New Graph

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

π = [2, 2, 1, 1, 3]

POSSIBLE RANKS

1 x 9
3 x 3

BASE DETERMINANT 351/4096, .8569335938e-1

NullSpace of Δ

{1, 2, 3, 4, 5}

Nullspace of A

` det(A) = ` -1/16

1 . Coloring, {}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 3 + τ ² ` )` , 9` (` - 1 + τ ` )` ² ` (` - 3 + τ ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 9` (` 3 + τ ² ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-20, -52, -5, -13, -65] . FixedPtCheck, [20, 52, 5, 13, 65]

det(A + τ Δ) = 1` (` - 1 + τ ` )` ² ` (` 1 + 3τ ² ` )`

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 0, 3/4] , [0, 3/4, 0, 0, 1/4] , [3/4, 1/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [-5/76, -65/228, -37/19, 44/171, 368/171] , [53/38, 119/114, -6/19, -568/171, 224/171] , [-12/19, -23/57, 44/57, -64/171, 128/171] , [-27/76, 409/228, 115/57, 116/171, -688/171] , [-17/76, -221/228, 11/19, 332/171, -208/171] ] $ x $ [ [5/2, 3/2, 3/2, 3/2, 2] , [15/8, 13/8, 15/8, 9/8, 5/2] , [75/32, 47/32, 45/32, 39/32, 41/16] , [291/128, 199/128, 225/128, 141/128, 37/16] , [1113/512, 719/512, 873/512, 597/512, 653/256] ] $

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

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

[y₁, y₂, 0, 0, y₃]

See Matrices

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

See Matrix

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

p' = - 1 - s + s ³ + s ⁴

» SYNC'D 45/512 , 0.08789062500

2 . Coloring, {2}

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

` See graph

` ` See pair graph

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

For τ=1/2, [4, 12, 1, 9, 13] . FixedPtCheck, [4, 12, 1, 9, 13]

det(A + τ Δ) = 1` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 1 + 3τ ² ` )`

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

[2, 2, 1, 1, 3]

+ \ ; - \ ; Δ

See Matrices

[y₁, y₂, -y₂, -y₁, 0]

p = s + 4s ³

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s + 4s ³

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

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

See Matrix

[y₂ - y₁, y₂, 0, y₁, y₂]

p = s ² - s ⁴ p' = s ² - s ³

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

See Matrix

[y₂, y₂ - y₁, y₁, 0, y₂]

p = - s ² + s ³ p = - s ² + s ⁴

« NOT SYNC'D »

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

Range of {ΩΔⁱ}: [-μ₂, -μ₁, μ₁, μ₂, 0]

rank of M is 5 , rank of N is 3

M N

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

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

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

τ= 9 , r'= 2/3

Ranges

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

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

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

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

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}

Rank 3

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

π₂ [1, 1, 0, 2, 0, 1, 2, 0, 1, 1]

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

wpp [2, 2, 2, 2, 1]

π₃ [0, 0, 1, 0, 1, 0, 0, 0, 1, 0]

u₃ [0, 0, 1, 0, 1, 0, 0, 0, 1, 1]

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

3 . Coloring, {3}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `18` (` - 1 + τ ` )` , -18` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` ² , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 3 + τ ` )`` (` 1 + τ ` )``]`

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

det(A + τ Δ) = 1` (` - 1 + τ ` )` ³ ` (` 1 + τ ` )`

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

[2, 2, 1, 1, 3]

+ \ ; - \ ; Δ

See Matrices

[y₁, y₂, 6 y₁ + 5 y₂ + 2 y₃, -7 y₁ - 6 y₂ - 3 y₃, y₃]

p = s + 2s ² - 4s ³ - 8s 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 2	2 vs 4

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

[0, y₂, 0, 0, y₁]

See Matrices

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

See Matrix

[2 y₁, y₂, 2 y₂, y₁, 0]

p = - s + s ³ p' = - s + s ³

» SYNC'D 9/128 , 0.07031250000

4 . Coloring, {4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` ² , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 3 + τ ² ` )``]`

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

det(A + τ Δ) = 1` (` - 1 + τ ` )` ³ ` (` 1 + τ ` )`

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 0, 3/4] , [0, 1/4, 0, 0, 3/4] , [3/4, 1/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [-93/1001, -262/3003, -2752/3003, 12928/9009, -2048/9009] , [1104/1001, -1249/3003, -5692/3003, -6224/9009, 18112/9009] , [142/1001, -235/3003, -996/1001, -13712/9009, 23104/9009] , [-173/1001, 3818/3003, 736/3003, -8672/9009, -2432/9009] , [-30/91, -17/273, 580/273, 272/819, -1600/819] ] $ x $ [ [5/2, 1, 3/2, 3/2, 5/2] , [9/4, 1, 15/8, 3/4, 25/8] , [45/16, 31/32, 27/16, 3/4, 89/32] , [321/128, 113/128, 135/64, 93/128, 355/128] , [1335/512, 7/8, 963/512, 339/512, 1523/512] ] $

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

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

[y₁, y₃, 0, 0, y₂]

See Matrices

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

[y₄, 0, y₃, y₂, y₁]

See Matrices

B = $ [ [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] , [0, 0, 0, 0, 1] , [1, 0, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 0, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ = $ [ [0, 1/27, -8/27, 10/27] , [1/2, 1/27, -8/27, -7/54] , [0, 10/27, 1/27, -8/27] , [0, 10/27, 1/27, -8/27] , [0, -8/27, 10/27, 1/27] ] $ x $ [ [3, 0, 2, 2, 2] , [2, 0, 3, 0, 4] , [4, 0, 2, 0, 3] , [3, 0, 4, 0, 2] ] $

» SYNC'D 9/128 , 0.07031250000

5 . Coloring, {5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -18` (` 3 + τ ² ` )`` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , 9` (` 3 + τ ² ` )`` (` - 1 + τ ` )` ² , -9` (` 3 + τ ² ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [180, 52, 45, 13, 195] . FixedPtCheck, [180, 52, 45, 13, 195]

det(A + τ Δ) = 1` (` - 1 + τ ` )` ³ ` (` 1 + τ ` )`

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 0, 3/4] , [0, 3/4, 0, 0, 1/4] , [1/4, 3/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [49/37, -8/111, -1180/333, -160/333, 320/111] , [-13/111, -16/333, -140/999, 32/37, -448/999] , [-28/111, 401/333, 40/999, -112/111, 128/999] , [8/37, 7/111, -392/333, -304/333, 640/333] , [-17/37, -38/111, 944/333, 128/333, -256/111] ] $ x $ [ [1, 3, 3/2, 3/2, 2] , [7/8, 21/8, 3/4, 9/4, 5/2] , [13/16, 57/16, 21/32, 63/32, 2] , [85/128, 381/128, 39/64, 171/64, 133/64] , [43/64, 57/16, 255/512, 1143/512, 521/256] ] $

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

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

[y₁, 0, 0, 0, y₂]

See Matrices

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

[0, y₄, y₂, y₃, y₁]

See Matrices

B = $ [ [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] , [0, 1, 0, 0, 0] , [0, 1, 0, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ = $ [ [1/2, -1/4, -5/18, 5/36] , [0, 0, -1/9, 2/9] , [0, 1/2, -1/9, -5/18] , [0, 0, 2/9, -1/9] , [0, 0, 2/9, -1/9] ] $ x $ [ [0, 4, 2, 2, 1] , [0, 3, 0, 4, 2] , [0, 6, 0, 3, 0] , [0, 3, 0, 6, 0] ] $

» SYNC'D 1/16 , 0.06250000000

6 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `18` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 18` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -9` (` - 1 + τ ` )` ² ` (` 3 + τ ² ` )` , 9` (` 1 + τ ` )` ³ ` (` - 3 + τ ` )` , 9` (` 1 + τ ` )`` (` 3 + τ ² ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-52, -180, -13, -135, -195] . FixedPtCheck, [52, 180, 13, 135, 195]

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

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 0, 1/4] , [0, 3/4, 0, 0, 1/4] , [3/4, 1/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [1/2, -19/18, -38/27, 88/27, -32/27] , [2/3, 7, -380/27, -272/27, 448/27] , [1, 16/9, -200/27, -128/27, 256/27] , [37/6, -43/6, -362/27, 232/27, 160/27] , [-17/6, -13/6, 466/27, 88/27, -416/27] ] $ x $ [ [3, 3/2, 3/2, 1/2, 5/2] , [3, 1, 9/4, 3/8, 19/8] , [111/32, 7/8, 9/4, 1/4, 69/32] , [423/128, 93/128, 333/128, 7/32, 275/128] , [57/16, 359/512, 1269/512, 93/512, 1063/512] ] $

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

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

[0, y₁, 0, y₃, y₂]

See Matrices

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

[y₂, y₃, y₁, 0, y₄]

See Matrices

B = $ [ [0, 0, 1, 0, 0] , [0, 0, 0, 0, 1] , [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [1, 0, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 0, 0] , [0, 0, 0, 0, 1] ] $ = $ [ [0, 0, -4/9, 5/9] , [0, 1, -4/9, -4/9] , [0, 0, 5/9, -4/9] , [1, -2, -4/9, 14/9] , [0, 0, 5/9, -4/9] ] $ x $ [ [4, 1, 2, 0, 2] , [4, 0, 4, 0, 1] , [5, 0, 4, 0, 0] , [4, 0, 5, 0, 0] ] $

» SYNC'D 7/64 , 0.1093750000

7 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` 3 + τ ` )`` (` - 1 + τ ` )` ² , -18` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 9` (` 3 + τ ` )`` (` - 1 + τ ` )` ³ , -9` (` 1 + τ ` )` ² ` (` 3 + τ ² ` )` , 9` (` 3 + τ ` )`` (` 3 + τ ² ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-28, -156, -7, -117, -91] . FixedPtCheck, [28, 156, 7, 117, 91]

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

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 3

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 0, 3/4] , [0, 1/4, 0, 0, 3/4] , [3/4, 1/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [-172/197, 871/591, 2468/591, 112/1773, -8384/1773] , [799/394, -4723/1182, -3334/591, -4136/1773, 17824/1773] , [41/394, -841/1182, 38/591, 3496/1773, -2336/1773] , [-487/197, 3844/591, 7304/591, 10048/1773, -38912/1773] , [139/394, -295/1182, -1870/591, -1832/1773, 7456/1773] ] $ x $ [ [5/2, 1, 3/2, 1/2, 7/2] , [3, 1, 15/8, 1/4, 23/8] , [21/8, 25/32, 9/4, 1/4, 99/32] , [369/128, 107/128, 63/32, 25/128, 399/128] , [1449/512, 53/64, 1107/512, 107/512, 1521/512] ] $

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

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

[y₁, y₂, 0, y₄, y₃]

See Matrices

R = $ [ [0, 0, 0, 0, 1] , [0, 0, 0, 1, 0] , [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 1, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 0, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ = $ [ [0, 1, -4/9, -4/9] , [0, 0, -4/9, 5/9] , [1, -2, -4/9, 14/9] , [0, 0, 5/9, -4/9] , [0, 0, 5/9, -4/9] ] $ x $ [ [1, 4, 0, 2, 2] , [0, 4, 0, 4, 1] , [0, 5, 0, 4, 0] , [0, 4, 0, 5, 0] ] $

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

[y₃, 0, y₂, 0, y₁]

See Matrices

» SYNC'D 7/64 , 0.1093750000

8 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` 1 + τ ` )` , 18` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 3 + τ ² ` )``]`

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

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

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 0, 3/4] , [0, 3/4, 0, 0, 1/4] , [1/4, 3/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [15/26, -7/78, 4/3, 56/117, -256/117] , [-1/6, -1/6, -4/9, 8/9, 0] , [7/39, 46/39, -10/9, -112/117, 32/39] , [9/13, -38/39, -14/3, -16/117, 608/117] , [-3/13, 4/39, 4/3, -64/117, -64/117] ] $ x $ [ [1, 3, 3/2, 1/2, 3] , [9/8, 21/8, 3/4, 3/4, 15/4] , [9/8, 27/8, 27/32, 21/32, 3] , [123/128, 351/128, 27/32, 27/32, 231/64] , [285/256, 855/256, 369/512, 351/512, 201/64] ] $

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

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

[y₁, 0, 0, y₃, y₂]

See Matrices

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

[0, y₁, y₂, 0, y₃]

See Matrices

» SYNC'D 3/16 , 0.1875000000

9 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , -18` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , 9` (` - 1 + τ ` )` ² ` (` 3 + τ ² ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -9` (` 3 + τ ² ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [52, 180, 13, 45, 195] . FixedPtCheck, [52, 180, 13, 45, 195]

det(A + τ Δ) = 1` (` - 1 + τ ` )` ³ ` (` 1 + τ ` )`

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 0, 1/4] , [0, 1/4, 0, 0, 3/4] , [3/4, 1/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [-13/111, -16/333, -140/999, 32/37, -448/999] , [49/37, -8/111, -1180/333, -160/333, 320/111] , [8/37, 7/111, -392/333, -304/333, 640/333] , [-28/111, 401/333, 40/999, -112/111, 128/999] , [-17/37, -38/111, 944/333, 128/333, -256/111] ] $ x $ [ [3, 1, 3/2, 3/2, 2] , [21/8, 7/8, 9/4, 3/4, 5/2] , [57/16, 13/16, 63/32, 21/32, 2] , [381/128, 85/128, 171/64, 39/64, 133/64] , [57/16, 43/64, 1143/512, 255/512, 521/256] ] $

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

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

[0, y₂, 0, 0, y₁]

See Matrices

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

[y₁, 0, y₃, y₄, y₂]

See Matrices

B = $ [ [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [1, 0, 0, 0, 0] , [0, 0, 0, 0, 1] , [1, 0, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 0, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ = $ [ [0, 0, -1/9, 2/9] , [1/2, -1/4, -5/18, 5/36] , [0, 0, 2/9, -1/9] , [0, 1/2, -1/9, -5/18] , [0, 0, 2/9, -1/9] ] $ x $ [ [4, 0, 2, 2, 1] , [3, 0, 4, 0, 2] , [6, 0, 3, 0, 0] , [3, 0, 6, 0, 0] ] $

» SYNC'D 1/16 , 0.06250000000

10 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` 1 + τ ` )` , 18` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` ² , 9` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

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

det(A + τ Δ) = 1` (` - 1 + τ ` )` ³ ` (` 1 + τ ` )`

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

[2, 2, 1, 1, 3]

+ \ ; - \ ; Δ

See Matrices

[y₂, y₁, -6 y₂ - 7 y₁ - 3 y₃, 5 y₂ + 6 y₁ + 2 y₃, y₃]

p = s + 2s ² - 4s ³ - 8s 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 2	2 vs 4

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

[y₁, 0, 0, 0, y₂]

See Matrices

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

See Matrix

[y₁, 2 y₂, y₂, 2 y₁, 0]

p = - s + s ³ p' = - s + s ³

» SYNC'D 9/128 , 0.07031250000

11 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `18` (` 1 + τ ` )` , -18` (` - 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` ² , 9` (` 3 + τ ² ` )``]`

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

det(A + τ Δ) = 1` (` - 1 + τ ` )` ³ ` (` 1 + τ ` )`

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4] , [1/4, 0, 0, 0, 3/4] , [0, 1/4, 0, 0, 3/4] , [1/4, 3/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [1104/1001, -1249/3003, -5692/3003, -6224/9009, 18112/9009] , [-93/1001, -262/3003, -2752/3003, 12928/9009, -2048/9009] , [-173/1001, 3818/3003, 736/3003, -8672/9009, -2432/9009] , [142/1001, -235/3003, -996/1001, -13712/9009, 23104/9009] , [-30/91, -17/273, 580/273, 272/819, -1600/819] ] $ x $ [ [1, 5/2, 3/2, 3/2, 5/2] , [1, 9/4, 3/4, 15/8, 25/8] , [31/32, 45/16, 3/4, 27/16, 89/32] , [113/128, 321/128, 93/128, 135/64, 355/128] , [7/8, 1335/512, 339/512, 963/512, 1523/512] ] $

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

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

[y₂, y₃, 0, 0, y₁]

See Matrices

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

[0, y₁, y₂, y₃, y₄]

See Matrices

B = $ [ [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] , [0, 0, 0, 0, 1] , [0, 1, 0, 0, 0] ] $ x $ [ [0, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ = $ [ [1/2, 1/27, -8/27, -7/54] , [0, 1/27, -8/27, 10/27] , [0, 10/27, 1/27, -8/27] , [0, 10/27, 1/27, -8/27] , [0, -8/27, 10/27, 1/27] ] $ x $ [ [0, 3, 2, 2, 2] , [0, 2, 0, 3, 4] , [0, 4, 0, 2, 3] , [0, 3, 0, 4, 2] ] $

» SYNC'D 9/128 , 0.07031250000

12 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` - 1 + τ ` )` ² ` (` 3 + τ ` )` , 18` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , 9` (` - 1 + τ ` )` ³ ` (` 3 + τ ` )` , 9` (` 1 + τ ` )` ³ ` (` - 3 + τ ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-28, -180, -7, -135, -105] . FixedPtCheck, [28, 180, 7, 135, 105]

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

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 0, 1/4] , [0, 1/4, 0, 0, 3/4] , [3/4, 1/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [1/2, 1/2, -32/9, -40/9, 64/9] , [-1/2, -7/6, 8, 8/9, -64/9] , [0, -1, 10/9, 32/9, -32/9] , [4, 1/3, -22, -64/9, 224/9] , [-1, 2/3, 4, 32/9, -64/9] ] $ x $ [ [3, 1, 3/2, 1/2, 3] , [27/8, 7/8, 9/4, 1/4, 9/4] , [27/8, 5/8, 81/32, 7/32, 9/4] , [459/128, 79/128, 81/32, 5/32, 135/64] , [891/256, 145/256, 1377/512, 79/512, 135/64] ] $

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

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

[0, y₃, 0, y₁, y₂]

See Matrices

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

[y₃, 0, y₂, 0, y₁]

See Matrices

» SYNC'D 3/16 , 0.1875000000

13 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` 3 + τ ² ` )` , -18` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 9` (` 3 + τ ² ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-52, -20, -13, -15, -65] . FixedPtCheck, [52, 20, 13, 15, 65]

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

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4] , [3/4, 0, 0, 0, 1/4] , [0, 3/4, 0, 0, 1/4] , [1/4, 3/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [3/5, 62/15, 48/5, -128/45, -512/45] , [-4/15, -19/5, -212/45, 208/45, 64/15] , [14/15, 39/5, 172/45, -368/45, -64/15] , [3, -14/3, -16, 32/9, 128/9] , [-6/5, -19/15, 4/5, 16/45, 64/45] ] $ x $ [ [3/2, 3, 3/2, 1/2, 5/2] , [7/4, 9/4, 9/8, 3/4, 25/8] , [13/8, 93/32, 21/16, 9/16, 83/32] , [209/128, 303/128, 39/32, 93/128, 391/128] , [859/512, 363/128, 627/512, 303/512, 1367/512] ] $

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

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

[y₁, 0, 0, y₃, y₂]

See Matrices

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

See Matrix

[y₂, 2 y₁, y₁, 0, 2 y₂]

p = - s + s ³ p' = - s + s ³

» SYNC'D 3/128 , 0.02343750000

14 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `18` (` 3 + τ ` )`` (` 1 + τ ` )` , 18` (` 3 + τ ² ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )` , 9` (` 3 + τ ² ` )`` (` 1 + τ ` )` , 9` (` 3 + τ ` )`` (` 3 + τ ² ` )``]`

For τ=1/2, [84, 52, 21, 39, 91] . FixedPtCheck, [84, 52, 21, 39, 91]

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

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 3

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 1/4, 3/4] , [1/4, 0, 0, 0, 3/4] , [0, 1/4, 0, 0, 3/4] , [1/4, 3/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [14/27, -7/243, 596/243, 16/243, -704/243] , [-7/27, -118/243, 80/243, 640/243, -512/243] , [5/27, 254/243, -304/243, 160/243, -128/243] , [20/27, -253/243, -1324/243, 208/243, 1216/243] , [-4/27, 83/243, 92/243, -560/243, 448/243] ] $ x $ [ [1, 5/2, 3/2, 1/2, 7/2] , [5/4, 11/4, 3/4, 5/8, 29/8] , [35/32, 23/8, 15/16, 11/16, 109/32] , [139/128, 349/128, 105/128, 23/32, 467/128] , [143/128, 1493/512, 417/512, 349/512, 1777/512] ] $

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

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

See Matrix

[2 y₂, y₁, 0, y₂, 2 y₁]

p = - s + s ³ p' = - s + s ³

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

[0, y₁, y₂, 0, y₃]

See Matrices

» SYNC'D 3/128 , 0.02343750000

15 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-18` (` 3 + τ ² ` )` , -18` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 9` (` - 1 + τ ` )` ² ` (` - 3 + τ ` )` , 9` (` 3 + τ ² ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-52, -20, -13, -5, -65] . FixedPtCheck, [52, 20, 13, 5, 65]

det(A + τ Δ) = 1` (` - 1 + τ ` )` ² ` (` 1 + 3τ ² ` )`

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

See Matrix for A+τΔ

bi =

$ [ [0, 0, 3/4, 0, 1/4] , [0, 0, 0, 3/4, 1/4] , [3/4, 0, 0, 0, 1/4] , [0, 1/4, 0, 0, 3/4] , [1/4, 3/4, 0, 0, 0] ] $ x $ [ [1, 0, 0, 0, 0] , [0, 1, 0, 0, 0] , [0, 0, 1, 0, 0] , [0, 0, 0, 1, 0] , [0, 0, 0, 0, 1] ] $ =
$ [ [53/38, 119/114, -6/19, -568/171, 224/171] , [-5/76, -65/228, -37/19, 44/171, 368/171] , [-27/76, 409/228, 115/57, 116/171, -688/171] , [-12/19, -23/57, 44/57, -64/171, 128/171] , [-17/76, -221/228, 11/19, 332/171, -208/171] ] $ x $ [ [3/2, 5/2, 3/2, 3/2, 2] , [13/8, 15/8, 9/8, 15/8, 5/2] , [47/32, 75/32, 39/32, 45/32, 41/16] , [199/128, 291/128, 141/128, 225/128, 37/16] , [719/512, 1113/512, 597/512, 873/512, 653/256] ] $

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

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

[y₁, y₂, 0, 0, y₃]

See Matrices

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

See Matrix

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

p' = - 1 - s + s ³ + s ⁴

» SYNC'D 45/512 , 0.08789062500

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

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

` See graph

` ` See pair graph

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

For τ=1/2, [-28, -20, -7, -15, -35] . FixedPtCheck, [28, 20, 7, 15, 35]

det(A + τ Δ) = 1` (` - 1 + τ ` )`` (` 1 + 3τ ² ` )`` (` 1 + τ ` )`

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

[2, 2, 1, 1, 3]

+ \ ; - \ ; Δ

See Matrices

[-y₁, -y₂, y₁, y₂, 0]

p = s - 4s ³

S+ \ ; S- \ ; NM

See Matrices

CmmCk true, true, true

p' = s - 4s ³

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

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

See Matrix

[y₂ + y₁, y₂, 0, y₁, y₂ + y₁]

p' = s - s ³ p = - s + s ³

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

See Matrix

[y₁, y₂, -y₁ + y₂, 0, y₂]

p' = - s + s ³ p = - s + s ³

« NOT SYNC'D »

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

Range of {ΩΔⁱ}: [-μ₁, -μ₂, μ₁, μ₂, 0]

rank of M is 5 , rank of N is 3

M N

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

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

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

τ= 9 , r'= 2/3

Ranges

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

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

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

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

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

Rank 3

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

π₂ [1, 0, 1, 2, 1, 0, 2, 0, 1, 1]

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

wpp [2, 2, 2, 2, 1]

π₃ [0, 0, 1, 0, 0, 1, 0, 1, 0, 0]

u₃ [0, 0, 1, 0, 0, 1, 0, 1, 0, 1]

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

SUMMARY

Graph Type
NOT CC

ν(A)
0

ν(Δ)
1

π
[2, 2, 1, 1, 3]

Dbly Stoch
false

SUMMARY
Graph Type		NOT CC
ν(A)		0
ν(Δ)		1
π		[2, 2, 1, 1, 3]
Dbly Stoch		false

SANDWICH
Total 0

No . Coloring Rank

SANDWICH		Total 0
No .	Coloring	Rank

RT GROUPS
Total 2

No . Coloring Rank Solv

1 {2} 3 Not Solvable

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

RT GROUPS		Total 2
No .	Coloring	Rank	Solv
1	{2}	3	Not Solvable
2	{2, 3, 4, 5}	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
12 0 12 , 12 13 , 9 2 16 16