 - τ + τ ² + τ ³ ` )`` (` 1 + τ ` )` , 6` (` 3 + τ ² ` )` ² , 12` (` 1 + τ ` )`` (` - 3 + τ ` )`` (` - 1 + τ ` )` , 12` (` 3 + 2τ + τ ² ` )`` (` - 1 + τ ` )` ² `]`

For τ=1/2, [45, 76, 138, 169, 60, 34] . FixedPtCheck, [45, 76, 138, 169, 60, 34]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	6 vs 6	6 vs 6	4 vs 5	5 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [-1979/936, -215/936, 1189/117, 2375/117, 1984/351, -11824/351] , [-593/468, 160/117, 110/117, 880/117, -928/351, -2048/351] , [91/36, -20/9, -34/9, -128/9, 224/27, 256/27] , [589/936, -671/936, -203/117, -337/117, -704/351, 2384/351] , [-751/936, 3101/936, -175/117, 355/117, -4480/351, 3088/351] , [145/936, -1187/936, 217/117, -253/117, 2560/351, -2032/351] ] $ x $ [ [1/2, 5/2, 3/2, 5/2, 2, 3] , [1/2, 2, 1, 13/4, 15/8, 27/8] , [15/32, 41/16, 19/16, 105/32, 51/32, 93/32] , [51/128, 165/64, 75/64, 399/128, 207/128, 399/128] , [207/512, 39/16, 69/64, 1677/512, 849/512, 1611/512] , [849/2048, 2619/1024, 1149/1024, 6633/2048, 3267/2048, 6291/2048] ] $

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

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

See Matrix

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

p = s ⁴ - s ⁵

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

[0, y₅, y₁, y₂, y₃, y₄]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [1, -2, 13/36, -47/36, 73/36] , [0, 0, -11/36, 1/36, 13/36] , [0, 1, 1/36, 13/36, -47/36] , [0, 0, 1/36, 13/36, -11/36] , [0, 0, -11/36, 1/36, 13/36] , [0, 0, 13/36, -11/36, 1/36] ] $ x $ [ [0, 3, 1, 2, 2, 4] , [0, 2, 0, 4, 1, 5] , [0, 4, 0, 5, 0, 3] , [0, 5, 0, 3, 0, 4] , [0, 3, 0, 4, 0, 5] ] $

» SYNC'D 63/512 , 0.1230468750

15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` - 1 + τ ` )` ² ` (` 3 + τ ² ` )` , 12` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 12` (`3 + 2τ + 4τ ² - 2τ ³ + τ ⁴ ` )` , 6` (` 1 + τ ` )`` (` 9 - τ - τ ² + τ ³ ` )` , -12` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , -12` (` - 1 + τ ` )`` (` 3 - 2τ + τ ² ` )`` (` 1 + τ ` )``]`

For τ=1/2, [13, 60, 154, 201, 52, 54] . FixedPtCheck, [13, 60, 154, 201, 52, 54]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	2 vs 4	5 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [99/100, 0, 0, -9/100, 3/100, 3/100] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-9/100, 0, 0, 19/100, 27/100, 27/100] , [3/100, 0, 0, 27/100, 91/100, -9/100] , [3/100, 0, 0, 27/100, -9/100, 91/100] ] $ =
$ [ [7/148, 6/37, -142/111, -96/37, 416/111] , [-183/1036, 340/259, 138/259, -1520/777, 96/259] , [-303/1036, -6/259, 738/259, 880/777, -928/259] , [801/1036, -138/259, -638/259, -160/259, 2272/777] , [81/1036, -142/259, -1474/777, 496/259, 416/777] , [-303/1036, -6/259, 738/259, 880/777, -928/259] ] $ x $ [ [3/2, 5/2, 3/2, 3/2, 3, 2] , [9/4, 3/2, 3/2, 3/2, 21/8, 21/8] , [63/32, 27/16, 33/16, 45/32, 99/32, 57/32] , [297/128, 99/64, 117/64, 177/128, 369/128, 261/128] , [1107/512, 207/128, 267/128, 693/512, 1485/512, 963/512] ] $

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

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

See Matrix

[0, y₁, y₂, 3 y₁ + y₂, 0, 2 y₁]

p = - s ² + s ³ p = - s ² + s ⁴

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

[y₁, y₄, y₂, 0, y₃, y₅]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 1, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, 0, 1/36, -11/36, 13/36] , [0, 1/3, 1/36, -11/36, 1/36] , [0, 0, 13/36, 1/36, -11/36] , [1/3, -2/9, -11/36, 1/36, 1/4] , [0, 0, -11/36, 13/36, 1/36] , [0, 0, 13/36, 1/36, -11/36] ] $ x $ [ [2, 3, 1, 0, 4, 2] , [4, 0, 2, 0, 3, 3] , [3, 0, 4, 0, 5, 0] , [5, 0, 3, 0, 4, 0] , [4, 0, 5, 0, 3, 0] ] $

» SYNC'D 3/64 , 0.04687500000

16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-3` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` ² , 6` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -6` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 3` (` 1 + τ ` )`` (` - 9 - τ + τ ² + τ ³ ` )` , -6` (` 3 + τ ` )`` (` - 1 + τ ` )` ² , 6` (` 3 + τ ² ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-21, -180, -60, -219, -28, -52] . FixedPtCheck, [21, 180, 60, 219, 28, 52]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	4 vs 4	4 vs 4	3 vs 3	3 vs 3

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 1/4, 3/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [11/20, 0, 0, -9/20, 3/20, 3/20] , [0, 1/10, 3/10, 0, 0, 0] , [0, 3/10, 9/10, 0, 0, 0] , [-9/20, 0, 0, 11/20, 3/20, 3/20] , [3/20, 0, 0, 3/20, 19/20, -1/20] , [3/20, 0, 0, 3/20, -1/20, 19/20] ] $ =
$ [ [1/4, -7/6, -1/3, 4/3] , [1/4, 11/6, 2/3, -8/3] , [-1/12, 1/18, -1/3, 4/9] , [1/4, -7/6, -1/3, 4/3] , [-1/12, 7/18, 2/3, -8/9] , [-1/12, 1/18, -1/3, 4/9] ] $ x $ [ [1/2, 1, 3, 3/2, 3, 3] , [3/4, 1/2, 3/2, 7/4, 9/2, 3] , [9/8, 5/8, 15/8, 5/4, 27/8, 15/4] , [27/32, 19/32, 57/32, 25/16, 135/32, 3] ] $

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

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

[y₁, y₃, 0, y₂, 0, 0]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 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, 1/6, -1/12] , [0, -1/12, 1/6] , [0, -1/12, 1/6] , [0, 1/6, -1/12] , [1/2, -1/12, -1/3] , [0, -1/12, 1/6] ] $ x $ [ [2, 4, 0, 6, 0, 0] , [0, 8, 0, 4, 0, 0] , [0, 4, 0, 8, 0, 0] ] $

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

[0, 0, y₃, 0, y₂, y₁]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] ] $ x $ [ [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, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [1/4, -1/12, -1/12] , [0, -1/12, 1/6] , [0, 1/6, -1/12] , [1/4, -1/12, -1/12] , [0, -1/12, 1/6] , [0, 1/6, -1/12] ] $ x $ [ [0, 0, 4, 0, 4, 4] , [0, 0, 0, 0, 8, 4] , [0, 0, 0, 0, 4, 8] ] $

» SYNC'D 1/4 , 0.2500000000

17 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` - 1 + τ ` )`` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -12` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 12` (` - 1 + τ ` )`` (` - 3 + τ ² ` )`` (` 1 + τ ` )` , 6` (` - 1 + τ ` )`` (` - 9 - τ + τ ² + τ ³ ` )` , -12` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -12` (` - 3 - τ - τ ² + τ ³ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [45, 52, 66, 73, 180, 174] . FixedPtCheck, [45, 52, 66, 73, 180, 174]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[-4 y₃ + 5 y₂ + 4 y₁, 3 y₃, -3 y₃, 3 y₂, 3 y₁, -8 y₂ - 7 y₁ + 4 y₃]

p = s - 2s ³ - 4s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 4	5 vs 5	5 vs 5	3 vs 4	4 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [19/20, 0, 0, -1/20, 3/20, 3/20] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-1/20, 0, 0, 19/20, 3/20, 3/20] , [3/20, 0, 0, 3/20, 11/20, -9/20] , [3/20, 0, 0, 3/20, -9/20, 11/20] ] $ =
$ [ [-415/1964, 480/491, 158/491, -1160/1473, -320/1473] , [69/3928, -195/3928, -219/491, -785/1473, 1612/1473] , [57/3928, -1015/3928, 673/491, 547/1473, -2084/1473] , [-211/1964, 37/491, -166/1473, 688/1473, -352/1473] , [2829/3928, -139/3928, -1123/491, -761/1473, 3244/1473] , [57/3928, -1015/3928, 673/491, 547/1473, -2084/1473] ] $ x $ [ [3/2, 5/2, 3/2, 9/2, 1, 1] , [3/4, 15/4, 9/4, 15/4, 5/8, 7/8] , [15/32, 3, 3/2, 165/32, 25/32, 35/32] , [75/128, 255/64, 105/64, 537/128, 83/128, 121/128] , [249/512, 843/256, 381/256, 2523/512, 331/512, 593/512] ] $

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

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

See Matrix

[0, y₁, 3 y₁, 0, y₃, y₂]

p = s ² - s ⁴

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

[y₂, y₃, y₄, y₁, 0, 0]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ] $ = $ [ [0, 1/2, -1/12, -1/3] , [0, 0, 1/6, -1/12] , [0, 0, 1/6, -1/12] , [0, 0, -1/12, 1/6] , [1/2, -1/4, -1/3, 1/6] , [0, 0, 1/6, -1/12] ] $ x $ [ [2, 3, 1, 6, 0, 0] , [0, 6, 2, 4, 0, 0] , [0, 4, 0, 8, 0, 0] , [0, 8, 0, 4, 0, 0] ] $

» SYNC'D 1/8 , 0.1250000000

18 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [9, 12, 4, 7, 12, 12] . FixedPtCheck, [9, 12, 4, 7, 12, 12]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[y₁, y₂, -y₂, -y₁ - y₂, y₂, 0]

p = s ² - 4s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s ² + 2s ³

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 4	2 vs 4	2 vs 4	2 vs 4	2 vs 3

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

See Matrix

[y₁, y₂, 0, 0, y₂, y₁]

p = - s + s ³ p' = - s + s ³

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

See Matrix

[0, 0, y₁ - y₂, y₁, 0, y₂]

p = - s ² + s ³

« NOT SYNC'D »

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

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

rank of M is 6 , rank of N is 3

M N

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

Check is ΩΔN zero? true, πΔ= [1, 2, -2, -3, 2, 0]

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

τ= 18 , r'= 1/2

Ranges

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 3, 5}

Rank 2

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

π₂ [0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0]

u₂ [2, 3, 0, 1, 3, 1, 2, 3, 1, 3, 2, 0, 1, 3, 2] (dim 1)

wpp [3, 3, 3, 3, 3, 3]

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

19 . Coloring, {2, 3, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-3` (` 1 + τ ` )`` (` - 1 + τ ` )` ² ` (` 3 + τ ` )` , -6` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 6` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 3` (` - 9 - 2τ - 8τ ² + 2τ ³ + τ ⁴ ` )` , 6` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )` , 6` (` 1 + τ ` )` ² ` (` - 3 + τ ` )``]`

For τ=1/2, [-21, -156, -52, -187, -84, -180] . FixedPtCheck, [21, 156, 52, 187, 84, 180]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	4 vs 5	4 vs 4	3 vs 4

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

[0, y₁, 0, y₃, y₂, y₄]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, -5/36, 1/36, 7/36] , [0, 7/36, -5/36, 1/36] , [1/2, -5/36, 1/36, -11/36] , [0, -5/36, 1/36, 7/36] , [0, 7/36, -5/36, 1/36] , [0, 1/36, 7/36, -5/36] ] $ x $ [ [0, 4, 0, 2, 2, 4] , [0, 2, 0, 4, 0, 6] , [0, 4, 0, 6, 0, 2] , [0, 6, 0, 2, 0, 4] ] $

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

See Matrix

[y₂, 0, y₃, y₁ - y₂ + y₃, y₁, 0]

p = - s ³ + s ⁴

» SYNC'D 1/4 , 0.2500000000

20 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` 3 + τ ` )`` (` 1 + τ ` )` ² , 36` (` 3 + 2τ + 3τ ² ` )` , 12` (` 3 + 2τ + τ ² ` )`` (` 1 + τ ` )` , 6` (` 9 + 7τ + 7τ ² + τ ³ ` )` , 12` (` 3 + τ ` )`` (` 1 + τ ` )` , 12` (` 1 + τ ` )`` (` 3 + τ ² ` )``]`

For τ=1/2, [63, 76, 102, 115, 84, 78] . FixedPtCheck, [63, 76, 102, 115, 84, 78]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	6 vs 6	6 vs 6	4 vs 6	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [865/144, 463/144, -49/12, -377/6, -688/9, 1208/9] , [-7/72, -103/72, 13/6, -7/3, 128/9, -112/9] , [-199/72, -295/72, 13/6, 137/3, 512/9, -880/9] , [193/144, 367/144, -49/12, -89/6, -304/9, 440/9] , [229/144, 667/144, -37/12, -173/6, -304/9, 536/9] , [-467/144, -653/144, 83/12, 235/6, 464/9, -808/9] ] $ x $ [ [1/2, 5/2, 3/2, 7/2, 2, 2] , [1/2, 11/4, 5/4, 15/4, 13/8, 17/8] , [13/32, 47/16, 21/16, 127/32, 47/32, 61/32] , [47/128, 197/64, 83/64, 507/128, 187/128, 235/128] , [187/512, 49/16, 81/64, 2053/512, 733/512, 955/512] , [733/2048, 3173/1024, 1307/1024, 8217/2048, 2899/2048, 3767/2048] ] $

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

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

See Matrix

[y₁, 6 y₁ - 7 y₂ - y₃ + 6 y₄, y₂, 5 y₁ - 6 y₂ + 5 y₄, y₃, y₄]

p' = 1 - s ⁴ p' = s - s ⁵

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

See Matrix

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

p = - s ⁴ + s ⁵

» SYNC'D 3/128 , 0.02343750000

21 . Coloring, {2, 4, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [1, 4, 10, 13, 4, 6] . FixedPtCheck, [1, 4, 10, 13, 4, 6]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	4 vs 4	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [91/100, 0, 0, -9/100, 3/100, 27/100] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-9/100, 0, 0, 91/100, 3/100, 27/100] , [3/100, 0, 0, 3/100, 99/100, -9/100] , [27/100, 0, 0, 27/100, -9/100, 19/100] ] $ =
$ [ [-493/308, -661/231, -722/693, 128/77, 2720/693] , [843/308, 547/77, 1090/231, -944/231, -800/77] , [-621/308, -375/77, -226/77, 976/231, 1312/231] , [1107/308, 481/77, 914/231, -1472/231, -1696/231] , [-123/44, -175/33, -422/99, 48/11, 800/99] , [-621/308, -375/77, -226/77, 976/231, 1312/231] ] $ x $ [ [3/2, 5/2, 3/2, 5/2, 3, 1] , [9/4, 9/4, 7/4, 5/2, 15/8, 11/8] , [45/32, 39/16, 37/16, 79/32, 75/32, 33/32] , [225/128, 141/64, 107/64, 341/128, 321/128, 153/128] , [963/512, 39/16, 127/64, 1213/512, 1101/512, 603/512] ] $

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

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

[0, y₂, y₁, y₄, 0, y₃]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [1, -4, -29/24, 103/24] , [0, 1, 7/24, -29/24] , [0, 0, -5/24, 7/24] , [0, 0, 7/24, -5/24] , [0, 1, 7/24, -29/24] , [0, 0, -5/24, 7/24] ] $ x $ [ [0, 1, 3, 4, 0, 4] , [0, 0, 4, 7, 0, 1] , [0, 0, 7, 5, 0, 0] , [0, 0, 5, 7, 0, 0] ] $

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

See Matrix

[7 y₁ - 5 y₂ + 7 y₄ - 5 y₃, 5 y₁, 5 y₂, 5 y₄, 5 y₃, 0]

p = - s - s ² + s ⁴ + s ⁵

» SYNC'D 63/512 , 0.1230468750

22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `3` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 6` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -6` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -3` (` 9 - τ - τ ² + τ ³ ` )`` (` 1 + τ ` )` , 6` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 6` (` - 3 - τ - 5τ ² + τ ³ ` )``]`

For τ=1/2, [-39, -180, -60, -201, -52, -148] . FixedPtCheck, [39, 180, 60, 201, 52, 148]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	4 vs 4	3 vs 4	4 vs 4	3 vs 4

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

[y₄, y₂, 0, y₃, 0, y₁]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 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, 1] ] $ = $ [ [0, 7/36, -5/36, 1/36] , [0, 1/36, 7/36, -5/36] , [0, -5/36, 1/36, 7/36] , [0, 7/36, -5/36, 1/36] , [1/2, -5/36, 1/36, -11/36] , [0, -5/36, 1/36, 7/36] ] $ x $ [ [2, 4, 0, 4, 0, 2] , [0, 6, 0, 2, 0, 4] , [0, 2, 0, 4, 0, 6] , [0, 4, 0, 6, 0, 2] ] $

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

See Matrix

[0, 0, y₁ - y₂ + y₃, y₁, y₃, y₂]

p = s ³ - s ⁴

» SYNC'D 1/4 , 0.2500000000

23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` 1 + τ ` )` ² ` (` 3 + τ ² ` )` , 12` (` 3 - τ + 5τ ² + τ ³ ` )` , 12` (` 1 + τ ` )`` (` 3 - τ + τ ² + τ ³ ` )` , 6` (`9 - 4τ + 6τ ² + 4τ ³ + τ ⁴ ` )` , 12` (` 1 + τ ` )`` (` 3 + τ ² ` )` , -12` (` 1 + τ ² ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [117, 124, 138, 145, 156, 70] . FixedPtCheck, [117, 124, 138, 145, 156, 70]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	5 vs 5	4 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [91/100, 0, 0, -9/100, 27/100, 3/100] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-9/100, 0, 0, 91/100, 27/100, 3/100] , [27/100, 0, 0, 27/100, 19/100, -9/100] , [3/100, 0, 0, 3/100, -9/100, 99/100] ] $ =
$ [ [-4127/5532, 7381/1383, 12358/1383, 3520/1383, -22112/1383] , [-119/5532, -509/1383, 670/1383, 3088/1383, -3104/1383] , [961/5532, -515/1383, -4202/1383, -1136/1383, 5728/1383] , [-2495/5532, 1717/1383, 4750/1383, -896/1383, -4832/1383] , [6769/5532, -4727/1383, -5570/1383, -1232/1383, 9952/1383] , [961/5532, -515/1383, -4202/1383, -1136/1383, 5728/1383] ] $ x $ [ [1/2, 5/2, 3/2, 7/2, 1, 3] , [1/4, 11/4, 5/4, 4, 9/8, 21/8] , [9/32, 49/16, 19/16, 115/32, 31/32, 93/32] , [31/128, 177/64, 71/64, 491/128, 131/128, 387/128] , [131/512, 47/16, 73/64, 1941/512, 529/512, 1455/512] ] $

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

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

[y₅, y₄, y₃, y₂, y₁, 0]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 1, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0] ] $ = $ [ [-17/132, -29/132, 67/132, -41/132, 31/132] , [31/132, -17/132, -29/132, 67/132, -41/132] , [67/132, -41/132, 31/132, -17/132, -29/132] , [-41/132, 31/132, -17/132, -29/132, 67/132] , [-29/132, 67/132, -41/132, 31/132, -17/132] , [67/132, -41/132, 31/132, -17/132, -29/132] ] $ x $ [ [2, 1, 3, 2, 4, 0] , [4, 2, 2, 1, 3, 0] , [3, 4, 1, 2, 2, 0] , [2, 3, 2, 4, 1, 0] , [1, 2, 4, 3, 2, 0] ] $

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

[0, y₄, y₁, y₂, 0, y₃]

See Matrices

B = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [1, -11/36, 1/36, -23/36] , [0, -11/36, 1/36, 13/36] , [0, 13/36, -11/36, 1/36] , [0, 1/36, 13/36, -11/36] , [0, -11/36, 1/36, 13/36] , [0, 13/36, -11/36, 1/36] ] $ x $ [ [0, 3, 1, 4, 0, 4] , [0, 4, 0, 5, 0, 3] , [0, 5, 0, 3, 0, 4] , [0, 3, 0, 4, 0, 5] ] $

» SYNC'D 75/512 , 0.1464843750

24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` - 3 - τ - 5τ ² + τ ³ ` )`` (` - 1 + τ ` )` , -36` (` 3 + 2τ + 3τ ² ` )`` (` - 1 + τ ` )` , 12` (`3 + 2τ + 4τ ² - 2τ ³ + τ ⁴ ` )` , 6` (`9 + 4τ + 6τ ² - 4τ ³ + τ ⁴ ` )` , -12` (` - 3 - τ - 5τ ² + τ ³ ` )` , 12` (` 1 + τ ² ` )`` (` 3 + τ ² ` )``]`

For τ=1/2, [37, 76, 154, 193, 148, 130] . FixedPtCheck, [37, 76, 154, 193, 148, 130]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	6 vs 6	6 vs 6	4 vs 5	5 vs 6

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [379/528, -259/144, 373/1188, -1175/594, 2128/297, -1288/297] , [-305/264, 203/72, -665/594, -89/297, -1600/297, 1552/297] , [-113/264, 11/72, -1337/594, -329/297, -1216/297, 2320/297] , [1051/528, -163/144, 1717/1188, -695/594, 1744/297, -2056/297] , [-833/528, 137/144, -2975/1188, 1957/594, -1328/297, 1304/297] , [167/528, -191/144, 4217/1188, 509/594, 464/297, -1448/297] ] $ x $ [ [3/2, 5/2, 3/2, 5/2, 2, 2] , [3/2, 9/4, 7/4, 9/4, 15/8, 19/8] , [45/32, 33/16, 27/16, 79/32, 71/32, 69/32] , [213/128, 141/64, 107/64, 297/128, 261/128, 269/128] , [783/512, 69/32, 117/64, 1193/512, 1021/512, 1107/512] , [3063/2048, 2181/1024, 1771/1024, 5019/2048, 4257/2048, 4333/2048] ] $

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

Omega Rank for R : cycles: {{3, 4, 5, 6}}, net cycles: 0 . order: 4

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

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

See Matrix

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

p' = - 1 + s - s ² + s ³ - s ⁴ + s ⁵

» SYNC'D 9/256 , 0.03515625000

25 . Coloring, {3, 5, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-3, -12, -4, -13, -4, -4] . FixedPtCheck, [3, 12, 4, 13, 4, 4]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₂ - y₁, -y₂, y₂, y₁, 0, y₂]

p = s ² - 4s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s ² - 2s ³

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 4	3 vs 5	3 vs 5	3 vs 4	2 vs 4

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

See Matrix

[y₂, y₂ + y₁ - y₃, 0, y₁, y₃, 0]

p = - s ³ + s ⁴

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

See Matrix

[0, 0, y₂, y₁, y₁, y₂]

p = - s + s ³ p' = - s + s ³

« NOT SYNC'D »

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

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

rank of M is 6 , rank of N is 2

M N

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

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

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

τ= 18 , r'= 1/2

Ranges

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

ker N, [-μ₃ - μ₄, μ₂, μ₃, μ₄, -μ₂ - μ₁, μ₁]
Range of N
[y₁, y₂, y₁, y₁, y₂, y₂]

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

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

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

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

` ` See level-2 partition graph.

Right Group

Coloring {3, 5, 6}

Rank 2

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

π₂ [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0]

u₂ [1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0] (dim 1)

wpp [3, 3, 3, 3, 3, 3]

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

26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` 1 + τ ` )`` (` 3 + τ ² ` )`` (` - 1 + τ ` )` , 12` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , 12` (` 1 + τ ` )`` (` - 3 - τ - τ ² + τ ³ ` )` , 6` (` 1 + τ ` )`` (` - 9 - τ + τ ² + τ ³ ` )` , 12` (` 3 + τ ² ` )`` (` - 1 + τ ` )` , -12` (` 3 + 2τ + τ ² ` )`` (` - 1 + τ ` )` ² `]`

For τ=1/2, [-39, -84, -174, -219, -52, -34] . FixedPtCheck, [39, 84, 174, 219, 52, 34]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	4 vs 4	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [11/20, 0, 0, -9/20, 3/20, 3/20] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-9/20, 0, 0, 11/20, 3/20, 3/20] , [3/20, 0, 0, 3/20, 19/20, -1/20] , [3/20, 0, 0, 3/20, -1/20, 19/20] ] $ =
$ [ [1343/2596, -533/649, -2330/649, 1376/1947, 6368/1947] , [11/118, 139/236, -64/59, 46/177, 40/177] , [-173/1298, 485/2596, 320/649, -2590/1947, 1688/1947] , [983/2596, -163/649, 734/649, 752/1947, -3040/1947] , [-199/1298, -455/2596, 128/649, 2858/1947, -2440/1947] , [-173/1298, 485/2596, 320/649, -2590/1947, 1688/1947] ] $ x $ [ [1/2, 5/2, 3/2, 3/2, 3, 3] , [3/4, 5/4, 3/4, 7/4, 27/8, 33/8] , [27/32, 3/2, 1, 49/32, 117/32, 111/32] , [117/128, 87/64, 65/64, 191/128, 429/128, 495/128] , [429/512, 345/256, 271/256, 799/512, 1875/512, 1809/512] ] $

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, 0, 0]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ] $ = $ [ [0, 1/2, -1/12, -1/3] , [0, 0, 1/6, -1/12] , [0, 0, 1/6, -1/12] , [0, 0, -1/12, 1/6] , [1/2, -1/4, -1/3, 1/6] , [0, 0, 1/6, -1/12] ] $ x $ [ [2, 1, 3, 6, 0, 0] , [0, 2, 6, 4, 0, 0] , [0, 0, 4, 8, 0, 0] , [0, 0, 8, 4, 0, 0] ] $

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

See Matrix

[0, 3 y₂, y₂, 0, y₁, y₃]

p = s ² - s ⁴

» SYNC'D 1/8 , 0.1250000000

27 . Coloring, {2, 3, 4, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-135, -124, -102, -91, -180, -138] . FixedPtCheck, [135, 124, 102, 91, 180, 138]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	5 vs 5	2 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] ] $ x $ [ [19/100, 0, 0, -9/100, 27/100, 27/100] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-9/100, 0, 0, 99/100, 3/100, 3/100] , [27/100, 0, 0, 3/100, 91/100, -9/100] , [27/100, 0, 0, 3/100, -9/100, 91/100] ] $ =
$ [ [27/4, -6, -130/3, 32/3, 32] , [3/4, 0, -6, -16/3, 32/3] , [-5/4, 14/3, 2, -16, 32/3] , [1/12, -2, 2, 32/3, -32/3] , [-5/4, -10/3, 62/3, 16, -32] , [-5/4, 14/3, 2, -16, 32/3] ] $ x $ [ [1/2, 5/2, 3/2, 9/2, 1, 2] , [1/4, 7/2, 3/2, 9/2, 7/8, 11/8] , [7/32, 55/16, 21/16, 153/32, 23/32, 49/32] , [23/128, 233/64, 87/64, 603/128, 91/128, 179/128] , [91/512, 229/64, 21/16, 2457/512, 353/512, 739/512] ] $

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

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

[y₃, y₁, y₂, 0, y₄, y₅]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 1, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, 1/16, -1/48, 11/48, -3/16] , [0, -3/16, 1/16, -1/48, 11/48] , [0, 11/48, -3/16, 1/16, -1/48] , [1/3, -3/16, 1/16, -1/48, -5/48] , [0, -1/48, 11/48, -3/16, 1/16] , [0, 11/48, -3/16, 1/16, -1/48] ] $ x $ [ [2, 1, 3, 0, 4, 2] , [4, 2, 0, 0, 5, 1] , [5, 4, 0, 0, 1, 2] , [1, 5, 0, 0, 2, 4] , [2, 1, 0, 0, 4, 5] ] $

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

See Matrix

[0, -3 y₁ + y₂, y₁, y₂, 0, 2 y₁]

p = - s ² + s ³ p = - s ² + s ⁴

» SYNC'D 15/64 , 0.2343750000

28 . Coloring, {2, 3, 4, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 36` (` - 1 + τ ` )`` (` 3 + 2τ + 3τ ² ` )` , 12` (` 1 + τ ² ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 6` (` - 9 - 2τ - 8τ ² + 2τ ³ + τ ⁴ ` )` , -12` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 12` (` 1 + τ ` )`` (` - 3 - τ - τ ² + τ ³ ` )``]`

For τ=1/2, [-39, -76, -150, -187, -156, -174] . FixedPtCheck, [39, 76, 150, 187, 156, 174]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	6 vs 6	6 vs 6	5 vs 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [3/4, 0, 0, 0, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [-51371/102104, 108877/102104, 33367/38289, -53453/38289, -159104/38289, 160816/38289] , [10375/51052, -17099/25526, -794/38289, -5060/38289, 167008/38289, -140096/38289] , [9943/51052, -21475/25526, 32926/38289, 86380/38289, 74080/38289, -165440/38289 ] , [-41971/102104, 64853/102104, -13001/38289, -10229/38289, -146432/38289, 164272/38289] , [43193/102104, 9065/102104, -22637/38289, 38311/38289, 164416/38289, -196496/38289] , [55865/102104, -6487/102104, -6677/38289, -77561/38289, -106304/38289, 175216/38289] ] $ x $ [ [3/2, 5/2, 3/2, 7/2, 2, 1] , [3/2, 3, 2, 13/4, 9/8, 9/8] , [27/32, 45/16, 31/16, 129/32, 43/32, 33/32] , [129/128, 207/64, 105/64, 489/128, 161/128, 133/128] , [483/512, 399/128, 219/128, 2005/512, 609/512, 575/512] , [1827/2048, 3249/1024, 1727/1024, 7991/2048, 2601/2048, 2205/2048] ] $

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

Omega Rank for R : cycles: {{3, 4, 5, 6}}, net cycles: 0 . order: 4

[0, y₁, y₂, y₃, y₄, y₅]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [1, -5/48, -17/48, 19/48, -41/48] , [0, 7/48, -5/48, -17/48, 19/48] , [0, -5/48, -17/48, 19/48, 7/48] , [0, -17/48, 19/48, 7/48, -5/48] , [0, 7/48, -5/48, -17/48, 19/48] , [0, 19/48, 7/48, -5/48, -17/48] ] $ x $ [ [0, 1, 3, 2, 2, 4] , [0, 0, 2, 4, 3, 3] , [0, 0, 4, 3, 2, 3] , [0, 0, 3, 3, 4, 2] , [0, 0, 3, 2, 3, 4] ] $

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

See Matrix

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

p = s ⁴ - s ⁵

» SYNC'D 29/512 , 0.05664062500

29 . Coloring, {2, 3, 5, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `1` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 2` (` 1 + τ ` )`` (` - 3 + τ ` )` , -2` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -1` (` 9 - 5τ + 3τ ² + τ ³ ` )` , 2` (` 3 + τ ` )`` (` - 1 + τ ` )` , -2` (` 3 + τ ² ` )``]`

For τ=1/2, [-21, -60, -20, -59, -28, -52] . FixedPtCheck, [21, 60, 20, 59, 28, 52]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[y₁, -y₂, y₂, -y₁, -y₃, y₃]

p = s ³ - 2s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 4	3 vs 5	3 vs 5	3 vs 5	1 vs 4

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

See Matrix

[-y₁ + y₃, y₃, 0, y₁, y₂, y₃ - y₂]

p = - s ³ + s ⁴ p = - s ³ + s ⁵

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

See Matrix

[0, 0, 2 y₁, 2 y₁, y₁, y₁]

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

« NOT SYNC'D »

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

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

rank of M is 5 , rank of N is 3

M N

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

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

ker M, [0, 0, 0, 0, -λ₁, λ₁]
Range M, [x₁, x₃, x₄, x₂, x₅, x₅]

τ= 12 , r'= 2/3

Ranges

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

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {2, 3, 5, 6}

Rank 3

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

π₂ [2, 0, 0, 1, 1, 0, 2, 2, 2, 4, 2, 2, 3, 3, 0]

u₂ [1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)

wpp [2, 2, 2, 2, 2, 2]

π₃ [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0]

u₃ [0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

Right Group
Coloring	{2, 3, 5, 6}
Rank	3
R,B	[2, 6, 5, 2, 1, 4], [3, 4, 4, 3, 6, 5]
π₂	[2, 0, 0, 1, 1, 0, 2, 2, 2, 4, 2, 2, 3, 3, 0]
u₂	[1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)
wpp	[2, 2, 2, 2, 2, 2]
π₃	[0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0]
u₃	[0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

30 . Coloring, {2, 4, 5, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -12` (` 3 + τ ` )`` (` - 1 + τ ` )` , 12` (` 3 - 2τ + τ ² ` )`` (` 1 + τ ` )` , 6` (` 9 - τ - τ ² + τ ³ ` )` , 12` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -12` (` 3 + τ ² ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [15, 28, 54, 67, 20, 26] . FixedPtCheck, [15, 28, 54, 67, 20, 26]

det(A + τ Δ) = 0

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	5 vs 5	5 vs 5	5 vs 5	3 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [91/820, 0, 0, -81/820, 27/820, 243/820] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [-81/820, 0, 0, 811/820, 3/820, 27/820] , [27/820, 0, 0, 3/820, 819/820, -9/820] , [243/820, 0, 0, 27/820, -9/820, 739/820] ] $ =
$ [ [51/4, 16, -66, -112/3, 224/3] , [3/20, -1, 18/5, 88/15, -128/15] , [-141/20, -13, 194/5, 424/15, -704/15] , [63/20, 8, -82/5, -272/15, 352/15] , [67/20, 7, -118/5, -248/15, 448/15] , [-141/20, -13, 194/5, 424/15, -704/15] ] $ x $ [ [1/2, 5/2, 3/2, 5/2, 3, 2] , [3/4, 2, 1, 11/4, 21/8, 23/8] , [21/32, 9/4, 5/4, 79/32, 93/32, 79/32] , [93/128, 129/64, 71/64, 335/128, 357/128, 351/128] , [357/512, 549/256, 307/256, 1267/512, 1479/512, 1329/512] ] $

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

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

[y₂, y₁, y₃, y₄, 0, y₅]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, 1/2, -1/4, -5/24, 1/24] , [0, 0, 1/2, -5/24, -5/24] , [0, 0, 0, 7/24, -5/24] , [0, 0, 0, -5/24, 7/24] , [1/2, -1/4, -3/8, 1/24, 1/6] , [0, 0, 0, 7/24, -5/24] ] $ x $ [ [2, 1, 3, 4, 0, 2] , [0, 2, 4, 5, 0, 1] , [0, 0, 5, 5, 0, 2] , [0, 0, 5, 7, 0, 0] , [0, 0, 7, 5, 0, 0] ] $

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

See Matrix

[0, y₃, -7 y₃ + 6 y₁ - y₂, -6 y₃ + 5 y₁, y₁, y₂]

p' = s ² - s ⁴ p = - s ² + s ⁴

» SYNC'D 3/128 , 0.02343750000

31 . Coloring, {3, 4, 5, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `6` (` - 3 - τ - 5τ ² + τ ³ ` )`` (` 1 + τ ` )` , -36` (` 3 - 2τ + 3τ ² ` )`` (` 1 + τ ` )` , 12` (` - 3 - τ - τ ² + τ ³ ` )`` (` 1 + τ ` )` , 6` (` 3 + τ ² ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 12` (` - 3 - τ - 5τ ² + τ ³ ` )` , 12` (` 1 + τ ² ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-111, -132, -174, -195, -148, -70] . FixedPtCheck, [111, 132, 174, 195, 148, 70]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
4 vs 4	6 vs 6	6 vs 6	5 vs 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 1/4, 3/4, 0, 0, 0] , [0, 0, 0, 1/4, 0, 3/4] , [0, 0, 0, 3/4, 1/4, 0] , [0, 3/4, 1/4, 0, 0, 0] , [1/4, 0, 0, 0, 0, 3/4] , [0, 0, 0, 1/4, 3/4, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [13091/7336, -62383/22008, -5183/2751, 3451/393, 12352/2751, -28208/2751] , [289/3668, -811/5502, -866/2751, 1564/393, -7040/2751, -2624/2751] , [-2447/3668, 13949/5502, -3938/2751, -2516/393, -704/2751, 17344/2751] , [3707/7336, -21367/22008, 4993/2751, 67/393, 2368/2751, -6320/2751] , [-2449/7336, 1037/22008, 1537/2751, -593/393, -12512/2751, 16144/2751] , [-239/1048, 1379/3144, -233/393, -281/393, 1504/393, -1040/393] ] $ x $ [ [1/2, 5/2, 3/2, 5/2, 2, 3] , [1/2, 2, 1, 5/2, 21/8, 27/8] , [21/32, 2, 1, 67/32, 89/32, 111/32] , [89/128, 111/64, 65/64, 271/128, 365/128, 459/128] , [365/512, 451/256, 269/256, 1071/512, 1507/512, 1761/512] , [1507/2048, 1789/1024, 1083/1024, 4277/2048, 5821/2048, 7227/2048] ] $

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

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

[y₁, y₄, y₅, y₃, y₂, 0]

See Matrices

R = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 1, 0, 0, 0] , [1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0] ] $ = $ [ [-107/492, 1/492, -23/492, 37/492, 133/492] , [133/492, -107/492, 1/492, -23/492, 37/492] , [-23/492, 37/492, 133/492, -107/492, 1/492] , [37/492, 133/492, -107/492, 1/492, -23/492] , [1/492, -23/492, 37/492, 133/492, -107/492] , [133/492, -107/492, 1/492, -23/492, 37/492] ] $ x $ [ [2, 1, 3, 4, 2, 0] , [2, 2, 4, 1, 3, 0] , [3, 2, 1, 2, 4, 0] , [4, 3, 2, 2, 1, 0] , [1, 4, 2, 3, 2, 0] ] $

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

See Matrix

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

p = - s ⁴ + s ⁵

» SYNC'D 35/256 , 0.1367187500

32 . Coloring, {2, 3, 4, 5, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `2` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 12` (` 3 - 2τ + 3τ ² ` )` , 4` (` 1 + τ ` )`` (` 3 - 2τ + τ ² ` )` , 2` (` 9 - 5τ + 3τ ² + τ ³ ` )` , 4` (` 3 + τ ² ` )` , 4` (` 3 - τ + τ ² + τ ³ ` )``]`

For τ=1/2, [39, 44, 54, 59, 52, 46] . FixedPtCheck, [39, 44, 54, 59, 52, 46]

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

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₂, -y₁, y₁, y₂, -y₃, y₃]

p = s - 2s ³ + 4s ⁴

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 4	4 vs 6	4 vs 6	4 vs 6	2 vs 5

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

See Matrix

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

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

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

See Matrix

[0, y₁, -y₁ + 2 y₂, 2 y₂, y₂, y₂]

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

« NOT SYNC'D »

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

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

rank of M is 6 , rank of N is 3

M N

$ [ [0, 2, 7, 0, 4, 5] , [2, 0, 0, 16, 10, 8] , [7, 0, 0, 11, 8, 10] , [0, 16, 11, 0, 14, 13] , [4, 10, 8, 14, 0, 0] , [5, 8, 10, 13, 0, 0] ] $ $ [ [0, 1, 1, 0, 1, 1] , [1, 0, 0, 1, 1, 1] , [1, 0, 0, 1, 1, 1] , [0, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 0, 0] , [1, 1, 1, 1, 0, 0] ] $

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

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

τ= 12 , r'= 2/3

Ranges

Action of R on ranges, [[2], [6], [1], [5], [4], [8], [3], [7]]
Action of B on ranges, [[8], [7], [8], [7], [6], [5], [6], [5]]
β({1, 2, 5}) = 1/27
β({1, 2, 6}) = 1/54
β({1, 3, 5}) = 2/27
β({1, 3, 6}) = 13/108
β({2, 4, 5}) = 13/54
β({2, 4, 6}) = 11/54
β({3, 4, 5}) = 4/27
β({3, 4, 6}) = 17/108

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

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

Rank 3

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

π₂ [2, 7, 0, 4, 5, 0, 16, 10, 8, 11, 8, 10, 14, 13, 0]

u₂ [1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)

wpp [2, 2, 2, 2, 2, 2]

π₃ [0, 0, 4, 2, 0, 8, 13, 0, 0, 0, 0, 0, 0, 26, 22, 0, 16, 17, 0, 0]

u₃ [0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

Right Group
Coloring	{2, 3, 4, 5, 6}
Rank	3
R,B	[2, 6, 5, 3, 1, 4], [3, 4, 4, 2, 6, 5]
π₂	[2, 7, 0, 4, 5, 0, 16, 10, 8, 11, 8, 10, 14, 13, 0]
u₂	[1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0] (dim 1)
wpp	[2, 2, 2, 2, 2, 2]
π₃	[0, 0, 4, 2, 0, 8, 13, 0, 0, 0, 0, 0, 0, 26, 22, 0, 16, 17, 0, 0]
u₃	[0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0]

SUMMARY

Graph Type
CC

ν(A)
2

ν(Δ)
2

π
[1, 2, 2, 3, 2, 2]

Dbly Stoch
false

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

SANDWICH
Total 2

No . Coloring Rank

1 {5} 2

2 {2, 3, 5} 2

SANDWICH		Total 2
No .	Coloring	Rank
1	{5}	2
2	{2, 3, 5}	2

RT GROUPS
Total 7

No . Coloring Rank Solv

1 {} 3 Not Solvable

2 {2, 3, 4, 5, 6} 3 Not Solvable

3 {4} 3 Not Solvable

4 {3} 2 Solvable

5 {3, 5, 6} 2 Solvable

6 {2, 6} 3 Not Solvable

7 {2, 3, 5, 6} 3 Not Solvable

RT GROUPS		Total 7
No .	Coloring	Rank	Solv
1	{}	3	Not Solvable
2	{2, 3, 4, 5, 6}	3	Not Solvable
3	{4}	3	Not Solvable
4	{3}	2	Solvable
5	{3, 5, 6}	2	Solvable
6	{2, 6}	3	Not Solvable
7	{2, 3, 5, 6}	3	Not Solvable

Δ-RANK'D SC'D !RK'D τ-RANK'D R/B RANK'D NOT SYNC'D Total Runs 2^n-1
22 0 21 , 21 12 , 10 9 32 32