New Graph

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

π = [1, 1, 1, 1, 1, 1]

POSSIBLE RANKS

1 x 6
2 x 3

BASE DETERMINANT 91/512, .1777343750

NullSpace of Δ

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

Nullspace of A

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

1 . Coloring, {}

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

` See graph

` ` See pair graph

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

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

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₁, y₁, -y₁, y₁, -y₁, y₁]

p = s ²

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s ² p' = s ³

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

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

See Matrix

[y₁, 0, y₁, 0, y₁, 0]

p = s - s ³ p' = s - s ²

Omega Rank for B : 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

« NOT SYNC'D »

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

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

rank of M is 6 , rank of N is 4

M N

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

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

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

τ= 12 , r'= 2/3

Ranges

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

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

Partitions

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

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

b1 = {5, 6} ` , ` b2 = {1, 2} ` , ` b3 = {2, 5} ` , ` b4 = {3, 4} ` , ` b5 = {1, 6}

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

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

` ` See level-3 partition graph.

Sandwich

Coloring {}

Rank 3

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

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

u₂ [1, 2, 2, 2, 1, 2, 2, 1, 2, 0, 2, 2, 2, 2, 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, 1, 0, 0, 0, 0, 0]

u₃ [1, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 1, 2, 1, 2, 0, 0, 0, 1, 1]

Sandwich
Coloring	{}
Rank	3
R,B	[3, 3, 5, 5, 1, 1], [2, 4, 6, 6, 4, 2]
π₂	[0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0]
u₂	[1, 2, 2, 2, 1, 2, 2, 1, 2, 0, 2, 2, 2, 2, 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, 1, 0, 0, 0, 0, 0]
u₃	[1, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 1, 2, 1, 2, 0, 0, 0, 1, 1]

2 . Coloring, {2}

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

` See graph

` ` See pair graph

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

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

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[y₁, -y₁, -y₂, y₂, y₁, -y₁]

p' = s ³ p = s ³

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 5	2 vs 5	2 vs 4	2 vs 4

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

See Matrix

[y₂, 0, y₁, y₂ - y₁, y₂, 0]

p = s ² - s ⁴ p' = s ² - s ³

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

See Matrix

[0, y₂, y₂ - y₁, y₁, 0, y₂]

p = - s ² + s ³ p = - s ² + s ⁴

« NOT SYNC'D »

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

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

rank of M is 5 , rank of N is 4

M N

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

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

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

τ= 12 , r'= 2/3

Ranges

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

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

Partitions

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

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

b1 = {5, 6} ` , ` b2 = {1, 2} ` , ` b3 = {2, 5} ` , ` b4 = {3, 4} ` , ` b5 = {1, 6}

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

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

` ` See level-3 partition graph.

Sandwich

Coloring {2}

Rank 3

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

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

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

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

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

u₃ [1, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 1, 2, 1, 2, 0, 0, 0, 1, 1]

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

3 . Coloring, {3}

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

` See graph

` ` See pair graph

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

For τ=1/2, [87, 41, 96, 20, 39, 77] . FixedPtCheck, [87, 41, 96, 20, 39, 77]

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	2 vs 4	4 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 3/4, 0, 0] , [1/4, 3/4, 0, 0, 0, 0] ] $ x $ [ [11/20, 3/20, -9/20, 3/20, 0, 0] , [3/20, 19/20, 3/20, -1/20, 0, 0] , [-9/20, 3/20, 11/20, 3/20, 0, 0] , [3/20, -1/20, 3/20, 19/20, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [11/60, -21/10, -209/180, 22/45, 124/45] , [-7/60, 7/10, -227/180, -14/45, 52/45] , [83/60, 7/10, 223/180, -74/45, -68/45] , [-19/60, -1/10, 241/180, -38/45, 4/45] , [-13/60, 9/5, -23/180, 34/45, -92/45] , [1/12, -1, -1/36, 14/9, -4/9] ] $ x $ [ [1/2, 3/2, 1/2, 3/2, 1, 1] , [1/2, 9/8, 1/2, 15/8, 3/4, 5/4] , [1/2, 21/16, 13/32, 45/32, 27/32, 49/32] , [19/32, 195/128, 29/64, 207/128, 21/32, 37/32] , [29/64, 21/16, 271/512, 837/512, 381/512, 679/512] ] $

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

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

See Matrix

[y₁ + y₂, 0, y₁ + y₂, 0, y₁, y₂]

p = - s ² + s ³ p = - s ² + s ⁴

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

[0, y₃, 0, y₂, y₁, y₄]

See Matrices

B = $ [ [0, 1, 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, 1, 0, 0] , [0, 1, 0, 0, 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, -5/18, 1/18, 7/18] , [0, 7/18, -5/18, 1/18] , [1, -5/18, 1/18, -11/18] , [0, 1/18, 7/18, -5/18] , [0, 7/18, -5/18, 1/18] , [0, -5/18, 1/18, 7/18] ] $ x $ [ [0, 2, 0, 2, 1, 1] , [0, 1, 0, 3, 0, 2] , [0, 2, 0, 1, 0, 3] , [0, 3, 0, 2, 0, 1] ] $

» SYNC'D 5/16 , 0.3125000000

4 . Coloring, {4}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-105, -39, -108, -32, -89, -51] . FixedPtCheck, [105, 39, 108, 32, 89, 51]

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	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 3/4, 0, 0] , [1/4, 3/4, 0, 0, 0, 0] ] $ x $ [ [11/20, 3/20, -9/20, 3/20, 0, 0] , [3/20, 19/20, 3/20, -1/20, 0, 0] , [-9/20, 3/20, 11/20, 3/20, 0, 0] , [3/20, -1/20, 3/20, 19/20, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [1, -5/6, -8/3, -40/9, 64/9] , [-1, 19/6, 2/3, 8/9, -32/9] , [1/3, -1/6, 40/9, 8/3, -64/9] , [1/3, -13/6, 10/9, -8/3, 32/9] , [-5/6, 2, -1/9, 40/9, -16/3] , [7/6, -2, -31/9, -8/9, 16/3] ] $ x $ [ [1/2, 3/2, 1/2, 3/2, 1, 1] , [1/2, 9/8, 1/2, 15/8, 5/4, 3/4] , [1/2, 15/16, 13/32, 57/32, 49/32, 27/32] , [19/32, 129/128, 23/64, 237/128, 23/16, 3/4] , [35/64, 129/128, 205/512, 939/512, 757/512, 375/512] ] $

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

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

See Matrix

[y₂ + y₁, 0, y₂ + y₁, 0, y₂, y₁]

p = - s ² + s ³ p = - s ² + s ⁴

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

See Matrix

[0, y₁ - y₂ + y₃, 0, y₁, y₂, y₃]

p = - s ³ + s ⁴

» SYNC'D 3/32 , 0.09375000000

5 . Coloring, {5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [48, 20, 51, 77, 96, 32] . FixedPtCheck, [48, 20, 51, 77, 96, 32]

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, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 1/4, 0, 0] , [1/4, 3/4, 0, 0, 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, 1/10, 3/10] , [0, 0, 0, 0, 3/10, 9/10] ] $ =
$ [ [0, 7/2, -2/3, 8/9, -32/9] , [0, -3/2, 5/3, -16/9, 16/9] , [0, 1/2, -1/3, 16/9, -16/9] , [0, 1/2, -1/3, 16/9, -16/9] , [3/2, -7/2, 5/6, -28/9, 40/9] , [-1/2, 1/2, -7/6, 4/9, 8/9] ] $ x $ [ [1, 3/2, 1/2, 1, 1/2, 3/2] , [3/4, 15/8, 5/8, 5/4, 3/8, 9/8] , [9/16, 45/32, 21/32, 3/2, 15/32, 45/32] , [45/64, 189/128, 63/128, 75/64, 69/128, 207/128] , [207/256, 891/512, 279/512, 159/128, 213/512, 639/512] ] $

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

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

See Matrix

[y₂ + y₁ - y₃, 0, y₂, y₁, y₃, 0]

p = s ³ - s ⁴

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

[y₂, y₁, 0, y₄, 0, y₃]

See Matrices

B = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 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, 7/18, -5/18, 1/18] , [0, 1/18, 7/18, -5/18] , [0, -5/18, 1/18, 7/18] , [0, -5/18, 1/18, 7/18] , [1, -5/18, 1/18, -11/18] , [0, 7/18, -5/18, 1/18] ] $ x $ [ [1, 2, 0, 1, 0, 2] , [0, 3, 0, 2, 0, 1] , [0, 1, 0, 3, 0, 2] , [0, 2, 0, 1, 0, 3] ] $

» SYNC'D 5/32 , 0.1562500000

6 . Coloring, {6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-80, -44, -93, -35, -96, -32] . FixedPtCheck, [80, 44, 93, 35, 96, 32]

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	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 3/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 1/10, 3/10] , [0, 0, 0, 0, 3/10, 9/10] ] $ =
$ [ [0, 11/6, 1/9, 32/9, -16/3] , [0, -25/6, 7/9, -16/9, 16/3] , [0, 5/2, -7/3, 16/9, -16/9] , [0, 5/2, -7/3, 16/9, -16/9] , [3/2, -7/3, 17/9, -8/3, 16/9] , [-1/2, -1/3, 17/9, -8/3, 16/9] ] $ x $ [ [1, 1, 1/2, 3/2, 1/2, 3/2] , [5/4, 9/8, 1/2, 9/8, 1/2, 3/2] , [5/4, 21/16, 19/32, 39/32, 13/32, 39/32] , [65/64, 159/128, 41/64, 165/128, 29/64, 87/64] , [145/128, 141/128, 289/512, 651/512, 247/512, 741/512] ] $

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

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

See Matrix

[-y₁ + y₂, y₁, y₂, 0, y₂, 0]

p = s ² - s ³ p' = - s ² + s ³

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

See Matrix

[y₁, y₁ + y₂ - y₃, 0, y₂, 0, y₃]

p = - s + s ² - s ³ + s ⁴

» SYNC'D 3/16 , 0.1875000000

7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-93, -41, -80, -44, -53, -71] . FixedPtCheck, [93, 41, 80, 44, 53, 71]

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	3 vs 5	5 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 3/4, 0, 0] , [1/4, 3/4, 0, 0, 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, 1] ] $ =
$ [ [461/246, 11/41, -83/123, -272/369, -80/369, -128/369] , [-31/246, 93/41, -83/123, -272/369, -80/369, -128/369] , [77/246, 15/41, 349/123, -356/369, -944/369, 64/369] , [-247/246, -79/41, 37/123, -104/369, 1648/369, -512/369] , [11/246, 49/41, 208/123, 1636/369, -416/369, -2240/369] , [-25/246, -89/41, -428/123, -632/369, -128/369, 2944/369] ] $ x $ [ [1/2, 3/2, 1, 1, 1, 1] , [1/2, 9/8, 5/4, 9/8, 1, 1] , [1/2, 9/8, 31/32, 33/32, 39/32, 37/32] , [19/32, 159/128, 31/32, 153/128, 63/64, 65/64] , [1/2, 309/256, 553/512, 537/512, 525/512, 583/512] , [277/512, 2517/2048, 1055/1024, 2193/2048, 549/512, 541/512] ] $

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

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

See Matrix

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

p = s ³ - s ⁵ p' = s ³ - s ⁴

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

[0, y₄, y₁, y₂, y₃, y₅]

See Matrices

B = $ [ [0, 1, 0, 0, 0, 0] , [0, 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] ] $ 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] ] $ = $ [ [5/6, -1/6, -1/6, -1/6, -1/6] , [-1/6, 5/6, -1/6, -1/6, -1/6] , [-1/6, -1/6, 5/6, -1/6, -1/6] , [-1/6, -1/6, -1/6, -1/6, 5/6] , [-1/6, -1/6, -1/6, 5/6, -1/6] , [5/6, -1/6, -1/6, -1/6, -1/6] ] $ x $ [ [0, 2, 1, 1, 1, 1] , [0, 1, 2, 1, 1, 1] , [0, 1, 1, 1, 2, 1] , [0, 1, 1, 2, 1, 1] , [0, 1, 1, 1, 1, 2] ] $

» SYNC'D 25/128 , 0.1953125000

8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

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

For τ=1/2, [33, 13, 28, 16, 25, 19] . FixedPtCheck, [33, 13, 28, 16, 25, 19]

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	3 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 3/4, 0, 0] , [1/4, 3/4, 0, 0, 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, 1] ] $ =
$ [ [15/8, 7/24, -7/12, -3/4, -2/9, -4/9] , [-1/8, 55/24, -7/12, -3/4, -2/9, -4/9] , [11/24, 25/24, 31/12, -37/36, -10/3, 4/9] , [-31/24, -77/24, 13/12, -7/36, 6, -20/9] , [1/6, -4/3, -6, -14/9, -16/9, 32/3] , [-1/12, 11/12, 7/2, 77/18, -4/9, -8] ] $ x $ [ [1/2, 3/2, 1, 1, 1, 1] , [1/2, 9/8, 5/4, 9/8, 1, 1] , [1/2, 9/8, 31/32, 33/32, 37/32, 39/32] , [19/32, 165/128, 31/32, 147/128, 65/64, 63/64] , [1/2, 303/256, 571/512, 555/512, 565/512, 519/512] , [271/512, 2325/2048, 1037/1024, 2301/2048, 559/512, 567/512] ] $

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

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

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

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

See Matrix

[0, -y₁ + 4 y₂ - y₃, y₁, y₂, y₂, y₃]

p' = s - s ⁴ p = s - s ⁴

» SYNC'D 5/64 , 0.07812500000

9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [48, 20, 41, 87, 96, 32] . FixedPtCheck, [48, 20, 41, 87, 96, 32]

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	3 vs 4	4 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 1/4, 0, 0] , [1/4, 3/4, 0, 0, 0, 0] ] $ x $ [ [91/100, 3/100, -9/100, 27/100, 0, 0] , [3/100, 99/100, 3/100, -9/100, 0, 0] , [-9/100, 3/100, 91/100, 27/100, 0, 0] , [27/100, -9/100, 27/100, 19/100, 0, 0] , [0, 0, 0, 0, 1/10, 3/10] , [0, 0, 0, 0, 3/10, 9/10] ] $ =
$ [ [-1, 7/6, -8/9, 8/9] , [1/2, -1/3, 16/9, -16/9] , [-1/2, -2/3, -4/9, 16/9] , [-1/2, -2/3, -4/9, 16/9] , [11/4, -7/12, 10/9, -28/9] , [-1/4, 13/12, -10/9, 4/9] ] $ x $ [ [1, 3/2, 1, 1/2, 1/2, 3/2] , [3/4, 15/8, 11/8, 1/2, 3/8, 9/8] , [9/16, 45/32, 51/32, 9/16, 15/32, 45/32] , [45/64, 189/128, 153/128, 15/32, 69/128, 207/128] ] $

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

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

See Matrix

[y₁ + y₃ - y₂, 0, y₁, y₃, y₂, 0]

p = - s ³ + s ⁴

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

[y₃, y₂, y₁, 0, 0, y₄]

See Matrices

B = $ [ [0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 0, 1] , [1, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 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, 0, 0] , [0, 0, 0, 0, 0, 1] ] $ = $ [ [0, 7/18, -5/18, 1/18] , [0, 1/18, 7/18, -5/18] , [0, -5/18, 1/18, 7/18] , [0, -5/18, 1/18, 7/18] , [1, -5/18, 1/18, -11/18] , [0, 7/18, -5/18, 1/18] ] $ x $ [ [1, 2, 1, 0, 0, 2] , [0, 3, 2, 0, 0, 1] , [0, 1, 3, 0, 0, 2] , [0, 2, 1, 0, 0, 3] ] $

» SYNC'D 5/32 , 0.1562500000

10 . Coloring, {2, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-80, -44, -71, -57, -96, -32] . FixedPtCheck, [80, 44, 71, 57, 96, 32]

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 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 3/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 1/10, 3/10] , [0, 0, 0, 0, 3/10, 9/10] ] $ =
$ [ [0, 11/6, 1/9, 32/9, -16/3] , [0, -25/6, 7/9, -16/9, 16/3] , [0, 5/2, -7/3, 16/9, -16/9] , [0, 5/2, -7/3, 16/9, -16/9] , [3/2, -7/3, 17/9, -8/3, 16/9] , [-1/2, -1/3, 17/9, -8/3, 16/9] ] $ x $ [ [1, 1, 1, 1, 1/2, 3/2] , [5/4, 9/8, 1, 5/8, 1/2, 3/2] , [5/4, 21/16, 37/32, 21/32, 13/32, 39/32] , [65/64, 159/128, 83/64, 81/128, 29/64, 87/64] , [145/128, 141/128, 607/512, 333/512, 247/512, 741/512] ] $

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

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

See Matrix

[-y₂ + y₁, y₂, y₃, -y₃ + y₁, y₁, 0]

p = s ³ - s ⁵ p' = s ³ - s ⁴

Omega Rank for B : cycles: {{1, 2, 3, 6}}, net cycles: 0 . order: 4

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

» SYNC'D 3/16 , 0.1875000000

11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-24, -12, -27, -5, -8, -24] . FixedPtCheck, [24, 12, 27, 5, 8, 24]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[y₁, y₂, 2 y₁ - y₃, -3 y₁ - y₂ + y₃, -y₃, y₃]

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
3 vs 4	4 vs 4	4 vs 4	1 vs 3	3 vs 3

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 3/4, 0, 0] , [1/4, 3/4, 0, 0, 0, 0] ] $ x $ [ [11/20, 3/20, -9/20, 3/20, 0, 0] , [3/20, 19/20, 3/20, -1/20, 0, 0] , [-9/20, 3/20, 11/20, 3/20, 0, 0] , [3/20, -1/20, 3/20, 19/20, 0, 0] , [0, 0, 0, 0, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [11/12, 5/36, -4/9, -4/9] , [-1/12, 41/36, -4/9, -4/9] , [-1/12, -7/36, 8/9, -4/9] , [-1/12, -7/36, 8/9, -4/9] , [-1/3, 1/18, -4/9, 8/9] , [2/3, -17/18, -4/9, 8/9] ] $ x $ [ [1/2, 3/2, 1/2, 3/2, 3/2, 1/2] , [1/2, 3/4, 1/2, 9/4, 3/2, 1/2] , [1/2, 3/4, 5/16, 27/16, 33/16, 11/16] , [11/16, 57/64, 5/16, 135/64, 3/2, 1/2] ] $

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

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

See Matrix

[y₁, 0, y₁, 0, 0, y₁]

p = - s + s ³ p = - s + s ²

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

[0, y₁, 0, y₂, y₃, 0]

See Matrices

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

» SYNC'D 1/4 , 0.2500000000

12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-45, -23, -51, -35, -39, -47] . FixedPtCheck, [45, 23, 51, 35, 39, 47]

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	3 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 1/4, 0, 0] , [1/4, 3/4, 0, 0, 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, 1] ] $ =
$ [ [76/105, -487/630, -278/315, -482/105, 1184/315, 608/315] , [-62/105, 809/630, -44/315, 34/105, -1408/315, 1184/315] , [148/105, -451/630, -44/315, 34/105, -1408/315, 1184/315] , [-74/105, -247/630, 1072/315, 298/105, 704/315, -2272/315] , [-13/30, 124/45, 97/45, 58/15, -16/45, -352/45] , [25/42, -136/63, -277/63, -58/21, 208/63, 352/63] ] $ x $ [ [1, 3/2, 1/2, 1, 1, 1] , [1, 3/2, 5/8, 11/8, 5/8, 7/8] , [11/16, 45/32, 5/8, 41/32, 13/16, 19/16] , [29/32, 45/32, 67/128, 161/128, 101/128, 143/128] , [223/256, 777/512, 37/64, 641/512, 181/256, 275/256] , [409/512, 747/512, 1223/2048, 2693/2048, 1529/2048, 2219/2048] ] $

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

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

See Matrix

[y₁, 0, -y₁ + 4 y₂ - y₃, y₂, y₂, y₃]

p' = - s + s ⁴ p = - s + s ⁴

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

See Matrix

[y₂, y₃, 0, y₃ - y₁, y₁, -y₂ + y₃]

p = s ³ - s ⁵ p' = s ³ - s ⁴

» SYNC'D 29/256 , 0.1132812500

13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-53, -71, -93, -29, -45, -77] . FixedPtCheck, [53, 71, 93, 29, 45, 77]

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	3 vs 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 3/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 1] ] $ =
$ [ [2/3, 1/6, 8/3, 2/9, 64/9, -32/3] , [-4/3, 1/6, -22/3, 14/9, -32/9, 32/3] , [2, -11/6, 14/3, -14/3, 32/9, -32/9] , [0, 1/6, 14/3, -14/3, 32/9, -32/9] , [-1/6, 8/3, -13/3, 34/9, -16/3, 32/9] , [-1/6, -4/3, -1/3, 34/9, -16/3, 32/9] ] $ x $ [ [1, 1, 1/2, 3/2, 1, 1] , [1, 1, 1/2, 3/2, 3/4, 5/4] , [9/8, 17/16, 1/2, 21/16, 3/4, 5/4] , [9/8, 37/32, 35/64, 87/64, 45/64, 71/64] , [129/128, 287/256, 73/128, 357/256, 3/4, 37/32] , [135/128, 535/512, 545/1024, 1437/1024, 795/1024, 1217/1024] ] $

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

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

See Matrix

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

p' = s ³ - s ⁴ p = s ³ - s ⁵

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

» SYNC'D 5/32 , 0.1562500000

14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-83, -41, -93, -77, -89, -81] . FixedPtCheck, [83, 41, 93, 77, 89, 81]

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, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [1/4, 3/4, 0, 0, 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, 1] ] $ =
$ [ [332/737, -3023/4422, 2066/2211, -6814/2211, 7520/2211, -1888/2211] , [-398/737, 4321/4422, -2308/2211, 1358/2211, -7168/2211, 7520/2211] , [1076/737, -4523/4422, -2308/2211, 1358/2211, -7168/2211, 7520/2211] , [-202/737, -2303/4422, 7880/2211, 230/2211, 6080/2211, -12064/2211] , [-683/1474, -1840/2211, -9817/2211, 4778/2211, -4432/2211, 12704/2211] , [541/1474, 4604/2211, 4487/2211, -910/2211, 5168/2211, -13792/2211] ] $ x $ [ [1, 3/2, 1/2, 1, 1, 1] , [1, 3/2, 5/8, 11/8, 7/8, 5/8] , [13/16, 39/32, 5/8, 43/32, 19/16, 13/16] , [35/32, 39/32, 65/128, 155/128, 149/128, 103/128] , [275/256, 729/512, 37/64, 617/512, 265/256, 175/256] , [485/512, 675/512, 1279/2048, 2717/2048, 2147/2048, 1505/2048] ] $

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

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

[y₅, 0, y₁, y₂, y₃, y₄]

See Matrices

R = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [1, 0, 0, 0, 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, 1] ] $ = $ [ [5/6, -1/6, -1/6, -1/6, -1/6] , [5/6, -1/6, -1/6, -1/6, -1/6] , [-1/6, 5/6, -1/6, -1/6, -1/6] , [-1/6, -1/6, -1/6, 5/6, -1/6] , [-1/6, -1/6, 5/6, -1/6, -1/6] , [-1/6, -1/6, -1/6, -1/6, 5/6] ] $ x $ [ [1, 0, 2, 1, 1, 1] , [1, 0, 1, 1, 2, 1] , [1, 0, 1, 2, 1, 1] , [1, 0, 1, 1, 1, 2] , [2, 0, 1, 1, 1, 1] ] $

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

» SYNC'D 35/256 , 0.1367187500

15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [75, 57, 99, 35, 83, 51] . FixedPtCheck, [75, 57, 99, 35, 83, 51]

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	3 vs 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 3/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 1] ] $ =
$ [ [2/3, 1/6, 0, -38/9, -64/9, 32/3] , [-4/3, 1/6, 6, 22/9, 32/9, -32/3] , [2, -11/6, -10/3, 10/3, -32/9, 32/9] , [0, 1/6, -10/3, 10/3, -32/9, 32/9] , [-1/6, -4/3, 7/3, -22/9, 16/3, -32/9] , [-1/6, 8/3, -5/3, -22/9, 16/3, -32/9] ] $ x $ [ [1, 1, 1/2, 3/2, 1, 1] , [1, 1, 1/2, 3/2, 5/4, 3/4] , [7/8, 15/16, 1/2, 27/16, 5/4, 3/4] , [7/8, 27/32, 29/64, 105/64, 89/64, 51/64] , [121/128, 219/256, 55/128, 429/256, 43/32, 3/4] , [115/128, 459/512, 461/1024, 1689/1024, 1397/1024, 759/1024] ] $

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

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

See Matrix

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

p = s ³ - s ⁴ p' = - s ³ + s ⁴

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

See Matrix

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

p = s ⁴ - s ⁵

» SYNC'D 3/32 , 0.09375000000

16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-2, -2, -3, -5, -6, -2] . FixedPtCheck, [2, 2, 3, 5, 6, 2]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

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

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
3 vs 4	4 vs 4	4 vs 4	3 vs 4	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 0, 0] ] $ x $ [ [99/100, 3/100, -9/100, 3/100, 0, 0] , [3/100, 91/100, 27/100, -9/100, 0, 0] , [-9/100, 27/100, 19/100, 27/100, 0, 0] , [3/100, -9/100, 27/100, 91/100, 0, 0] , [0, 0, 0, 0, 1/10, 3/10] , [0, 0, 0, 0, 3/10, 9/10] ] $ =
$ [ [-33/16, 25/16, -1, 5/3] , [27/16, -35/16, 3, -7/3] , [-21/16, 13/16, -7/3, 3] , [-21/16, 13/16, -7/3, 3] , [21/8, -9/8, 2, -10/3] , [11/8, 1/8, 2/3, -2] ] $ x $ [ [3/2, 1, 1/2, 1, 1/2, 3/2] , [3/2, 3/2, 5/8, 7/8, 3/8, 9/8] , [9/8, 45/32, 3/4, 39/32, 3/8, 9/8] , [9/8, 9/8, 81/128, 147/128, 63/128, 189/128] ] $

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

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

See Matrix

[0, y₃, y₃ - y₁ + y₂, y₁, y₂, 0]

p = - s ³ + s ⁴

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

See Matrix

[y₁, y₁ + y₂ - y₃, 0, y₂, 0, y₃]

p = - s + s ² - s ³ + s ⁴

» SYNC'D 1/8 , 0.1250000000

17 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-24, -12, -21, -11, -8, -24] . FixedPtCheck, [24, 12, 21, 11, 8, 24]

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	4 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 3/4, 0, 0] , [1/4, 3/4, 0, 0, 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, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [0, -13/12, -115/36, -4/9, 44/9] , [0, 5/4, -7/36, -4/9, -4/9] , [0, -1/12, 41/36, -4/9, -4/9] , [0, -1/12, 41/36, -4/9, -4/9] , [-1/2, -1/6, -17/18, 8/9, 8/9] , [3/2, 1/6, 37/18, 8/9, -40/9] ] $ x $ [ [1/2, 3/2, 1, 1, 3/2, 1/2] , [1/2, 3/4, 5/4, 3/2, 3/2, 1/2] , [1/2, 3/4, 11/16, 21/16, 33/16, 11/16] , [11/16, 57/64, 11/16, 111/64, 3/2, 1/2] , [1/2, 57/64, 215/256, 345/256, 465/256, 155/256] ] $

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

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

See Matrix

[y₂, 0, y₁, -y₁ + y₂, 0, y₂]

p = - s ² + s ⁴ p = - s ² + s ³

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

[0, y₁, y₄, y₂, y₃, 0]

See Matrices

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

» SYNC'D 1/4 , 0.2500000000

18 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [47, 23, 41, 57, 53, 45] . FixedPtCheck, [47, 23, 41, 57, 53, 45]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₁ - y₃, -y₂ + y₃, y₁, y₂, -y₃, y₃]

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

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

See Matrix

[y₂, 0, y₂, 3 y₂ - y₁, y₁, y₂]

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

Omega Rank for B : cycles: {{1, 2, 3, 5}}, net cycles: 0 . order: 4

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

« NOT SYNC'D »

Nullspace of {Ω&Deltaⁱ} :
[x₁, 3 x₁, 4 x₁, 4 x₁]
For A+2Δ : [-3 y₁ - 2 y₂, 9 y₁ + 8 y₂, -3 y₁ - 2 y₂, y₁, -y₂, y₂]
For A-2Δ : [-3 y₁ - 2 y₂, y₁, -3 y₁ - 2 y₂, 9 y₁ + 8 y₂, -y₂, y₂]

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

rank of M is 2 , rank of N is 2

M N

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

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

ker M, [λ₃, λ₂, -λ₃ - λ₄, -λ₂ - λ₁, λ₁, λ₄]
Range M, [x₂, x₁, x₂, x₁, x₁, x₂]

τ= 18 , r'= 1/2

Ranges

Action of R on ranges, [[6], [7], [6], [8], [2], [9], [8], [3], [2]]
Action of B on ranges, [[4], [5], [1], [7], [4], [9], [3], [5], [1]]
β({1, 2}) = 1/9
β({1, 4}) = 1/9
β({1, 5}) = 1/9
β({2, 3}) = 1/9
β({2, 6}) = 1/9
β({3, 4}) = 1/9
β({3, 5}) = 1/9
β({4, 6}) = 1/9
β({5, 6}) = 1/9

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

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

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

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 {2, 3, 5}

Rank 2

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

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

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

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

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

19 . Coloring, {2, 3, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `1 , 1 , 1 , 1 , 1 , 1`]`

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

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

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

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

+ \ ; - \ ; Δ

See Matrices

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

p = s

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s p' = s ² p' = s ³

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

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

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁]

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

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

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁]

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

« NOT SYNC'D »

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

Range of {ΩΔⁱ}: [0, 0, 0, 0, 0, 0]

rank of M is 6 , rank of N is 6

M N

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

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

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

τ= 6 , r'= 5/6

Ranges

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

ker N, [0, 0, 0, 0, 0, 0]
Range of N
[y₁, y₂, y₅, y₃, y₄, y₆]

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

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

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

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

` ` See level-6 partition graph.

Right Group

Coloring {2, 3, 6}

Rank 6

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

π₂ [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

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

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

π₆ [1]

u₆ [1]

Right Group
Coloring	{2, 3, 6}
Rank	6
R,B	[3, 4, 6, 5, 1, 2], [2, 3, 5, 6, 4, 1]
π₂	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
u₂	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 1)
wpp	[1, 1, 1, 1, 1, 1]
π₆	[1]
u₆	[1]

20 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [81, 41, 71, 87, 75, 83] . FixedPtCheck, [81, 41, 71, 87, 75, 83]

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, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [1/4, 3/4, 0, 0, 0, 0] ] $ x $ [ [91/100, 3/100, -9/100, 27/100, 0, 0] , [3/100, 99/100, 3/100, -9/100, 0, 0] , [-9/100, 3/100, 91/100, 27/100, 0, 0] , [27/100, -9/100, 27/100, 19/100, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [-23/84, -65/84, -37/42, -44/21, 88/21] , [-29/84, 169/84, 89/42, -20/21, -8/3] , [-23/84, -137/84, -1/42, 52/21, -8/21] , [139/84, 1/84, 89/42, -20/21, -8/3] , [25/84, 115/84, -55/42, 52/21, -8/3] , [-5/84, -83/84, -85/42, -20/21, 88/21] ] $ x $ [ [1, 3/2, 1, 1/2, 1, 1] , [1, 3/2, 11/8, 5/8, 5/8, 7/8] , [11/16, 45/32, 11/8, 17/32, 13/16, 19/16] , [29/32, 45/32, 157/128, 71/128, 95/128, 149/128] , [217/256, 795/512, 41/32, 275/512, 185/256, 271/256] ] $

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

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

[y₅, 0, y₄, y₃, y₁, y₂]

See Matrices

R = $ [ [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, 1, 0, 0] , [1, 0, 0, 0, 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, 1] ] $ = $ [ [-1/6, -1/6, -1/6, 5/6, -1/6] , [5/6, -1/6, -1/6, -1/6, -1/6] , [-1/6, -1/6, -1/6, -1/6, 5/6] , [-1/6, 5/6, -1/6, -1/6, -1/6] , [5/6, -1/6, -1/6, -1/6, -1/6] , [-1/6, -1/6, 5/6, -1/6, -1/6] ] $ x $ [ [1, 0, 1, 2, 1, 1] , [1, 0, 1, 1, 1, 2] , [2, 0, 1, 1, 1, 1] , [1, 0, 2, 1, 1, 1] , [1, 0, 1, 1, 2, 1] ] $

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

See Matrix

[y₃, y₂, y₁, 0, y₂ - y₁, -y₃ + y₂]

p = - s ³ + s ⁵ p = - s ³ + s ⁴

» SYNC'D 9/64 , 0.1406250000

21 . Coloring, {2, 4, 6}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `1 , 1 , 1 , 1 , 1 , 1`]`

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

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

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

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

+ \ ; - \ ; Δ

See Matrices

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

p = s

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s p' = s ² p' = s ³

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

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

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁]

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

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

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁]

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

« NOT SYNC'D »

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

Range of {ΩΔⁱ}: [0, 0, 0, 0, 0, 0]

rank of M is 6 , rank of N is 6

M N

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

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

τ= 6 , r'= 5/6

Ranges

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

ker N, [0, 0, 0, 0, 0, 0]
Range of N
[y₃, y₄, y₁, y₂, y₆, y₅]

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

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

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

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

` ` See level-6 partition graph.

Right Group

Coloring {2, 4, 6}

Rank 6

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

π₂ [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

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

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

π₆ [1]

u₆ [1]

Right Group
Coloring	{2, 4, 6}
Rank	6
R,B	[3, 4, 5, 6, 1, 2], [2, 3, 6, 5, 4, 1]
π₂	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
u₂	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 2)
wpp	[1, 1, 1, 1, 1, 1]
π₆	[1]
u₆	[1]

22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

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

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

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	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 1/10, 3/10] , [0, 0, 0, 0, 3/10, 9/10] ] $ =
$ [ [0, 47/16, -37/48, 5/3, -11/3] , [0, -53/16, 45/16, -7/3, 3] , [0, 27/16, -35/16, 3, -7/3] , [0, 27/16, -35/16, 3, -7/3] , [3/2, -15/8, 15/8, -10/3, 2] , [-1/2, -9/8, 11/24, -2, 10/3] ] $ x $ [ [3/2, 1, 1, 1/2, 1/2, 3/2] , [3/2, 3/2, 9/8, 3/8, 3/8, 9/8] , [9/8, 45/32, 3/2, 15/32, 3/8, 9/8] , [9/8, 9/8, 171/128, 57/128, 63/128, 189/128] , [189/128, 621/512, 9/8, 207/512, 57/128, 171/128] ] $

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

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

See Matrix

[0, y₂, y₂, y₁, y₁, 0]

p = - s ² + s ⁴ p = - s ² + s ³

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

See Matrix

[y₁ - y₂ + y₃, y₁, y₂, 0, 0, y₃]

p = - s + s ² - s ³ + s ⁴

» SYNC'D 1/8 , 0.1250000000

23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-80, -44, -93, -35, -32, -96] . FixedPtCheck, [80, 44, 93, 35, 32, 96]

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	2 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [1/4, 3/4, 0, 0, 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, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [0, 1/6, -16/9, -16/9, 32/9] , [0, 5/2, 11/9, -16/9, -16/9] , [0, -5/6, 17/9, 8/9, -16/9] , [0, -5/6, 17/9, 8/9, -16/9] , [-1/2, -7/6, -23/18, 20/9, 8/9] , [3/2, 1/6, -35/18, -4/9, 8/9] ] $ x $ [ [1, 3/2, 1/2, 1, 3/2, 1/2] , [5/4, 9/8, 5/8, 3/2, 9/8, 3/8] , [15/16, 39/32, 19/32, 9/8, 51/32, 17/32] , [85/64, 141/128, 69/128, 21/16, 165/128, 55/128] , [275/256, 675/512, 311/512, 147/128, 711/512, 237/512] ] $

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

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

[y₁, 0, y₂, y₃, 0, y₄]

See Matrices

R = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [1, 0, 0, 0, 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, 1] ] $ = $ [ [0, -5/18, 1/18, 7/18] , [0, -5/18, 1/18, 7/18] , [0, 7/18, -5/18, 1/18] , [0, 7/18, -5/18, 1/18] , [1, -5/18, 1/18, -11/18] , [0, 1/18, 7/18, -5/18] ] $ x $ [ [1, 0, 2, 1, 0, 2] , [2, 0, 1, 0, 0, 3] , [3, 0, 2, 0, 0, 1] , [1, 0, 3, 0, 0, 2] ] $

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

See Matrix

[y₂, y₁, 0, y₂, y₁, 0]

p = - s + s ³ p' = - s + s ³

» SYNC'D 15/32 , 0.4687500000

24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [48, 84, 99, 29, 32, 96] . FixedPtCheck, [48, 84, 99, 29, 32, 96]

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	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 3/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [0, 1/2, -19/9, -32/9, 16/3] , [0, 5/2, 11/9, 16/9, -16/3] , [0, -3/2, 5/3, -16/9, 16/9] , [0, -3/2, 5/3, -16/9, 16/9] , [-1/2, 1, -11/9, 8/3, -16/9] , [3/2, -1, -11/9, 8/3, -16/9] ] $ x $ [ [1, 1, 1/2, 3/2, 3/2, 1/2] , [3/4, 7/8, 1/2, 15/8, 3/2, 1/2] , [3/4, 11/16, 13/32, 57/32, 57/32, 19/32] , [57/64, 91/128, 23/64, 237/128, 105/64, 35/64] , [105/128, 103/128, 205/512, 903/512, 849/512, 283/512] ] $

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

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

See Matrix

[y₁, -y₁ + y₂, y₂, 0, 0, y₂]

p = s ² - s ³ p' = - s ² + s ³

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

See Matrix

[y₁ - y₂ + y₃, y₁, 0, y₂, y₃, 0]

p = - s ³ + s ⁴

» SYNC'D 1/16 , 0.06250000000

25 . Coloring, {3, 5, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [23, 41, 48, 44, 45, 47] . FixedPtCheck, [23, 41, 48, 44, 45, 47]

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 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 0, 0] ] $ x $ [ [99/100, 3/100, -9/100, 3/100, 0, 0] , [3/100, 91/100, 27/100, -9/100, 0, 0] , [-9/100, 27/100, 19/100, 27/100, 0, 0] , [3/100, -9/100, 27/100, 91/100, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [1/9, -43/18, 10/3, -40/9, 32/9] , [-5/9, 77/18, -8/3, 56/9, -64/9] , [19/9, -79/18, 10/3, -40/9, 32/9] , [-1, -83/18, 4/3, -8/3, 64/9] , [1/18, 14/3, -11/3, 40/9, -16/3] , [5/18, 22/9, -5/3, 8/9, -16/9] ] $ x $ [ [3/2, 1, 1/2, 1, 1, 1] , [3/2, 11/8, 5/8, 1, 5/8, 7/8] , [9/8, 43/32, 23/32, 19/16, 23/32, 29/32] , [39/32, 137/128, 79/128, 19/16, 107/128, 137/128] , [183/128, 605/512, 293/512, 259/256, 389/512, 535/512] ] $

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

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

See Matrix

[0, -y₁ + 4 y₂ - y₃, y₁, y₂, y₂, y₃]

p = - s + s ⁴ p' = - s + s ⁴

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

» SYNC'D 25/256 , 0.09765625000

26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

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

For τ=1/2, [41, 71, 84, 80, 83, 81] . FixedPtCheck, [41, 71, 84, 80, 83, 81]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

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

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	5 vs 5	5 vs 5	5 vs 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 0, 0] ] $ x $ [ [99/100, 3/100, -9/100, 3/100, 0, 0] , [3/100, 91/100, 27/100, -9/100, 0, 0] , [-9/100, 27/100, 19/100, 27/100, 0, 0] , [3/100, -9/100, 27/100, 91/100, 0, 0] , [0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 1] ] $ =
$ [ [-1/6, 47/6, -25/6, 22/3, -32/3] , [-1/6, -61/6, 47/6, -32/3, 40/3] , [11/6, 35/6, -25/6, 22/3, -32/3] , [-7/6, 47/6, -43/6, 34/3, -32/3] , [1/3, -26/3, 35/6, -32/3, 40/3] , [1/3, -8/3, 11/6, -14/3, 16/3] ] $ x $ [ [3/2, 1, 1/2, 1, 1, 1] , [3/2, 11/8, 5/8, 1, 7/8, 5/8] , [9/8, 41/32, 23/32, 5/4, 29/32, 23/32] , [39/32, 131/128, 77/128, 19/16, 143/128, 109/128] , [189/128, 577/512, 287/512, 67/64, 533/512, 383/512] ] $

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

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

[0, y₁, y₂, y₃, y₄, y₅]

See Matrices

R = $ [ [0, 0, 1, 0, 0, 0] , [0, 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] ] $ 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] ] $ = $ [ [5/6, -1/6, -1/6, -1/6, -1/6] , [5/6, -1/6, -1/6, -1/6, -1/6] , [-1/6, 5/6, -1/6, -1/6, -1/6] , [-1/6, -1/6, -1/6, 5/6, -1/6] , [-1/6, -1/6, 5/6, -1/6, -1/6] , [-1/6, -1/6, -1/6, -1/6, 5/6] ] $ x $ [ [0, 1, 2, 1, 1, 1] , [0, 1, 1, 1, 2, 1] , [0, 1, 1, 2, 1, 1] , [0, 1, 1, 1, 1, 2] , [0, 2, 1, 1, 1, 1] ] $

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

» SYNC'D 55/256 , 0.2148437500

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

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

` See graph

` ` See pair graph

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

For τ=1/2, [-80, -44, -71, -57, -32, -96] . FixedPtCheck, [80, 44, 71, 57, 32, 96]

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	2 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [1/4, 3/4, 0, 0, 0, 0] ] $ x $ [ [91/100, 3/100, -9/100, 27/100, 0, 0] , [3/100, 99/100, 3/100, -9/100, 0, 0] , [-9/100, 3/100, 91/100, 27/100, 0, 0] , [27/100, -9/100, 27/100, 19/100, 0, 0] , [0, 0, 0, 0, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [5/3, 25/18, -10/9, -16/9] , [-5/6, 17/9, 8/9, -16/9] , [-5/6, -13/9, 14/9, 8/9] , [-5/6, -13/9, 14/9, 8/9] , [-1/12, -31/36, -10/9, 20/9] , [23/12, 17/36, -16/9, -4/9] ] $ x $ [ [1, 3/2, 1, 1/2, 3/2, 1/2] , [5/4, 9/8, 11/8, 3/4, 9/8, 3/8] , [15/16, 39/32, 37/32, 9/16, 51/32, 17/32] , [85/64, 141/128, 147/128, 45/64, 165/128, 55/128] ] $

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

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

[y₁, 0, y₂, y₃, 0, y₄]

See Matrices

R = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [1, 0, 0, 0, 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, 1] ] $ = $ [ [0, -5/18, 1/18, 7/18] , [1/2, -5/18, 1/18, -1/9] , [0, 7/18, -5/18, 1/18] , [0, 7/18, -5/18, 1/18] , [1/2, -5/18, 1/18, -1/9] , [0, 1/18, 7/18, -5/18] ] $ x $ [ [1, 0, 1, 2, 0, 2] , [2, 0, 1, 0, 0, 3] , [3, 0, 2, 0, 0, 1] , [1, 0, 3, 0, 0, 2] ] $

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

See Matrix

[y₁, y₂, y₁, 0, y₂, 0]

p = - s + s ³ p' = - s + s ³

» SYNC'D 15/32 , 0.4687500000

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

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

` See graph

` ` See pair graph

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

For τ=1/2, [48, 84, 57, 71, 32, 96] . FixedPtCheck, [48, 84, 57, 71, 32, 96]

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 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 3/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [0, 1/2, -19/9, -32/9, 16/3] , [0, 5/2, 11/9, 16/9, -16/3] , [0, -3/2, 5/3, -16/9, 16/9] , [0, -3/2, 5/3, -16/9, 16/9] , [-1/2, 1, -11/9, 8/3, -16/9] , [3/2, -1, -11/9, 8/3, -16/9] ] $ x $ [ [1, 1, 1, 1, 3/2, 1/2] , [3/4, 7/8, 1, 11/8, 3/2, 1/2] , [3/4, 11/16, 27/32, 43/32, 57/32, 19/32] , [57/64, 91/128, 45/64, 193/128, 105/64, 35/64] , [105/128, 103/128, 387/512, 721/512, 849/512, 283/512] ] $

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

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

See Matrix

[-y₁ + y₃, y₁, -y₂ + y₃, y₂, 0, y₃]

p = - s ³ + s ⁴ p = - s ³ + s ⁵

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

See Matrix

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

p = - s ⁴ + s ⁵

» SYNC'D 1/16 , 0.06250000000

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

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

` See graph

` ` See pair graph

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

For τ=1/2, [-29, -41, -32, -84, -71, -45] . FixedPtCheck, [29, 41, 32, 84, 71, 45]

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	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 1] ] $ =
$ [ [-69/58, 17/29, -299/29, 220/29, -1040/87, 448/29] , [3/2, -1, 17, -12, 16, -64/3] , [-93/58, 33/29, -403/29, 352/29, -400/29, 1408/87] , [47/58, -41/29, -299/29, 220/29, -1040/87, 448/29] , [45/58, -59/29, 340/29, -320/29, 448/29, -1280/87] , [41/58, 79/29, 168/29, -124/29, 544/87, -320/29] ] $ x $ [ [3/2, 1, 1, 1/2, 1, 1] , [3/2, 11/8, 9/8, 1/2, 7/8, 5/8] , [9/8, 41/32, 45/32, 9/16, 31/32, 21/32] , [39/32, 129/128, 159/128, 9/16, 153/128, 99/128] , [189/128, 567/512, 543/512, 141/256, 549/512, 375/512] , [693/512, 2643/2048, 2457/2048, 279/512, 1911/2048, 1389/2048] ] $

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

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

See Matrix

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

p = s ⁴ - s ⁵

Omega Rank for B : cycles: {{1, 2, 3, 5}}, net cycles: 0 . order: 4

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

» SYNC'D 27/256 , 0.1054687500

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

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

` See graph

` ` See pair graph

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

For τ=1/2, [35, 71, 44, 96, 57, 83] . FixedPtCheck, [35, 71, 44, 96, 57, 83]

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

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

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

+ \ ; - \ ; Δ

See Matrices

[-y₁, y₂, -y₂, y₁, -y₂, 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	6 vs 6	6 vs 6	3 vs 5	4 vs 5

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 1/4, 3/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 1] ] $ =
$ [ [-2/3, 1/2, 14/3, -31/9, 40/9, -16/3] , [2/3, -5/6, -14/3, 37/9, -8, 80/9] , [-1, 5/6, 4, -1, 56/9, -80/9] , [4/3, -3/2, 14/3, -31/9, 40/9, -16/3] , [1/6, 8/3, -14/3, 34/9, -16/3, 32/9] , [1/2, -5/3, -4, 0, -16/9, 64/9] ] $ x $ [ [3/2, 1, 1, 1/2, 1, 1] , [3/2, 11/8, 9/8, 1/2, 5/8, 7/8] , [9/8, 43/32, 45/32, 1/2, 21/32, 31/32] , [39/32, 139/128, 165/128, 1/2, 93/128, 151/128] , [183/128, 619/512, 573/512, 29/64, 357/512, 559/512] , [687/512, 2755/2048, 2589/2048, 61/128, 1269/2048, 1951/2048] ] $

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

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

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{1, 2, 3, 6}}, net cycles: 0 . order: 4

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

» SYNC'D 47/256 , 0.1835937500

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

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

` See graph

` ` See pair graph

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

For τ=1/2, [8, 20, 21, 11, 8, 24] . FixedPtCheck, [8, 20, 21, 11, 8, 24]

det(A + τ Δ) = 0

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

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

+ \ ; - \ ; Δ

See Matrices

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

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
3 vs 4	4 vs 4	4 vs 4	4 vs 4	3 vs 4

See Matrix for A+τΔ

bi =

$ [ [0, 3/4, 1/4, 0, 0, 0] , [0, 0, 1/4, 3/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 0, 0] ] $ x $ [ [99/100, 3/100, -9/100, 3/100, 0, 0] , [3/100, 91/100, 27/100, -9/100, 0, 0] , [-9/100, 27/100, 19/100, 27/100, 0, 0] , [3/100, -9/100, 27/100, 91/100, 0, 0] , [0, 0, 0, 0, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [7/12, 19/108, 52/27, -68/27] , [-29/12, 55/108, -20/27, 76/27] , [19/12, -185/108, 28/27, -20/27] , [19/12, -185/108, 28/27, -20/27] , [-2/3, 77/54, -56/27, 40/27] , [1/3, 71/54, -32/27, -8/27] ] $ x $ [ [3/2, 1, 1/2, 1, 3/2, 1/2] , [3/2, 5/4, 5/8, 9/8, 9/8, 3/8] , [9/8, 39/32, 11/16, 39/32, 21/16, 7/16] , [21/16, 61/64, 75/128, 159/128, 183/128, 61/128] ] $

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

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

[0, y₁, y₂, y₃, 0, y₄]

See Matrices

R = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 1, 0, 0, 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] ] $ = $ [ [0, -5/18, 1/18, 7/18] , [0, -5/18, 1/18, 7/18] , [0, 7/18, -5/18, 1/18] , [0, 7/18, -5/18, 1/18] , [1, -5/18, 1/18, -11/18] , [0, 1/18, 7/18, -5/18] ] $ x $ [ [0, 1, 2, 1, 0, 2] , [0, 2, 1, 0, 0, 3] , [0, 3, 2, 0, 0, 1] , [0, 1, 3, 0, 0, 2] ] $

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

See Matrix

[y₁ - y₂ + y₃, y₁, 0, y₂, y₃, 0]

p = - s + s ² - s ³ + s ⁴

» SYNC'D 3/8 , 0.3750000000

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

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

` See graph

` ` See pair graph

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

For τ=1/2, [8, 20, 11, 21, 8, 24] . FixedPtCheck, [8, 20, 11, 21, 8, 24]

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, 3/4, 1/4, 0, 0, 0] , [0, 0, 3/4, 1/4, 0, 0] , [0, 0, 0, 0, 3/4, 1/4] , [0, 0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 1/4, 0, 0] , [3/4, 1/4, 0, 0, 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, 9/10, 3/10] , [0, 0, 0, 0, 3/10, 1/10] ] $ =
$ [ [0, -1/12, 187/108, -68/27, 28/27] , [0, 19/12, 7/108, 76/27, -116/27] , [0, -29/12, 55/108, -20/27, 76/27] , [0, -29/12, 55/108, -20/27, 76/27] , [-1/2, 3/2, -109/54, 40/27, -8/27] , [3/2, 11/6, -43/54, -8/27, -56/27] ] $ x $ [ [3/2, 1, 1, 1/2, 3/2, 1/2] , [3/2, 5/4, 9/8, 5/8, 9/8, 3/8] , [9/8, 39/32, 21/16, 19/32, 21/16, 7/16] , [21/16, 61/64, 153/128, 81/128, 183/128, 61/128] , [183/128, 565/512, 267/256, 305/512, 351/256, 117/256] ] $

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

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

[0, y₃, y₄, y₂, 0, y₁]

See Matrices

R = $ [ [0, 0, 1, 0, 0, 0] , [0, 0, 0, 1, 0, 0] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 0, 1] , [0, 0, 0, 1, 0, 0] , [0, 1, 0, 0, 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, -5/18, 1/18, -11/18] , [0, -5/18, 1/18, 7/18] , [0, 7/18, -5/18, 1/18] , [0, 7/18, -5/18, 1/18] , [0, -5/18, 1/18, 7/18] , [0, 1/18, 7/18, -5/18] ] $ x $ [ [0, 1, 1, 2, 0, 2] , [0, 2, 0, 1, 0, 3] , [0, 3, 0, 2, 0, 1] , [0, 1, 0, 3, 0, 2] ] $

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

See Matrix

[y₁ - y₃ + y₂, y₁, y₃, 0, y₂, 0]

p = - s + s ² - s ³ + s ⁴

» SYNC'D 3/8 , 0.3750000000

SUMMARY

Graph Type
CC

ν(A)
2

ν(Δ)
2

π
[1, 1, 1, 1, 1, 1]

Dbly Stoch
true

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

SANDWICH
Total 2

No . Coloring Rank

1 {} 3

2 {2} 3

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

RT GROUPS
Total 3

No . Coloring Rank Solv

1 {2, 3, 6} 6 ["group", Not Solvable]

2 {2, 3, 5} 2 Not Solvable

3 {2, 4, 6} 6 ["group", Not Solvable]

RT GROUPS		Total 3
No .	Coloring	Rank	Solv
1	{2, 3, 6}	6	["group", Not Solvable]
2	{2, 3, 5}	2	Not Solvable
3	{2, 4, 6}	6	["group", Not Solvable]

CC Colorings
Total 1

No . Coloring Sandwich,Rank

1 {} true, 3

CC Colorings		Total 1
No .	Coloring	Sandwich,Rank
1	{}	true, 3

Δ-RANK'D SC'D !RK'D τ-RANK'D R/B RANK'D NOT SYNC'D Total Runs 2^n-1
22 0 24 , 27 7 , 6 5 32 32