New Graph

             [3, 3, 1, 1, 7, 7, 5, 5], [6, 8, 8, 6, 2, 4, 4, 2]

π = [1, 1, 1, 1, 1, 1, 1, 1]

POSSIBLE RANKS

1 x 8
2 x 4

BASE DETERMINANT 4236243/134217728, .3156246990e-1

NullSpace of Δ

{1, 3, 6, 8}, {2, 4, 5, 7}

Nullspace of A

[{2, 4},{5, 7}] `,` [{1, 3},{6, 8}]

1 . Coloring, {}

Ωp(Δ)=0: p' = s ³ p' = s ⁵ p' = s ⁴ p = s ² p' = s ²

R: [3, 3, 1, 1, 7, 7, 5, 5]
B: [6, 8, 8, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

1 vs 6 1 vs 6 1 vs 6 1 vs 4 1 vs 4

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

Omega Rank for R : cycles: {{5, 7}, {1, 3}} order: 2

See Matrix

[y₁, 0, y₁, 0, y₁, 0, y₁, 0]

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

Omega Rank for B : cycles: {{4, 6}, {2, 8}} order: 2

See Matrix

[0, y₁, 0, y₁, 0, y₁, 0, y₁]

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

` See 4-level graph

M \ ; N

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

τ= 16 , r'= 3/4

R: [3, 3, 1, 1, 7, 7, 5, 5]
B: [6, 8, 8, 6, 2, 4, 4, 2]

Ranges

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

Cycles: R , {{5, 7}, {1, 3}}, B , {{4, 6}, {2, 8}}

β({1, 3, 5, 7}) = 1/2
β({2, 4, 6, 8}) = 1/2

Partitions

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

α([{1, 2}, {7, 8}, {5, 6}, {3, 4}]) = 1/2
α([{6, 7}, {5, 8}, {2, 3}, {1, 4}]) = 1/2

b1 = {1, 2} ` , ` b2 = {6, 7} ` , ` b3 = {5, 8} ` , ` b4 = {7, 8} ` , ` b5 = {5, 6} ` , ` b6 = {2, 3} ` , ` b7 = {3, 4} ` , ` b8 = {1, 4}

Action of R and B on the blocks of the partitions: = [7, 5, 4, 5, 4, 1, 1, 7] [3, 8, 6, 6, 8, 3, 2, 2]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

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

` ` See level-4 partition graph.

Sandwich

Coloring {}

Rank 4

R,B [3, 3, 1, 1, 7, 7, 5, 5], [6, 8, 8, 6, 2, 4, 4, 2]

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

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

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

π₄ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

u₄ [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Sandwich
Coloring	{}
Rank	4
R,B	[3, 3, 1, 1, 7, 7, 5, 5], [6, 8, 8, 6, 2, 4, 4, 2]
π₂	[0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0]
u₂	[1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1] (dim 3)
wpp	[2, 2, 2, 2, 2, 2, 2, 2]
π₄	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u₄	[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]

2 . Coloring, {2}

R: [3, 8, 1, 1, 7, 7, 5, 5]
B: [6, 3, 8, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	3 vs 5	3 vs 5

Omega Rank for R : cycles: {{5, 7}, {1, 3}} order: 2

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴

Omega Rank for B : cycles: {{4, 6}, {2, 3, 8}} order: 6

See Matrix

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

p' = s - s ⁴ p = - s + s ⁴

3 . Coloring, {3}

R: [3, 3, 8, 1, 7, 7, 5, 5]
B: [6, 8, 1, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	4 vs 5	3 vs 5

Omega Rank for R : cycles: {{5, 7}} order: 4

See Matrix

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

p = - s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

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

p' = - s ² + s ⁴ p = - s ² + s ⁴

4 . Coloring, {4}

R: [3, 3, 1, 6, 7, 7, 5, 5]
B: [6, 8, 8, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	3 vs 5	3 vs 5

Omega Rank for R : cycles: {{5, 7}, {1, 3}} order: 2

See Matrix

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

p = - s ² + s ⁴ p' = s ² - s ⁴

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}} order: 6

See Matrix

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

p = - s + s ⁴ p' = s - s ⁴

5 . Coloring, {5}

R: [3, 3, 1, 1, 2, 7, 5, 5]
B: [6, 8, 8, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	4 vs 5	3 vs 5

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

See Matrix

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

p = s ⁴ - s ⁵

Omega Rank for B : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

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

p' = - s ² + s ⁴ p = - s ² + s ⁴

6 . Coloring, {6}

R: [3, 3, 1, 1, 7, 4, 5, 5]
B: [6, 8, 8, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	3 vs 5	3 vs 5

Omega Rank for R : cycles: {{1, 3}, {5, 7}} order: 2

See Matrix

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

p = s ² - s ⁴ p' = s ² - s ⁴

Omega Rank for B : cycles: {{2, 8}, {4, 6, 7}} order: 6

See Matrix

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

p = - s + s ⁴ p' = s - s ⁴

7 . Coloring, {7}

R: [3, 3, 1, 1, 7, 7, 4, 5]
B: [6, 8, 8, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	4 vs 5	3 vs 5

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

See Matrix

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

p = - s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴

8 . Coloring, {8}

R: [3, 3, 1, 1, 7, 7, 5, 2]
B: [6, 8, 8, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 3 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	3 vs 5	3 vs 5

Omega Rank for R : cycles: {{1, 3}, {5, 7}} order: 2

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴

Omega Rank for B : cycles: {{4, 6}, {2, 5, 8}} order: 6

See Matrix

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

p = s - s ⁴ p' = - s + s ⁴

9 . Coloring, {2, 3}

R: [3, 8, 8, 1, 7, 7, 5, 5]
B: [6, 3, 1, 6, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

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

Omega Rank for R : cycles: {{5, 7}} order: 4

See Matrix

[y₁, 0, y₂, 0, y₃, 0, y₄, y₅]

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

See Matrix

[y₅, y₁, y₂, y₃, 0, y₄, 0, 0]

10 . Coloring, {2, 4}

Ωp(Δ)=0: p' = s + 4s ⁵ p = s + 4s ⁵ p' = s ² - 2s ⁴ + 4s ⁵ p' = s ³ - 2s ⁴ + 2s ⁵

R: [3, 8, 1, 6, 7, 7, 5, 5]
B: [6, 3, 8, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 6 3 vs 7 3 vs 7 2 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 6	3 vs 7	3 vs 7	2 vs 6	3 vs 6

Omega Rank for R : cycles: {{1, 3}, {5, 7}} order: 2

See Matrix

[y₁, 0, y₁, 0, 3 y₁ - y₂, y₂, 3 y₁ - y₂, y₂]

p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁵ p = - s ² + s ⁶

Omega Rank for B : cycles: {{1, 4, 6}, {2, 3, 8}} order: 3

See Matrix

[y₁, y₂, y₁, y₂, 0, y₃, 0, y₃]

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

M \ ; N

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

τ= 32 , r'= 1/2

R: [3, 8, 1, 6, 7, 7, 5, 5]
B: [6, 3, 8, 1, 2, 4, 4, 2]

Ranges

Action of R on ranges, [[1], [4], [3], [3]]
Action of B on ranges, [[4], [1], [2], [2]]

Cycles: R , {{1, 3}, {5, 7}}, B , {{1, 4, 6}, {2, 3, 8}}

β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

Partitions

Action of R on partitions, [[7], [4], [3], [5], [5], [4], [3], [7]]
Action of B on partitions, [[2], [8], [8], [2], [1], [6], [6], [1]]

α([{1, 2, 5, 8}, {3, 4, 6, 7}]) = 1/8
α([{3, 4, 5, 8}, {1, 2, 6, 7}]) = 1/8
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/8
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 1/8
α([{1, 4, 7, 8}, {2, 3, 5, 6}]) = 1/8
α([{2, 3, 5, 8}, {1, 4, 6, 7}]) = 1/8
α([{2, 3, 7, 8}, {1, 4, 5, 6}]) = 1/8
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/8

b1 = {3, 4, 7, 8} ` , ` b2 = {1, 2, 5, 6} ` , ` b3 = {1, 4, 7, 8} ` , ` b4 = {2, 3, 5, 6} ` , ` b5 = {3, 4, 5, 8} ` , ` b6 = {1, 2, 6, 7} ` , ` b7 = {1, 2, 7, 8} ` , ` b8 = {3, 4, 5, 6} ` , ` b9 = {2, 3, 5, 8} ` , ` b10 = {1, 4, 6, 7} ` , ` b11 = {1, 4, 5, 8} ` , ` b12 = {2, 3, 6, 7} ` , ` b13 = {2, 3, 7, 8} ` , ` b14 = {1, 4, 5, 6} ` , ` b15 = {1, 2, 5, 8} ` , ` b16 = {3, 4, 6, 7}

Action of R and B on the blocks of the partitions: = [2, 1, 4, 3, 7, 8, 4, 3, 7, 8, D, E, 2, 1, D, E] [C, B, 10, F, C, B, 5, 6, 9, A, 10, F, 9, A, 5, 6]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 4}

Rank 2

R,B [3, 8, 1, 6, 7, 7, 5, 5], [6, 3, 8, 1, 2, 4, 4, 2]

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

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

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

11 . Coloring, {2, 5}

R: [3, 8, 1, 1, 2, 7, 5, 5]
B: [6, 3, 8, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 5 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	5 vs 6	3 vs 6

Omega Rank for R : cycles: {{2, 5, 8}, {1, 3}} order: 6

See Matrix

[3 y₁, 5 y₁ + 5 y₂ - 3 y₃ - 3 y₄ - 3 y₅, 3 y₂, 0, 3 y₃, 0, 3 y₄, 3 y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{4, 6}, {2, 3, 8}} order: 6

See Matrix

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

p = s ² - s ⁶ p' = s ³ - s ⁵ p' = s ² - s ⁴

12 . Coloring, {2, 6}

Ωp(Δ)=0: p = s + 4s ⁴ p' = s + 4s ⁴ p' = s ² + 4s ⁵

R: [3, 8, 1, 1, 7, 4, 5, 5]
B: [6, 3, 8, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 2 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 6	4 vs 8	4 vs 8	2 vs 6	3 vs 6

Omega Rank for R : cycles: {{1, 3}, {5, 7}} order: 2

See Matrix

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

p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁵ p = - s ² + s ⁶

Omega Rank for B : cycles: {{4, 6, 7}, {2, 3, 8}} order: 3

See Matrix

[0, y₃, y₁, y₂, 0, y₃, y₁, y₂]

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

M \ ; N

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

τ= 32 , r'= 1/2

R: [3, 8, 1, 1, 7, 4, 5, 5]
B: [6, 3, 8, 6, 2, 7, 4, 2]

Ranges

Action of R on ranges, [[3], [4], [1], [1]]
Action of B on ranges, [[2], [3], [4], [2]]

Cycles: R , {{1, 3}, {5, 7}}, B , {{4, 6, 7}, {2, 3, 8}}

β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

Partitions

Action of R on partitions, [[3], [2], [2], [4], [4]]
Action of B on partitions, [[5], [4], [1], [1], [4]]

α([{3, 5, 6, 8}, {1, 2, 4, 7}]) = 1/4
α([{2, 3, 4, 5}, {1, 6, 7, 8}]) = 1/8
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 1/8
α([{1, 3, 4, 6}, {2, 5, 7, 8}]) = 3/8
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/8

b1 = {3, 5, 6, 8} ` , ` b2 = {1, 2, 4, 7} ` , ` b3 = {2, 3, 4, 5} ` , ` b4 = {1, 6, 7, 8} ` , ` b5 = {1, 2, 7, 8} ` , ` b6 = {3, 4, 5, 6} ` , ` b7 = {1, 3, 4, 6} ` , ` b8 = {2, 5, 7, 8} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [5, 6, 4, 3, 3, 4, 7, 8, 7, 8] [9, A, 8, 7, 1, 2, 2, 1, 8, 7]
with invariant measure [2, 2, 1, 1, 1, 1, 3, 3, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 6}

Rank 2

R,B [3, 8, 1, 1, 7, 4, 5, 5], [6, 3, 8, 6, 2, 7, 4, 2]

π₂ [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, 0, 0, 0]

u₂ [2, 2, 1, 4, 2, 2, 3, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 3, 3, 2, 3, 4, 2, 2, 1, 3, 2, 1] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

13 . Coloring, {2, 7}

R: [3, 8, 1, 1, 7, 7, 4, 5]
B: [6, 3, 8, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 6 vs 6 5 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	6 vs 6	5 vs 6

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

See Matrix

[y₄, 0, y₅, y₆, y₁, 0, y₂, y₃]

Omega Rank for B : cycles: {{4, 6}, {2, 3, 8}} order: 6

See Matrix

[0, -3 y₃ + 5 y₄ - 3 y₂ + 5 y₁ - 3 y₅, 3 y₃, 3 y₄, 3 y₂, 3 y₁, 0, 3 y₅]

p = s ² + s ³ - s ⁵ - s ⁶

14 . Coloring, {2, 8}

Ωp(Δ)=0: p = s + 4s ⁴ + 8s ⁵ - 16s ⁶ p' = s - 4s ³ - 4s ⁴ + 8s ⁵ p' = s ² + 2s ³ - 4s ⁵

R: [3, 8, 1, 1, 7, 7, 5, 2]
B: [6, 3, 8, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 2 vs 6 1 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 6	4 vs 8	4 vs 8	2 vs 6	1 vs 6

Omega Rank for R : cycles: {{2, 8}, {1, 3}, {5, 7}} order: 2

See Matrix

[3 y₁ - y₂, y₁, y₂, 0, y₂, 0, 3 y₁ - y₂, y₁]

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

Omega Rank for B : cycles: {{4, 6}, {2, 3, 5, 8}} order: 4

See Matrix

[0, y₁, y₁, 2 y₁, y₁, 2 y₁, 0, y₁]

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

M \ ; N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $ $ [ [0, 7, 6, 3, 5, 8, 11, 4] , [7, 0, 5, 6, 6, 5, 4, 11] , [6, 5, 0, 3, 11, 8, 5, 6] , [3, 6, 3, 0, 8, 11, 8, 5] , [5, 6, 11, 8, 0, 3, 6, 5] , [8, 5, 8, 11, 3, 0, 3, 6] , [11, 4, 5, 8, 6, 3, 0, 7] , [4, 11, 6, 5, 5, 6, 7, 0] ] $

τ= 32 , r'= 1/2

R: [3, 8, 1, 1, 7, 7, 5, 2]
B: [6, 3, 8, 6, 2, 4, 4, 5]

Ranges

Action of R on ranges, [[3], [2], [1], [1]]
Action of B on ranges, [[4], [3], [2], [4]]

Cycles: R , {{2, 8}, {1, 3}, {5, 7}}, B , {{4, 6}, {2, 3, 5, 8}}

β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

Partitions

Action of R on partitions, [[1], [4], [3], [2], [1], [3]]
Action of B on partitions, [[6], [6], [2], [2], [4], [5]]

α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 1/11
α([{2, 5, 6, 7}, {1, 3, 4, 8}]) = 3/11
α([{1, 5, 6, 8}, {2, 3, 4, 7}]) = 2/11
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 2/11
α([{1, 2, 4, 5}, {3, 6, 7, 8}]) = 1/11
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 2/11

b1 = {1, 5, 6, 8} ` , ` b2 = {2, 3, 4, 7} ` , ` b3 = {1, 2, 4, 5} ` , ` b4 = {3, 6, 7, 8} ` , ` b5 = {1, 4, 5, 8} ` , ` b6 = {2, 3, 6, 7} ` , ` b7 = {1, 2, 3, 4} ` , ` b8 = {5, 6, 7, 8} ` , ` b9 = {1, 2, 5, 6} ` , ` b10 = {2, 5, 6, 7} ` , ` b11 = {3, 4, 7, 8} ` , ` b12 = {1, 3, 4, 8}

Action of R and B on the blocks of the partitions: = [2, 1, B, 9, 2, 1, C, A, B, 8, 9, 7] [C, A, 8, 7, 4, 3, A, C, 5, 5, 6, 6]
with invariant measure [2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 1, 3]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 8}

Rank 2

R,B [3, 8, 1, 1, 7, 7, 5, 2], [6, 3, 8, 6, 2, 4, 4, 5]

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

u₂ [7, 6, 3, 5, 8, 11, 4, 5, 6, 6, 5, 4, 11, 3, 11, 8, 5, 6, 8, 11, 8, 5, 3, 6, 5, 3, 6, 7] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

Sandwich
Coloring	{2, 8}
Rank	2
R,B	[3, 8, 1, 1, 7, 7, 5, 2], [6, 3, 8, 6, 2, 4, 4, 5]
π₂	[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u₂	[7, 6, 3, 5, 8, 11, 4, 5, 6, 6, 5, 4, 11, 3, 11, 8, 5, 6, 8, 11, 8, 5, 3, 6, 5, 3, 6, 7] (dim 1)
wpp	[4, 4, 4, 4, 4, 4, 4, 4]

15 . Coloring, {3, 4}

R: [3, 3, 8, 6, 7, 7, 5, 5]
B: [6, 8, 1, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

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

Omega Rank for R : cycles: {{5, 7}} order: 4

See Matrix

[0, 0, 2 y₂, 0, y₄, y₂, y₃, y₁]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}} order: 6

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

16 . Coloring, {3, 5}

Ωp(Δ)=0: p = s - 4s ⁴ - 8s ⁵ p' = s - 4s ⁴ - 8s ⁵ p' = s ² + 2s ³ + 4s ⁴ + 4s ⁵

R: [3, 3, 8, 1, 2, 7, 5, 5]
B: [6, 8, 1, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 2 vs 6 2 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 6	4 vs 8	4 vs 8	2 vs 6	2 vs 6

Omega Rank for R : cycles: {{2, 3, 5, 8}} order: 4

See Matrix

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

p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁵ p = - s ² + s ⁶

Omega Rank for B : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

[y₁, y₂, 0, -y₁ + 3 y₂, 0, -y₁ + 3 y₂, y₁, y₂]

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

M \ ; N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $ $ [ [0, 2, 4, 2, 3, 5, 7, 5] , [2, 0, 4, 4, 3, 3, 5, 7] , [4, 4, 0, 5, 7, 2, 3, 3] , [2, 4, 5, 0, 2, 7, 5, 3] , [3, 3, 7, 2, 0, 5, 4, 4] , [5, 3, 2, 7, 5, 0, 2, 4] , [7, 5, 3, 5, 4, 2, 0, 2] , [5, 7, 3, 3, 4, 4, 2, 0] ] $

τ= 32 , r'= 1/2

R: [3, 3, 8, 1, 2, 7, 5, 5]
B: [6, 8, 1, 6, 7, 4, 4, 2]

Ranges

Action of R on ranges, [[3], [3], [2], [1]]
Action of B on ranges, [[4], [2], [1], [4]]

Cycles: R , {{2, 3, 5, 8}}, B , {{2, 8}, {4, 6}}

β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

Partitions

Action of R on partitions, [[5], [4], [1], [5], [4], [2]]
Action of B on partitions, [[5], [3], [6], [3], [5], [6]]

α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/14
α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 1/14
α([{2, 5, 6, 7}, {1, 3, 4, 8}]) = 1/7
α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 3/14
α([{1, 2, 4, 5}, {3, 6, 7, 8}]) = 5/14
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/7

b1 = {1, 2, 3, 6} ` , ` b2 = {4, 5, 7, 8} ` , ` b3 = {1, 2, 4, 5} ` , ` b4 = {3, 6, 7, 8} ` , ` b5 = {1, 4, 5, 8} ` , ` b6 = {2, 3, 6, 7} ` , ` b7 = {1, 2, 3, 4} ` , ` b8 = {5, 6, 7, 8} ` , ` b9 = {1, 2, 5, 6} ` , ` b10 = {2, 5, 6, 7} ` , ` b11 = {3, 4, 7, 8} ` , ` b12 = {1, 3, 4, 8}

Action of R and B on the blocks of the partitions: = [3, 4, 2, 1, B, 9, 3, 4, 2, 8, 1, 7] [C, A, 4, 3, 6, 5, 4, 3, C, 5, A, 6]
with invariant measure [3, 3, 5, 5, 2, 2, 1, 1, 1, 2, 1, 2]

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

` ` See level-2 partition graph.

Sandwich

Coloring {3, 5}

Rank 2

R,B [3, 3, 8, 1, 2, 7, 5, 5], [6, 8, 1, 6, 7, 4, 4, 2]

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

u₂ [2, 4, 2, 3, 5, 7, 5, 4, 4, 3, 3, 5, 7, 5, 7, 2, 3, 3, 2, 7, 5, 3, 5, 4, 4, 2, 4, 2] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

17 . Coloring, {3, 6}

R: [3, 3, 8, 1, 7, 4, 5, 5]
B: [6, 8, 1, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 6 vs 6 5 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	6 vs 6	5 vs 6

Omega Rank for R : cycles: {{5, 7}} order: 6

See Matrix

[y₆, 0, y₃, y₄, y₅, 0, y₁, y₂]

Omega Rank for B : cycles: {{2, 8}, {4, 6, 7}} order: 6

See Matrix

[5 y₁ - 3 y₅ - 3 y₄ - 3 y₂ + 5 y₃, 3 y₁, 0, 3 y₅, 0, 3 y₄, 3 y₂, 3 y₃]

p = - s ² - s ³ + s ⁵ + s ⁶

18 . Coloring, {3, 7}

Ωp(Δ)=0: p = s - 4s ⁴ p' = s - 4s ⁴ p' = s ² - 4s ⁵

R: [3, 3, 8, 1, 7, 7, 4, 5]
B: [6, 8, 1, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 3 vs 6 2 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 6	4 vs 8	4 vs 8	3 vs 6	2 vs 6

Omega Rank for R : cycles: {{1, 3, 4, 5, 7, 8}} order: 6

See Matrix

[y₁, 0, y₂, y₃, y₁, 0, y₂, y₃]

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

Omega Rank for B : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

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

p = - s ² + s ⁶ p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁵

M \ ; N

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

τ= 32 , r'= 1/2

R: [3, 3, 8, 1, 7, 7, 4, 5]
B: [6, 8, 1, 6, 2, 4, 5, 2]

Ranges

Action of R on ranges, [[3], [3], [4], [1]]
Action of B on ranges, [[2], [4], [1], [2]]

Cycles: R , {{1, 3, 4, 5, 7, 8}}, B , {{2, 8}, {4, 6}}

β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

Partitions

Action of R on partitions, [[2], [4], [5], [5], [2]]
Action of B on partitions, [[3], [1], [3], [5], [5]]

α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 1/8
α([{1, 2, 3, 8}, {4, 5, 6, 7}]) = 1/4
α([{1, 3, 4, 6}, {2, 5, 7, 8}]) = 1/8
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/8
α([{1, 2, 4, 7}, {3, 5, 6, 8}]) = 3/8

b1 = {1, 2, 3, 8} ` , ` b2 = {4, 5, 6, 7} ` , ` b3 = {1, 2, 4, 7} ` , ` b4 = {3, 5, 6, 8} ` , ` b5 = {1, 3, 4, 6} ` , ` b6 = {1, 4, 6, 7} ` , ` b7 = {2, 5, 7, 8} ` , ` b8 = {2, 3, 5, 8} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [9, A, 2, 1, 3, 2, 4, 1, 3, 4] [8, 6, 4, 3, 5, 5, 7, 7, 4, 3]
with invariant measure [2, 2, 3, 3, 1, 1, 1, 1, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {3, 7}

Rank 2

R,B [3, 3, 8, 1, 7, 7, 4, 5], [6, 8, 1, 6, 2, 4, 5, 2]

π₂ [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, 0, 0, 0]

u₂ [1, 2, 1, 4, 3, 2, 3, 2, 2, 3, 4, 2, 2, 3, 2, 2, 4, 1, 3, 2, 1, 4, 1, 2, 1, 2, 2, 3] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

19 . Coloring, {3, 8}

R: [3, 3, 8, 1, 7, 7, 5, 2]
B: [6, 8, 1, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 5 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	5 vs 6	3 vs 6

Omega Rank for R : cycles: {{2, 3, 8}, {5, 7}} order: 6

See Matrix

[-3 y₁ - 3 y₂ + 5 y₃ + 5 y₄ - 3 y₅, 3 y₁, 3 y₂, 0, 3 y₃, 0, 3 y₄, 3 y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{4, 6}, {2, 5, 8}} order: 6

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴ p = - s ² + s ⁶

20 . Coloring, {4, 5}

R: [3, 3, 1, 6, 2, 7, 5, 5]
B: [6, 8, 8, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 6 vs 6 5 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	6 vs 6	5 vs 6

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

See Matrix

[y₂, y₁, y₅, 0, y₆, y₄, y₃, 0]

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}} order: 6

See Matrix

[5 y₁ - 3 y₂ - 3 y₃ - 3 y₄ + 5 y₅, 3 y₁, 0, 3 y₂, 0, 3 y₃, 3 y₄, 3 y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

21 . Coloring, {4, 6}

Ωp(Δ)=0: p = s + 4s ⁴ + 8s ⁵ - 16s ⁶ p' = s - 4s ³ - 4s ⁴ + 8s ⁵ p' = s ² + 2s ³ - 4s ⁵

R: [3, 3, 1, 6, 7, 4, 5, 5]
B: [6, 8, 8, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 2 vs 6 1 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 6	4 vs 8	4 vs 8	2 vs 6	1 vs 6

Omega Rank for R : cycles: {{4, 6}, {1, 3}, {5, 7}} order: 2

See Matrix

[y₂, 0, -y₂ + 3 y₁, y₁, -y₂ + 3 y₁, y₁, y₂, 0]

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

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6, 7}} order: 4

See Matrix

[y₁, 2 y₁, 0, y₁, 0, y₁, y₁, 2 y₁]

p = s - s ⁶ p' = s ² - s ⁵ p' = s - s ⁵ p' = s ³ - s ⁵ p' = s ⁴ - s ⁵

M \ ; N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $ $ [ [0, 3, 6, 5, 5, 6, 11, 8] , [3, 0, 3, 6, 8, 5, 8, 11] , [6, 3, 0, 7, 11, 4, 5, 8] , [5, 6, 7, 0, 4, 11, 6, 5] , [5, 8, 11, 4, 0, 7, 6, 3] , [6, 5, 4, 11, 7, 0, 5, 6] , [11, 8, 5, 6, 6, 5, 0, 3] , [8, 11, 8, 5, 3, 6, 3, 0] ] $

τ= 32 , r'= 1/2

R: [3, 3, 1, 6, 7, 4, 5, 5]
B: [6, 8, 8, 1, 2, 7, 4, 2]

Ranges

Action of R on ranges, [[3], [3], [1], [4]]
Action of B on ranges, [[4], [2], [2], [1]]

Cycles: R , {{4, 6}, {1, 3}, {5, 7}}, B , {{2, 8}, {1, 4, 6, 7}}

β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

Partitions

Action of R on partitions, [[4], [5], [6], [4], [2], [6]]
Action of B on partitions, [[5], [3], [1], [3], [2], [2]]

α([{2, 3, 4, 7}, {1, 5, 6, 8}]) = 1/11
α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 3/11
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 2/11
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/11
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 2/11
α([{1, 2, 4, 5}, {3, 6, 7, 8}]) = 2/11

b1 = {1, 2, 3, 6} ` , ` b2 = {2, 3, 4, 7} ` , ` b3 = {1, 5, 6, 8} ` , ` b4 = {4, 5, 7, 8} ` , ` b5 = {1, 4, 5, 8} ` , ` b6 = {2, 3, 6, 7} ` , ` b7 = {3, 4, 7, 8} ` , ` b8 = {1, 2, 5, 6} ` , ` b9 = {1, 2, 4, 5} ` , ` b10 = {1, 2, 3, 4} ` , ` b11 = {3, 6, 7, 8} ` , ` b12 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [A, 8, 7, C, B, 9, 8, 7, B, 1, 9, 4] [5, C, A, 6, 2, 3, 6, 5, 4, 4, 1, 1]
with invariant measure [3, 1, 1, 3, 2, 2, 1, 1, 2, 2, 2, 2]

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

` ` See level-2 partition graph.

Sandwich

Coloring {4, 6}

Rank 2

R,B [3, 3, 1, 6, 7, 4, 5, 5], [6, 8, 8, 1, 2, 7, 4, 2]

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

u₂ [3, 6, 5, 5, 6, 11, 8, 3, 6, 8, 5, 8, 11, 7, 11, 4, 5, 8, 4, 11, 6, 5, 7, 6, 3, 5, 6, 3] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

Sandwich
Coloring	{4, 6}
Rank	2
R,B	[3, 3, 1, 6, 7, 4, 5, 5], [6, 8, 8, 1, 2, 7, 4, 2]
π₂	[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
u₂	[3, 6, 5, 5, 6, 11, 8, 3, 6, 8, 5, 8, 11, 7, 11, 4, 5, 8, 4, 11, 6, 5, 7, 6, 3, 5, 6, 3] (dim 1)
wpp	[4, 4, 4, 4, 4, 4, 4, 4]

22 . Coloring, {4, 7}

R: [3, 3, 1, 6, 7, 7, 4, 5]
B: [6, 8, 8, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 5 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	5 vs 6	3 vs 6

Omega Rank for R : cycles: {{1, 3}, {4, 6, 7}} order: 6

See Matrix

[3 y₄, 0, 3 y₃, 3 y₅, 5 y₄ + 5 y₃ - 3 y₅ - 3 y₁ - 3 y₂, 3 y₁, 3 y₂, 0]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}} order: 6

See Matrix

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

p' = s ³ - s ⁵ p = s ² - s ⁶ p' = s ² - s ⁴

23 . Coloring, {4, 8}

Ωp(Δ)=0: p' = s ² + 4s ⁵ p = s + 4s ⁴ p' = s + 4s ⁴

R: [3, 3, 1, 6, 7, 7, 5, 2]
B: [6, 8, 8, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 2 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 6	4 vs 8	4 vs 8	2 vs 6	3 vs 6

Omega Rank for R : cycles: {{1, 3}, {5, 7}} order: 2

See Matrix

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

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

Omega Rank for B : cycles: {{2, 5, 8}, {1, 4, 6}} order: 3

See Matrix

[y₁, y₂, 0, y₃, y₁, y₂, 0, y₃]

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

M \ ; N

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

τ= 32 , r'= 1/2

R: [3, 3, 1, 6, 7, 7, 5, 2]
B: [6, 8, 8, 1, 2, 4, 4, 5]

Ranges

Action of R on ranges, [[3], [3], [1], [2]]
Action of B on ranges, [[2], [4], [4], [1]]

Cycles: R , {{1, 3}, {5, 7}}, B , {{2, 5, 8}, {1, 4, 6}}

β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

Partitions

Action of R on partitions, [[1], [3], [1], [4], [4]]
Action of B on partitions, [[4], [5], [2], [2], [4]]

α([{1, 2, 4, 7}, {3, 5, 6, 8}]) = 1/8
α([{1, 6, 7, 8}, {2, 3, 4, 5}]) = 1/4
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 1/8
α([{1, 2, 3, 8}, {4, 5, 6, 7}]) = 3/8
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/8

b1 = {1, 2, 4, 7} ` , ` b2 = {3, 5, 6, 8} ` , ` b3 = {1, 6, 7, 8} ` , ` b4 = {2, 3, 4, 5} ` , ` b5 = {1, 2, 7, 8} ` , ` b6 = {3, 4, 5, 6} ` , ` b7 = {1, 2, 3, 8} ` , ` b8 = {4, 5, 6, 7} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {5, 6, 7, 8}

Action of R and B on the blocks of the partitions: = [2, 1, 6, 5, 2, 1, 7, 8, 7, 8] [8, 7, 9, A, 4, 3, 4, 3, 8, 7]
with invariant measure [1, 1, 2, 2, 1, 1, 3, 3, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {4, 8}

Rank 2

R,B [3, 3, 1, 6, 7, 7, 5, 2], [6, 8, 8, 1, 2, 4, 4, 5]

π₂ [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, 0, 0, 0]

u₂ [1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 3, 2, 2, 2, 3, 4, 2, 1, 2, 2, 4, 1, 2, 3, 1, 2, 2] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

24 . Coloring, {5, 6}

R: [3, 3, 1, 1, 2, 4, 5, 5]
B: [6, 8, 8, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

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

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

See Matrix

[y₄, y₃, y₁, y₂, 2 y₂, 0, 0, 0]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {4, 6, 7}} order: 6

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

25 . Coloring, {5, 7}

Ωp(Δ)=0: p' = s ³ - 2s ⁵ p = s - 4s ⁵ p' = s - 4s ⁵ p' = s ² - 2s ⁴

R: [3, 3, 1, 1, 2, 7, 4, 5]
B: [6, 8, 8, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 6 3 vs 7 3 vs 7 3 vs 6 2 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 6	3 vs 7	3 vs 7	3 vs 6	2 vs 6

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

See Matrix

[y₂, y₁, y₂, y₁, y₃, 0, y₃, 0]

p' = s ⁴ - s ⁵ p' = s ³ - s ⁵ p = s ³ - s ⁶

Omega Rank for B : cycles: {{2, 8}, {4, 6}, {5, 7}} order: 2

See Matrix

[0, 3 y₁ - y₂, 0, 3 y₁ - y₂, y₁, y₂, y₁, y₂]

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

M \ ; N

τ= 32 , r'= 1/2

R: [3, 3, 1, 1, 2, 7, 4, 5]
B: [6, 8, 8, 6, 7, 4, 5, 2]

Ranges

Action of R on ranges, [[1], [1], [2], [3]]
Action of B on ranges, [[4], [4], [3], [2]]

Cycles: R , {{1, 3}}, B , {{2, 8}, {4, 6}, {5, 7}}

β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

Partitions

Action of R on partitions, [[3], [5], [7], [1], [3], [1], [7], [5]]
Action of B on partitions, [[8], [6], [4], [4], [6], [2], [2], [8]]

α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/8
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 1/8
α([{3, 4, 5, 8}, {1, 2, 6, 7}]) = 1/8
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/8
α([{3, 4, 6, 7}, {1, 2, 5, 8}]) = 1/8
α([{1, 4, 5, 6}, {2, 3, 7, 8}]) = 1/8
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 1/8
α([{1, 4, 7, 8}, {2, 3, 5, 6}]) = 1/8

b1 = {1, 4, 7, 8} ` , ` b2 = {2, 3, 5, 6} ` , ` b3 = {1, 4, 5, 8} ` , ` b4 = {1, 4, 5, 6} ` , ` b5 = {2, 3, 6, 7} ` , ` b6 = {2, 3, 7, 8} ` , ` b7 = {3, 4, 7, 8} ` , ` b8 = {1, 2, 7, 8} ` , ` b9 = {3, 4, 5, 6} ` , ` b10 = {1, 2, 5, 6} ` , ` b11 = {3, 4, 5, 8} ` , ` b12 = {1, 2, 6, 7} ` , ` b13 = {3, 4, 6, 7} ` , ` b14 = {1, 4, 6, 7} ` , ` b15 = {2, 3, 5, 8} ` , ` b16 = {1, 2, 5, 8}

Action of R and B on the blocks of the partitions: = [D, 10, 7, 7, A, A, C, 9, 8, B, 8, 9, C, D, 10, B] [2, 1, 5, E, 3, F, 2, F, E, 1, 5, 3, 4, 4, 6, 6]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {5, 7}

Rank 2

R,B [3, 3, 1, 1, 2, 7, 4, 5], [6, 8, 8, 6, 7, 4, 5, 2]

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

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

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

26 . Coloring, {5, 8}

R: [3, 3, 1, 1, 2, 7, 5, 2]
B: [6, 8, 8, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

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

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

See Matrix

[y₁, y₄, y₃, 0, y₂, 0, y₅, 0]

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

See Matrix

[0, 0, 0, y₄, y₅, y₂, y₁, y₃]

27 . Coloring, {6, 7}

R: [3, 3, 1, 1, 7, 4, 4, 5]
B: [6, 8, 8, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

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

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

See Matrix

[y₁, 0, y₅, y₂, y₃, 0, y₄, 0]

Omega Rank for B : cycles: {{2, 8}} order: 4

See Matrix

[0, y₁, 0, 0, y₄, y₃, y₂, y₅]

28 . Coloring, {6, 8}

Ωp(Δ)=0: p' = s + 4s ⁵ p' = s ² - 2s ⁴ + 4s ⁵ p = s + 4s ⁵ p' = s ³ - 2s ⁴ + 2s ⁵

R: [3, 3, 1, 1, 7, 4, 5, 2]
B: [6, 8, 8, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 6 3 vs 7 3 vs 7 2 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
2 vs 6	3 vs 7	3 vs 7	2 vs 6	3 vs 6

Omega Rank for R : cycles: {{1, 3}, {5, 7}} order: 2

See Matrix

[-y₂ + 3 y₁, y₂, -y₂ + 3 y₁, y₂, y₁, 0, y₁, 0]

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

Omega Rank for B : cycles: {{4, 6, 7}, {2, 5, 8}} order: 3

See Matrix

[0, y₁, 0, y₁, y₂, y₃, y₂, y₃]

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

M \ ; N

τ= 32 , r'= 1/2

R: [3, 3, 1, 1, 7, 4, 5, 2]
B: [6, 8, 8, 6, 2, 7, 4, 5]

Ranges

Action of R on ranges, [[1], [1], [3], [2]]
Action of B on ranges, [[4], [4], [2], [3]]

Cycles: R , {{1, 3}, {5, 7}}, B , {{4, 6, 7}, {2, 5, 8}}

β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

Partitions

Action of R on partitions, [[6], [5], [6], [4], [4], [7], [7], [5]]
Action of B on partitions, [[8], [2], [1], [1], [8], [2], [3], [3]]

α([{1, 4, 5, 6}, {2, 3, 7, 8}]) = 1/8
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 1/8
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/8
α([{3, 4, 5, 8}, {1, 2, 6, 7}]) = 1/8
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 1/8
α([{3, 4, 6, 7}, {1, 2, 5, 8}]) = 1/8
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 1/8
α([{1, 4, 7, 8}, {2, 3, 5, 6}]) = 1/8

Action of R and B on the blocks of the partitions: = [9, 8, D, D, 10, 10, A, B, C, 7, C, B, A, 9, 8, 7] [5, 3, 6, 1, 4, 2, 5, 2, 1, 3, 6, 4, E, E, F, F]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {6, 8}

Rank 2

R,B [3, 3, 1, 1, 7, 4, 5, 2], [6, 8, 8, 6, 2, 7, 4, 5]

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

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

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

29 . Coloring, {7, 8}

R: [3, 3, 1, 1, 7, 7, 4, 2]
B: [6, 8, 8, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

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

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

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, 2 y₂, 0]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{4, 6}, {2, 5, 8}} order: 6

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

30 . Coloring, {2, 3, 4}

R: [3, 8, 8, 6, 7, 7, 5, 5]
B: [6, 3, 1, 1, 2, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 5 vs 5

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	3 vs 5	5 vs 5

Omega Rank for R : cycles: {{5, 7}} order: 4

See Matrix

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

p = s ³ - s ⁵ p' = s ³ - s ⁴

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

See Matrix

[y₅, y₄, y₃, y₂, 0, y₁, 0, 0]

31 . Coloring, {2, 3, 5}

R: [3, 8, 8, 1, 2, 7, 5, 5]
B: [6, 3, 1, 6, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 4 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	5 vs 6	4 vs 6

Omega Rank for R : cycles: {{2, 5, 8}} order: 3

See Matrix

[y₄, y₁, y₂, 0, y₃, 0, y₄, y₅]

p = - s ³ + s ⁶

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

See Matrix

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

p = s ⁴ - s ⁶ p' = s ⁴ - s ⁵

32 . Coloring, {2, 3, 6}

R: [3, 8, 8, 1, 7, 4, 5, 5]
B: [6, 3, 1, 6, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	5 vs 6	6 vs 6

Omega Rank for R : cycles: {{5, 7}} order: 6

See Matrix

[y₅, 0, y₃, y₄, y₂, 0, y₁, -y₅ + y₃ + y₄ + y₂ - y₁]

p = s ⁵ - s ⁶

Omega Rank for B : cycles: {{4, 6, 7}} order: 6

See Matrix

[y₁, y₂, y₃, y₄, 0, y₆, y₅, 0]

33 . Coloring, {2, 3, 7}

R: [3, 8, 8, 1, 7, 7, 4, 5]
B: [6, 3, 1, 6, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 6 vs 6 5 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	6 vs 6	5 vs 6

Omega Rank for R : cycles: {{1, 3, 4, 5, 7, 8}} order: 6

See Matrix

[y₁, 0, y₂, y₃, y₄, 0, y₅, y₆]

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

See Matrix

[y₃, y₂, y₃ + y₂ + y₁ - y₄ - y₅, y₁, y₄, y₅, 0, 0]

p = - s ⁵ + s ⁶

34 . Coloring, {2, 3, 8}

R: [3, 8, 8, 1, 7, 7, 5, 2]
B: [6, 3, 1, 6, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	4 vs 6	5 vs 6

Omega Rank for R : cycles: {{2, 8}, {5, 7}} order: 4

See Matrix

[4 y₁ + 4 y₂ - 5 y₃ - y₄, y₁, y₂, 0, y₃, 0, 3 y₁ + 3 y₂ - 4 y₃, y₄]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

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

See Matrix

[-y₁ + y₄ - y₃ + y₂ + y₅, y₁, y₄, y₃, y₂, y₅, 0, 0]

p = - s ⁵ + s ⁶

35 . Coloring, {2, 4, 5}

R: [3, 8, 1, 6, 2, 7, 5, 5]
B: [6, 3, 8, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 7 4 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	3 vs 7	4 vs 7

Omega Rank for R : cycles: {{1, 3}, {2, 5, 8}} order: 6

See Matrix

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

p = - s ³ + s ⁵ p = - s ³ + s ⁶ p = - s ³ + s ⁷ p = - s ³ + s ⁴

Omega Rank for B : cycles: {{1, 4, 6}, {2, 3, 8}} order: 3

See Matrix

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

p = - s ² + s ⁵ p' = - s ³ + s ⁶ p' = - s ² + s ⁵

36 . Coloring, {2, 4, 6}

R: [3, 8, 1, 6, 7, 4, 5, 5]
B: [6, 3, 8, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	2 vs 7	3 vs 7

Omega Rank for R : cycles: {{4, 6}, {1, 3}, {5, 7}} order: 2

See Matrix

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

p = s ² - s ⁴ p' = s ² - s ³ p' = - s ³ + s ⁴ p' = - s ³ + s ⁵ p' = - s ³ + s ⁶

Omega Rank for B : cycles: {{2, 3, 8}, {1, 4, 6, 7}}

See Matrix

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

p = - s + s ⁷ p = - s + s ⁴ p' = s - s ⁴ p' = s ² - s ⁵

37 . Coloring, {2, 4, 7}

R: [3, 8, 1, 6, 7, 7, 4, 5]
B: [6, 3, 8, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 7 4 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	3 vs 7	4 vs 7

Omega Rank for R : cycles: {{4, 6, 7}, {1, 3}} order: 6

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵ p = - s ³ + s ⁶ p = - s ³ + s ⁷

Omega Rank for B : cycles: {{2, 3, 8}, {1, 4, 6}} order: 3

See Matrix

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

p' = s ² - s ⁵ p' = s ³ - s ⁶ p = s ² - s ⁵

38 . Coloring, {2, 4, 8}

R: [3, 8, 1, 6, 7, 7, 5, 2]
B: [6, 3, 8, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	2 vs 7	3 vs 7

Omega Rank for R : cycles: {{2, 8}, {1, 3}, {5, 7}} order: 2

See Matrix

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

p' = - s ² + s ⁶ p' = - s ² + s ⁵ p' = - s ² + s ⁴ p' = - s ² + s ³ p = s ² - s ³

Omega Rank for B : cycles: {{1, 4, 6}, {2, 3, 5, 8}}

See Matrix

[y₃, y₂, y₂, y₁, y₂, -y₃ + 4 y₂ - y₁, 0, y₂]

p = - s + s ⁴ p' = - s + s ⁴ p' = - s ² + s ⁵ p = - s + s ⁷

39 . Coloring, {2, 5, 6}

R: [3, 8, 1, 1, 2, 4, 5, 5]
B: [6, 3, 8, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 3 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	7 vs 7	7 vs 7	4 vs 6	3 vs 6

Omega Rank for R : cycles: {{1, 3}, {2, 5, 8}} order: 6

See Matrix

[y₁ + y₃, 2 y₁ + 2 y₃ - y₂ - y₄, y₁, y₃, y₂, 0, 0, y₄]

p = - s ² + s ⁵ p' = - s ² + s ⁵

Omega Rank for B : cycles: {{4, 6, 7}, {2, 3, 8}} order: 3

See Matrix

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

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

40 . Coloring, {2, 5, 7}

R: [3, 8, 1, 1, 2, 7, 4, 5]
B: [6, 3, 8, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 4 vs 7 2 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 6	8 vs 8	8 vs 8	4 vs 7	2 vs 7

Omega Rank for R : cycles: {{1, 3}, {2, 5, 8}} order: 6

See Matrix

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

p = s ³ - s ⁵ p' = - s ³ + s ⁵ p' = - s ⁴ + s ⁶

Omega Rank for B : cycles: {{4, 6}, {5, 7}, {2, 3, 8}} order: 6

See Matrix

[0, y₂, y₂, y₁, y₂, 3 y₂ - y₁, y₂, y₂]

p' = - s ⁴ + s ⁶ p' = - s ³ + s ⁵ p' = s - s ³ p' = s ² - s ⁴ p = s - s ⁵

41 . Coloring, {2, 5, 8}

R: [3, 8, 1, 1, 2, 7, 5, 2]
B: [6, 3, 8, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R : cycles: {{2, 8}, {1, 3}} order: 4

See Matrix

[y₁, y₂, -4 y₁ + 3 y₂ + 3 y₄, 0, -5 y₁ + 4 y₂ + 4 y₄ - y₃, 0, y₄, y₃]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

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

See Matrix

[0, 0, -y₁ - y₂ + y₃ + y₄ + y₅, y₁, y₂, y₃, y₄, y₅]

p = - s ⁵ + s ⁶

42 . Coloring, {2, 6, 7}

R: [3, 8, 1, 1, 7, 4, 4, 5]
B: [6, 3, 8, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

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

See Matrix

[y₃, 0, y₂, y₁, y₃ - y₂ - y₁ + y₅ + y₄, 0, y₅, y₄]

p = s ⁵ - s ⁶

Omega Rank for B : cycles: {{2, 3, 8}} order: 6

See Matrix

[0, y₆, y₅, 0, y₂, y₃, y₁, y₄]

43 . Coloring, {2, 6, 8}

R: [3, 8, 1, 1, 7, 4, 5, 2]
B: [6, 3, 8, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Omega Rank for R : cycles: {{2, 8}, {1, 3}, {5, 7}} order: 2

See Matrix

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

p' = s ³ - s ⁴ p = s ² - s ⁵ p' = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁵ p' = s ² - s ⁴

Omega Rank for B : cycles: {{4, 6, 7}, {2, 3, 5, 8}}

See Matrix

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

p' = s ² - s ⁵ p' = s ³ - s ⁶ p = s - s ⁷ p' = s - s ⁴

44 . Coloring, {2, 7, 8}

R: [3, 8, 1, 1, 7, 7, 4, 2]
B: [6, 3, 8, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

Omega Rank for R : cycles: {{2, 8}, {1, 3}} order: 4

See Matrix

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

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{4, 6}, {2, 3, 5, 8}} order: 4

See Matrix

[0, y₁, y₄, y₃, 4 y₁ - y₄ - 5 y₃ + 4 y₂, 3 y₁ - 4 y₃ + 3 y₂, 0, y₂]

p' = - s + s ⁵ p = - s + s ⁵

45 . Coloring, {3, 4, 5}

R: [3, 3, 8, 6, 2, 7, 5, 5]
B: [6, 8, 1, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 6 vs 6 2 vs 6

Omega Rank for R : cycles: {{2, 3, 5, 8}} order: 4

See Matrix

[0, y₂, y₁, 0, y₆, y₅, y₄, y₃]

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}} order: 6

See Matrix

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

p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁵ p = - s ² + s ⁶

46 . Coloring, {3, 4, 6}

R: [3, 3, 8, 6, 7, 4, 5, 5]
B: [6, 8, 1, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

Omega Rank for R : cycles: {{4, 6}, {5, 7}} order: 4

See Matrix

[0, 0, 6 y₂ - y₁ - y₃ - y₄, y₂, y₁, y₂, y₃, y₄]

p = s ³ - s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6, 7}} order: 4

See Matrix

[y₂, y₁, 0, 4 y₂ - 5 y₁ - y₃ + 4 y₄, 0, y₃, y₄, 3 y₂ - 4 y₁ + 3 y₄]

p' = - s + s ⁵ p = - s + s ⁵

47 . Coloring, {3, 4, 7}

R: [3, 3, 8, 6, 7, 7, 4, 5]
B: [6, 8, 1, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 6 vs 6 4 vs 6

Omega Rank for R : cycles: {{4, 6, 7}} order: 6

See Matrix

[0, 0, y₁, y₃, y₄, y₂, y₅, y₆]

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}} order: 6

See Matrix

[2 y₃ + 2 y₄ - y₂ - y₁, y₃ + y₄, 0, y₂, y₃, y₁, 0, y₄]

p = - s ² + s ⁵ p' = - s ² + s ⁵

48 . Coloring, {3, 4, 8}

R: [3, 3, 8, 6, 7, 7, 5, 2]
B: [6, 8, 1, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 3 vs 6

Omega Rank for R : cycles: {{2, 3, 8}, {5, 7}} order: 6

See Matrix

[0, -y₁ + 2 y₂ - y₄, y₁, 0, y₃, -y₃ + y₂, y₂, y₄]

p' = - s ² + s ⁵ p = - s ² + s ⁵

Omega Rank for B : cycles: {{2, 5, 8}, {1, 4, 6}} order: 3

See Matrix

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

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

49 . Coloring, {3, 5, 6}

R: [3, 3, 8, 1, 2, 4, 5, 5]
B: [6, 8, 1, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 6 vs 6 2 vs 6

Omega Rank for R : cycles: {{2, 3, 5, 8}} order: 4

See Matrix

[y₁, y₄, y₂, y₃, y₅, 0, 0, y₆]

Omega Rank for B : cycles: {{2, 8}, {4, 6, 7}} order: 6

See Matrix

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

p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁵ p = - s ² + s ⁶

50 . Coloring, {3, 5, 7}

R: [3, 3, 8, 1, 2, 7, 4, 5]
B: [6, 8, 1, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 4 vs 7 2 vs 7

Omega Rank for R : cycles: {{2, 3, 5, 8}} order: 4

See Matrix

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

p = - s ⁴ + s ⁷ p = - s ⁴ + s ⁵ p = - s ⁴ + s ⁶

Omega Rank for B : cycles: {{2, 8}, {4, 6}, {5, 7}} order: 2

See Matrix

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

p' = - s ² + s ³ p' = - s ² + s ⁴ p' = - s ² + s ⁵ p' = - s ² + s ⁶ p = s ² - s ³

51 . Coloring, {3, 5, 8}

R: [3, 3, 8, 1, 2, 7, 5, 2]
B: [6, 8, 1, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 4 vs 6

Omega Rank for R : cycles: {{2, 3, 8}} order: 3

See Matrix

[y₄, y₃, y₁, 0, y₂, 0, y₄, y₅]

p = - s ³ + s ⁶

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

See Matrix

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

p = - s ⁴ + s ⁶ p = - s ⁴ + s ⁵

52 . Coloring, {3, 6, 7}

R: [3, 3, 8, 1, 7, 4, 4, 5]
B: [6, 8, 1, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 6 vs 6 5 vs 6

Omega Rank for R : cycles: {{1, 3, 4, 5, 7, 8}} order: 6

See Matrix

[y₁, 0, y₄, y₂, y₃, 0, y₅, y₆]

Omega Rank for B : cycles: {{2, 8}} order: 6

See Matrix

[-y₁ + y₂ + y₃ - y₄ + y₅, y₁, 0, 0, y₂, y₃, y₄, y₅]

p = - s ⁵ + s ⁶

53 . Coloring, {3, 6, 8}

R: [3, 3, 8, 1, 7, 4, 5, 2]
B: [6, 8, 1, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 7 4 vs 7

Omega Rank for R : cycles: {{2, 3, 8}, {5, 7}} order: 6

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵ p = - s ³ + s ⁷ p = - s ³ + s ⁶

Omega Rank for B : cycles: {{4, 6, 7}, {2, 5, 8}} order: 3

See Matrix

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

p = s ² - s ⁵ p' = - s ² + s ⁵ p' = - s ³ + s ⁶

54 . Coloring, {3, 7, 8}

R: [3, 3, 8, 1, 7, 7, 4, 2]
B: [6, 8, 1, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 6 vs 6 4 vs 6

Omega Rank for R : cycles: {{2, 3, 8}} order: 6

See Matrix

[y₁, y₂, y₅, y₃, 0, 0, y₆, y₄]

Omega Rank for B : cycles: {{4, 6}, {2, 5, 8}} order: 6

See Matrix

[-y₃ + y₁, 2 y₁ - y₂ - y₄, 0, y₃, y₂, y₁, 0, y₄]

p = - s ² + s ⁵ p' = - s ² + s ⁵

55 . Coloring, {4, 5, 6}

R: [3, 3, 1, 6, 2, 4, 5, 5]
B: [6, 8, 8, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

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

See Matrix

[y₄, y₃, y₂, y₁, -y₄ - y₃ - y₂ + 6 y₁, y₁, 0, 0]

p' = - s ³ + s ⁵ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6, 7}} order: 4

See Matrix

[-5 y₁ + 4 y₂ + 4 y₃ - y₄, y₁, 0, y₂, 0, y₃, y₄, -4 y₁ + 3 y₂ + 3 y₃]

p = - s + s ⁵ p' = - s + s ⁵

56 . Coloring, {4, 5, 7}

R: [3, 3, 1, 6, 2, 7, 4, 5]
B: [6, 8, 8, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 4 vs 7 2 vs 7

Omega Rank for R : cycles: {{1, 3}, {4, 6, 7}} order: 6

See Matrix

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

p = - s ³ + s ⁵ p' = - s ³ + s ⁵ p = - s ³ + s ⁷

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}, {5, 7}} order: 6

See Matrix

[y₂, 3 y₂ - y₁, 0, y₂, y₂, y₂, y₂, y₁]

p = - s + s ⁵ p = - s + s ⁷ p = - s + s ³ p' = - s + s ⁵ p' = - s + s ³

57 . Coloring, {4, 5, 8}

R: [3, 3, 1, 6, 2, 7, 5, 2]
B: [6, 8, 8, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

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

See Matrix

[y₁, -y₁ + y₂ + y₃ + y₄ - y₅, y₂, 0, y₃, y₄, y₅, 0]

p = - s ⁵ + s ⁶

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

See Matrix

[y₃, 0, 0, y₂, y₁, y₄, y₅, y₆]

58 . Coloring, {4, 6, 7}

R: [3, 3, 1, 6, 7, 4, 4, 5]
B: [6, 8, 8, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

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

See Matrix

[y₁, 0, -4 y₁ + 3 y₃ + 3 y₄, -5 y₁ - y₂ + 4 y₃ + 4 y₄, y₂, y₃, y₄, 0]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 8}} order: 6

See Matrix

[-y₁ + y₂ + y₃ - y₄ + y₅, y₁, 0, 0, y₂, y₃, y₄, y₅]

p = - s ⁵ + s ⁶

59 . Coloring, {4, 6, 8}

R: [3, 3, 1, 6, 7, 4, 5, 2]
B: [6, 8, 8, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 3 vs 7

Omega Rank for R : cycles: {{4, 6}, {1, 3}, {5, 7}} order: 2

See Matrix

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

p' = - s ² + s ³ p' = - s ² + s ⁴ p' = - s ² + s ⁵ p' = - s ² + s ⁶ p = s ² - s ³

Omega Rank for B : cycles: {{2, 5, 8}, {1, 4, 6, 7}}

See Matrix

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

p = - s + s ⁴ p' = - s + s ⁴ p' = - s ² + s ⁵ p = - s + s ⁷

60 . Coloring, {4, 7, 8}

R: [3, 3, 1, 6, 7, 7, 4, 2]
B: [6, 8, 8, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 3 vs 6

Omega Rank for R : cycles: {{1, 3}, {4, 6, 7}} order: 6

See Matrix

[y₁, y₂, y₁ + y₂, 2 y₁ + 2 y₂ - y₃ - y₄, 0, y₃, y₄, 0]

p = - s ² + s ⁵ p' = - s ² + s ⁵

Omega Rank for B : cycles: {{2, 5, 8}, {1, 4, 6}} order: 3

See Matrix

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

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

61 . Coloring, {5, 6, 7}

R: [3, 3, 1, 1, 2, 4, 4, 5]
B: [6, 8, 8, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

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

See Matrix

[y₄, y₃, y₂, 2 y₁, y₁, 0, 0, 0]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {5, 7}} order: 2

See Matrix

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

p' = s ² - s ⁴ p = s ² - s ⁴

62 . Coloring, {5, 6, 8}

R: [3, 3, 1, 1, 2, 4, 5, 2]
B: [6, 8, 8, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 5 vs 5

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

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{4, 6, 7}} order: 3

See Matrix

[0, 0, 0, y₄, y₅, y₁, y₂, y₃]

63 . Coloring, {5, 7, 8}

R: [3, 3, 1, 1, 2, 7, 4, 2]
B: [6, 8, 8, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 5 3 vs 5

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

See Matrix

[y₃, 2 y₄, y₁, y₂, 0, 0, y₄, 0]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{4, 6}, {5, 7}} order: 2

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴

64 . Coloring, {6, 7, 8}

R: [3, 3, 1, 1, 7, 4, 4, 2]
B: [6, 8, 8, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 5 vs 5

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

See Matrix

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

p = - s ³ + s ⁵ p = - s ³ + s ⁴

Omega Rank for B : cycles: {{2, 5, 8}} order: 3

See Matrix

[0, y₁, 0, 0, y₂, y₃, y₄, y₅]

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

R: [3, 8, 8, 6, 2, 7, 5, 5]
B: [6, 3, 1, 1, 7, 4, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R : cycles: {{2, 5, 8}} order: 3

See Matrix

[0, y₁, y₅, 0, y₂, y₅, y₃, y₄]

p = s ³ - s ⁶

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

See Matrix

[y₃, y₄, y₂, y₁, 0, y₅, y₄, 0]

p = s ³ - s ⁶

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

R: [3, 8, 8, 6, 7, 4, 5, 5]
B: [6, 3, 1, 1, 2, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 6 6 vs 6

Omega Rank for R : cycles: {{4, 6}, {5, 7}} order: 4

See Matrix

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

p' = s ⁴ - s ⁵ p = s ³ - s ⁶ p' = s ³ - s ⁵

Omega Rank for B : cycles: {{1, 4, 6, 7}} order: 4

See Matrix

[y₁, y₂, y₃, y₄, 0, y₅, y₆, 0]

67 . Coloring, {2, 3, 4, 7}

R: [3, 8, 8, 6, 7, 7, 4, 5]
B: [6, 3, 1, 1, 2, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 6 vs 6 6 vs 6

Omega Rank for R : cycles: {{4, 6, 7}} order: 6

See Matrix

[0, 0, y₅, y₆, y₂, y₃, y₄, y₁]

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

See Matrix

[y₃, y₂, y₁, y₆, y₄, y₅, 0, 0]

68 . Coloring, {2, 3, 4, 8}

R: [3, 8, 8, 6, 7, 7, 5, 2]
B: [6, 3, 1, 1, 2, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 8}, {5, 7}} order: 2

See Matrix

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

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

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

See Matrix

[y₁, y₂, y₃, y₄, y₅, y₆, 0, 0]

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

Ωp(Δ)=0: p = s ² p' = s ³ p' = s ² p' = s ⁴ p' = s ⁵

R: [3, 8, 8, 1, 2, 4, 5, 5]
B: [6, 3, 1, 6, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

1 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 5, 8}} order: 6

See Matrix

[y₄, y₅, y₁, y₂, y₃, 0, 0, y₆]

Omega Rank for B : cycles: {{4, 6, 7}} order: 6

See Matrix

[y₂, y₁, y₅, y₆, 0, y₃, y₄, 0]

70 . Coloring, {2, 3, 5, 7}

Ωp(Δ)=0: p = s - 2s ³ + 8s ⁶

R: [3, 8, 8, 1, 2, 7, 4, 5]
B: [6, 3, 1, 6, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 7 vs 7 4 vs 7

Omega Rank for R : cycles: {{2, 5, 8}} order: 6

See Matrix

[y₄, y₂, y₃, y₁, y₇, 0, y₆, y₅]

Omega Rank for B : cycles: {{4, 6}, {5, 7}} order: 4

See Matrix

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

p' = s ⁴ - s ⁶ p' = s ⁵ - s ⁶ p = s ⁴ - s ⁷

71 . Coloring, {2, 3, 5, 8}

Ωp(Δ)=0: p = s ² - 4s ⁶ p' = s ³ + 2s ⁵ p' = s ² + 2s ⁴

R: [3, 8, 8, 1, 2, 7, 5, 2]
B: [6, 3, 1, 6, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 3 vs 6 3 vs 6 3 vs 6 3 vs 6

Omega Rank for R : cycles: {{2, 8}} order: 4

See Matrix

[y₂, y₃, y₁, 0, y₁, 0, y₂, y₃]

p = - s ³ + s ⁶ p = - s ³ + s ⁴ p = - s ³ + s ⁵

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

See Matrix

[y₃, 0, y₁, y₂, y₁, y₂, y₃, 0]

p = s ³ - s ⁶ p' = s ³ - s ⁵ p' = s ⁴ - s ⁵

M \ ; N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $ $ [ [0, 5, 6, 3, 3, 6, 9, 4] , [5, 0, 3, 4, 6, 5, 4, 9] , [6, 3, 0, 5, 9, 4, 3, 6] , [3, 4, 5, 0, 4, 9, 6, 5] , [3, 6, 9, 4, 0, 5, 6, 3] , [6, 5, 4, 9, 5, 0, 3, 4] , [9, 4, 3, 6, 6, 3, 0, 5] , [4, 9, 6, 5, 3, 4, 5, 0] ] $

τ= 32 , r'= 1/2

R: [3, 8, 8, 1, 2, 7, 5, 2]
B: [6, 3, 1, 6, 7, 4, 4, 5]

Ranges

Action of R on ranges, [[3], [2], [2], [1]]
Action of B on ranges, [[4], [3], [1], [4]]

Cycles: R , {{2, 8}}, B , {{4, 6}}

β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

Partitions

Action of R on partitions, [[6], [5], [3], [3], [4], [4], [5], [6]]
Action of B on partitions, [[6], [2], [8], [2], [6], [4], [8], [4]]

α([{2, 4, 5, 7}, {1, 3, 6, 8}]) = 0/1
α([{1, 2, 4, 5}, {3, 6, 7, 8}]) = 2/9
α([{2, 3, 4, 7}, {1, 5, 6, 8}]) = 2/9
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 2/9
α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 1/9
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/9
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 0/1
α([{1, 3, 4, 8}, {2, 5, 6, 7}]) = 1/9

b1 = {1, 2, 3, 6} ` , ` b2 = {2, 4, 5, 7} ` , ` b3 = {1, 3, 6, 8} ` , ` b4 = {1, 2, 4, 5} ` , ` b5 = {3, 6, 7, 8} ` , ` b6 = {2, 3, 4, 7} ` , ` b7 = {1, 5, 6, 8} ` , ` b8 = {1, 4, 5, 8} ` , ` b9 = {2, 3, 6, 7} ` , ` b10 = {1, 2, 3, 4} ` , ` b11 = {5, 6, 7, 8} ` , ` b12 = {3, 4, 7, 8} ` , ` b13 = {4, 5, 7, 8} ` , ` b14 = {1, 2, 5, 6} ` , ` b15 = {1, 3, 4, 8} ` , ` b16 = {2, 5, 6, 7}

Action of R and B on the blocks of the partitions: = [8, B, A, D, 1, 7, 6, 6, 7, 8, 9, 1, 9, D, A, B] [A, B, A, 5, 4, 10, F, 5, 4, 9, 8, 10, B, F, 9, 8]
with invariant measure [1, 0, 0, 2, 2, 2, 2, 2, 2, 1, 1, 0, 1, 0, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 3, 5, 8}

Rank 2

R,B [3, 8, 8, 1, 2, 7, 5, 2], [6, 3, 1, 6, 7, 4, 4, 5]

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

u₂ [5, 6, 3, 3, 6, 9, 4, 3, 4, 6, 5, 4, 9, 5, 9, 4, 3, 6, 4, 9, 6, 5, 5, 6, 3, 3, 4, 5] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

72 . Coloring, {2, 3, 6, 7}

Ωp(Δ)=0: p' = s ² + 2s ⁴ p' = s ³ + 2s ⁵ p = s ² - 4s ⁶

R: [3, 8, 8, 1, 7, 4, 4, 5]
B: [6, 3, 1, 6, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 3 vs 6 3 vs 6 3 vs 6 3 vs 6

Omega Rank for R : cycles: {{1, 3, 4, 5, 7, 8}} order: 6

See Matrix

[y₁, 0, y₂, y₃, y₁, 0, y₂, y₃]

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

Omega Rank for B : cycles: {{1, 2, 3, 5, 6, 7}} order: 6

See Matrix

[y₂, y₃, y₁, 0, y₂, y₃, y₁, 0]

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

M \ ; N

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

τ= 32 , r'= 1/2

R: [3, 8, 8, 1, 7, 4, 4, 5]
B: [6, 3, 1, 6, 2, 7, 5, 2]

Ranges

Action of R on ranges, [[3], [4], [4], [1]]
Action of B on ranges, [[2], [3], [1], [2]]

Cycles: R , {{1, 3, 4, 5, 7, 8}}, B , {{1, 2, 3, 5, 6, 7}}

β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

Partitions

Action of R on partitions, [[3], [1], [2], [2]]
Action of B on partitions, [[2], [4], [2], [3]]

α([{4, 5, 6, 7}, {1, 2, 3, 8}]) = 1/5
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 2/5
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/5
α([{2, 5, 7, 8}, {1, 3, 4, 6}]) = 1/5

b1 = {1, 4, 6, 7} ` , ` b2 = {1, 2, 3, 4} ` , ` b3 = {5, 6, 7, 8} ` , ` b4 = {4, 5, 6, 7} ` , ` b5 = {1, 2, 3, 8} ` , ` b6 = {2, 3, 5, 8} ` , ` b7 = {2, 5, 7, 8} ` , ` b8 = {1, 3, 4, 6}

Action of R and B on the blocks of the partitions: = [4, 1, 6, 3, 2, 5, 6, 1] [8, 6, 1, 1, 6, 7, 3, 2]
with invariant measure [2, 1, 1, 1, 1, 2, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 3, 6, 7}

Rank 2

R,B [3, 8, 8, 1, 7, 4, 4, 5], [6, 3, 1, 6, 2, 7, 5, 2]

π₂ [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, 0, 0, 0]

u₂ [3, 2, 1, 5, 2, 3, 4, 1, 4, 2, 5, 4, 1, 3, 3, 4, 5, 2, 4, 1, 2, 5, 3, 2, 1, 1, 4, 3] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

73 . Coloring, {2, 3, 6, 8}

Ωp(Δ)=0: p = s - 2s ³ - 8s ⁶

R: [3, 8, 8, 1, 7, 4, 5, 2]
B: [6, 3, 1, 6, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 4 vs 7 7 vs 7

Omega Rank for R : cycles: {{2, 8}, {5, 7}} order: 4

See Matrix

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

p = - s ⁴ + s ⁵ p = - s ⁴ + s ⁶ p = - s ⁴ + s ⁷

Omega Rank for B : cycles: {{4, 6, 7}} order: 6

See Matrix

[y₁, y₂, y₇, y₄, y₅, y₆, y₃, 0]

74 . Coloring, {2, 3, 7, 8}

Ωp(Δ)=0: p = s ⁴ p' = s ⁴ p' = s ⁵

R: [3, 8, 8, 1, 7, 7, 4, 2]
B: [6, 3, 1, 6, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 8}} order: 6

See Matrix

[y₃, y₂, y₁, y₆, 0, 0, y₅, y₄]

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

See Matrix

[y₆, y₅, y₄, y₃, y₂, y₁, 0, 0]

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

Ωp(Δ)=0: p = s ² + s ³ - 4s ⁶

R: [3, 8, 1, 6, 2, 4, 5, 5]
B: [6, 3, 8, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 3 vs 7 4 vs 7

Omega Rank for R : cycles: {{4, 6}, {1, 3}, {2, 5, 8}} order: 6

See Matrix

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

p' = s ³ - s ⁶ p' = s ² - s ⁵ p' = s - s ⁴ p = s - s ⁷

Omega Rank for B : cycles: {{1, 4, 6, 7}, {2, 3, 8}}

See Matrix

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

p = - s + s ⁵ p' = - s + s ⁵ p' = - s ² + s ⁶

76 . Coloring, {2, 4, 5, 7}

Ωp(Δ)=0: p' = s ⁵ p = s p' = s p' = s ² p' = s ⁴ p' = s ³

R: [3, 8, 1, 6, 2, 7, 4, 5]
B: [6, 3, 8, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

0 vs 6 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R : cycles: {{1, 3}, {4, 6, 7}, {2, 5, 8}} order: 6

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁, y₁, y₁]

p' = - s + s ⁴ p' = - s + s ⁷ p' = - s + s ⁶ p' = - s + s ⁵ p' = - s + s ³ p' = - s + s ² p' = 1 - s

Omega Rank for B : cycles: {{1, 4, 6}, {5, 7}, {2, 3, 8}} order: 6

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁, y₁, y₁]

p' = - 1 + s p' = - 1 + s ² p' = - 1 + s ³ p' = - 1 + s ⁴ p' = - 1 + s ⁵ p' = - 1 + s ⁶ p' = - 1 + s ⁷

` See 8-level graph

M \ ; N

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

τ= 8 , r'= 7/8

R: [3, 8, 1, 6, 2, 7, 4, 5]
B: [6, 3, 8, 1, 7, 4, 5, 2]

Ranges

Action of R on ranges, [[1]]
Action of B on ranges, [[1]]

Cycles: R , {{1, 3}, {4, 6, 7}, {2, 5, 8}}, B , {{1, 4, 6}, {5, 7}, {2, 3, 8}}

β({1, 2, 3, 4, 5, 6, 7, 8}) = 1/1

Partitions
α([{8}, {1}, {3}, {4}, {2}, {5}, {6}, {7}]) = 1/1

b1 = {8} ` , ` b2 = {1} ` , ` b3 = {3} ` , ` b4 = {4} ` , ` b5 = {2} ` , ` b6 = {5} ` , ` b7 = {6} ` , ` b8 = {7}

Action of R and B on the blocks of the partitions: = [5, 3, 2, 8, 6, 1, 4, 7] [3, 4, 5, 7, 1, 8, 2, 6]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

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

` ` See level-8 partition graph.

Right Group

Coloring {2, 4, 5, 7}

Rank 8

R,B [3, 8, 1, 6, 2, 7, 4, 5], [6, 3, 8, 1, 7, 4, 5, 2]

π₂ [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 2)

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

π₈ [1]

u₈ [1]

77 . Coloring, {2, 4, 5, 8}

Ωp(Δ)=0: p = s - 2s ³ - 8s ⁶

R: [3, 8, 1, 6, 2, 7, 5, 2]
B: [6, 3, 8, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 4 vs 7 7 vs 7

Omega Rank for R : cycles: {{2, 8}, {1, 3}} order: 4

See Matrix

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

p = - s ⁴ + s ⁶ p = - s ⁴ + s ⁷ p = - s ⁴ + s ⁵

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

See Matrix

[y₁, 0, y₂, y₃, y₆, y₄, y₅, y₇]

78 . Coloring, {2, 4, 6, 7}

Ωp(Δ)=0: p = s - 2s ³ - 8s ⁶

R: [3, 8, 1, 6, 7, 4, 4, 5]
B: [6, 3, 8, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 4 vs 7 7 vs 7

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

See Matrix

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

p' = - s ⁵ + s ⁶ p = s ⁴ - s ⁶ p' = s ⁴ - s ⁵

Omega Rank for B : cycles: {{2, 3, 8}} order: 6

See Matrix

[y₁, y₂, y₄, 0, y₃, y₅, y₆, y₇]

79 . Coloring, {2, 4, 6, 8}

Ωp(Δ)=0: p' = s ³ p' = s ⁴ p' = s ⁵ p' = s ² p' = s p = s

R: [3, 8, 1, 6, 7, 4, 5, 2]
B: [6, 3, 8, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

0 vs 6 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R : cycles: {{2, 8}, {4, 6}, {1, 3}, {5, 7}} order: 2

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁, y₁, y₁]

p' = - s + s ⁷ p' = - s + s ⁶ p' = - s + s ⁵ p' = - s + s ³ p' = - s + s ⁴ p' = - s + s ² p' = 1 - s

Omega Rank for B : cycles: {{1, 4, 6, 7}, {2, 3, 5, 8}} order: 4

See Matrix

[y₁, y₁, y₁, y₁, y₁, y₁, y₁, y₁]

p' = - 1 + s ⁴ p' = - 1 + s ⁵ p' = - 1 + s ⁶ p' = - 1 + s p' = - 1 + s ² p' = - 1 + s ³ p' = - 1 + s ⁷

` See 8-level graph

M \ ; N

τ= 8 , r'= 7/8

R: [3, 8, 1, 6, 7, 4, 5, 2]
B: [6, 3, 8, 1, 2, 7, 4, 5]

Ranges

Action of R on ranges, [[1]]
Action of B on ranges, [[1]]

Cycles: R , {{2, 8}, {4, 6}, {1, 3}, {5, 7}}, B , {{1, 4, 6, 7}, {2, 3, 5, 8}}

β({1, 2, 3, 4, 5, 6, 7, 8}) = 1/1

Partitions
α([{8}, {1}, {3}, {4}, {2}, {5}, {6}, {7}]) = 1/1

b1 = {8} ` , ` b2 = {1} ` , ` b3 = {3} ` , ` b4 = {4} ` , ` b5 = {2} ` , ` b6 = {5} ` , ` b7 = {6} ` , ` b8 = {7}

Action of R and B on the blocks of the partitions: = [5, 3, 2, 7, 1, 8, 4, 6] [3, 4, 5, 8, 6, 1, 2, 7]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

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

` ` See level-8 partition graph.

Right Group

Coloring {2, 4, 6, 8}

Rank 8

R,B [3, 8, 1, 6, 7, 4, 5, 2], [6, 3, 8, 1, 2, 7, 4, 5]

π₂ [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 4)

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

π₈ [1]

u₈ [1]

80 . Coloring, {2, 4, 7, 8}

Ωp(Δ)=0: p = s ² + s ³ - 4s ⁶

R: [3, 8, 1, 6, 7, 7, 4, 2]
B: [6, 3, 8, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 3 vs 7 4 vs 7

Omega Rank for R : cycles: {{2, 8}, {4, 6, 7}, {1, 3}} order: 6

See Matrix

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

p = - s + s ⁴ p' = - s + s ⁴ p' = - s ² + s ⁵ p = - s + s ⁷

Omega Rank for B : cycles: {{1, 4, 6}, {2, 3, 5, 8}}

See Matrix

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

p = - s + s ⁵ p' = - s + s ⁵ p' = - s ² + s ⁶

81 . Coloring, {2, 5, 6, 7}

R: [3, 8, 1, 1, 2, 4, 4, 5]
B: [6, 3, 8, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 6 3 vs 6

Omega Rank for R : cycles: {{1, 3}, {2, 5, 8}} order: 6

See Matrix

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

p' = - s ² + s ⁴ p = - s ² + s ⁶ p = - s ² + s ⁴

Omega Rank for B : cycles: {{2, 3, 8}, {5, 7}} order: 6

See Matrix

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

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

82 . Coloring, {2, 5, 6, 8}

R: [3, 8, 1, 1, 2, 4, 5, 2]
B: [6, 3, 8, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 8}, {1, 3}} order: 2

See Matrix

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

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

Omega Rank for B : cycles: {{4, 6, 7}} order: 6

See Matrix

[0, 0, y₆, y₅, y₄, y₃, y₂, y₁]

83 . Coloring, {2, 5, 7, 8}

R: [3, 8, 1, 1, 2, 7, 4, 2]
B: [6, 3, 8, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 4 vs 6 4 vs 6

Omega Rank for R : cycles: {{2, 8}, {1, 3}} order: 4

See Matrix

[y₁, y₂, 4 y₁ - 5 y₂ - y₃ + 4 y₄, y₃, 0, 0, y₄, 3 y₁ - 4 y₂ + 3 y₄]

p = s ³ - s ⁵ p' = s ³ - s ⁵

Omega Rank for B : cycles: {{4, 6}, {5, 7}} order: 4

See Matrix

[0, 0, -5 y₁ - y₃ + 4 y₂ + 4 y₄, y₁, y₃, -4 y₁ + 3 y₂ + 3 y₄, y₂, y₄]

p = s ³ - s ⁵ p' = s ³ - s ⁵

84 . Coloring, {2, 6, 7, 8}

R: [3, 8, 1, 1, 7, 4, 4, 2]
B: [6, 3, 8, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 8}, {1, 3}} order: 4

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁶ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 3, 5, 8}} order: 4

See Matrix

[0, y₄, y₁, 0, y₂, y₃, y₅, y₆]

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

Ωp(Δ)=0: p' = s ³ - 2s ⁵ p = s ² - 4s ⁶ p' = s ² - 2s ⁴

R: [3, 3, 8, 6, 2, 4, 5, 5]
B: [6, 8, 1, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 3 vs 6 3 vs 6 2 vs 6 2 vs 6

Omega Rank for R : cycles: {{4, 6}, {2, 3, 5, 8}} order: 4

See Matrix

[0, y₂, -y₂ + 3 y₁, y₁, -y₂ + 3 y₁, y₁, 0, y₂]

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

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6, 7}} order: 4

See Matrix

[y₂, y₁, 0, -y₂ + 3 y₁, 0, -y₂ + 3 y₁, y₂, y₁]

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

M \ ; N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $ $ [ [0, 3, 4, 5, 5, 4, 9, 6] , [3, 0, 5, 6, 4, 3, 6, 9] , [4, 5, 0, 3, 9, 6, 5, 4] , [5, 6, 3, 0, 6, 9, 4, 3] , [5, 4, 9, 6, 0, 3, 4, 5] , [4, 3, 6, 9, 3, 0, 5, 6] , [9, 6, 5, 4, 4, 5, 0, 3] , [6, 9, 4, 3, 5, 6, 3, 0] ] $

τ= 32 , r'= 1/2

R: [3, 3, 8, 6, 2, 4, 5, 5]
B: [6, 8, 1, 1, 7, 7, 4, 2]

Ranges

Action of R on ranges, [[3], [3], [2], [4]]
Action of B on ranges, [[4], [2], [1], [1]]

Cycles: R , {{4, 6}, {2, 3, 5, 8}}, B , {{2, 8}, {1, 4, 6, 7}}

β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

Partitions

Action of R on partitions, [[4], [6], [5], [3], [6], [4]]
Action of B on partitions, [[2], [5], [6], [1], [6], [1]]

α([{1, 3, 4, 8}, {2, 5, 6, 7}]) = 2/9
α([{2, 3, 4, 7}, {1, 5, 6, 8}]) = 1/9
α([{1, 2, 4, 5}, {3, 6, 7, 8}]) = 1/9
α([{1, 2, 3, 6}, {4, 5, 7, 8}]) = 2/9
α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/9
α([{3, 4, 7, 8}, {1, 2, 5, 6}]) = 2/9

b1 = {1, 2, 3, 6} ` , ` b2 = {1, 3, 4, 8} ` , ` b3 = {2, 5, 6, 7} ` , ` b4 = {2, 3, 4, 7} ` , ` b5 = {1, 5, 6, 8} ` , ` b6 = {4, 5, 7, 8} ` , ` b7 = {1, 2, 4, 5} ` , ` b8 = {3, 6, 7, 8} ` , ` b9 = {1, 2, 3, 4} ` , ` b10 = {5, 6, 7, 8} ` , ` b11 = {3, 4, 7, 8} ` , ` b12 = {1, 2, 5, 6}

Action of R and B on the blocks of the partitions: = [7, 1, 6, C, B, 8, A, 9, C, B, 1, 6] [2, 4, 5, A, 9, 3, B, C, B, C, 3, 2]
with invariant measure [2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2]

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

` ` See level-2 partition graph.

Sandwich

Coloring {3, 4, 5, 6}

Rank 2

R,B [3, 3, 8, 6, 2, 4, 5, 5], [6, 8, 1, 1, 7, 7, 4, 2]

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

u₂ [3, 4, 5, 5, 4, 9, 6, 5, 6, 4, 3, 6, 9, 3, 9, 6, 5, 4, 6, 9, 4, 3, 3, 4, 5, 5, 6, 3] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

86 . Coloring, {3, 4, 5, 7}

Ωp(Δ)=0: p = - s ² + s ³ + 4s ⁶

R: [3, 3, 8, 6, 2, 7, 4, 5]
B: [6, 8, 1, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 4 vs 7 3 vs 7

Omega Rank for R : cycles: {{4, 6, 7}, {2, 3, 5, 8}}

See Matrix

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

p = - s + s ⁵ p' = - s + s ⁵ p' = - s ² + s ⁶

Omega Rank for B : cycles: {{2, 8}, {1, 4, 6}, {5, 7}} order: 6

See Matrix

[y₁, y₃, 0, y₂, y₃, -y₁ - y₂ + 4 y₃, y₃, y₃]

p = s - s ⁴ p' = - s + s ⁴ p' = - s ² + s ⁵ p' = - s ³ + s ⁶

87 . Coloring, {3, 4, 5, 8}

Ωp(Δ)=0: p' = s ² p' = s ³ p' = s ⁴ p = s ² p' = s ⁵

R: [3, 3, 8, 6, 2, 7, 5, 2]
B: [6, 8, 1, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

1 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 3, 8}} order: 6

See Matrix

[0, y₁, y₂, 0, y₃, y₄, y₅, y₆]

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

See Matrix

[y₂, 0, 0, y₃, y₁, y₆, y₄, y₅]

88 . Coloring, {3, 4, 6, 7}

Ωp(Δ)=0: p = s ⁴ p' = s ⁴ p' = s ⁵

R: [3, 3, 8, 6, 7, 4, 4, 5]
B: [6, 8, 1, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 6 vs 6 6 vs 6 6 vs 6 6 vs 6

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

See Matrix

[0, 0, y₂, y₁, y₄, y₅, y₆, y₃]

Omega Rank for B : cycles: {{2, 8}} order: 6

See Matrix

[y₂, y₁, 0, 0, y₃, y₆, y₅, y₄]

89 . Coloring, {3, 4, 6, 8}

Ωp(Δ)=0: p = s ² + s ³ - 4s ⁶

R: [3, 3, 8, 6, 7, 4, 5, 2]
B: [6, 8, 1, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 6 7 vs 7 7 vs 7 3 vs 7 4 vs 7

Omega Rank for R : cycles: {{4, 6}, {2, 3, 8}, {5, 7}} order: 6

See Matrix

[0, y₁, y₂, y₃, y₃, y₃, y₃, -y₁ - y₂ + 4 y₃]

p = s - s ⁴ p' = - s + s ⁴ p' = - s ² + s ⁵ p' = - s ³ + s ⁶

Omega Rank for B : cycles: {{1, 4, 6, 7}, {2, 5, 8}}

See Matrix

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

p' = - s + s ⁵ p' = - s ² + s ⁶ p = - s + s ⁵

90 . Coloring, {3, 4, 7, 8}

Ωp(Δ)=0: p = s ² - 4s ⁶ p' = s ³ + 2s ⁵ p' = s ² + 2s ⁴

R: [3, 3, 8, 6, 7, 7, 4, 2]
B: [6, 8, 1, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 3 vs 6 3 vs 6 3 vs 6 3 vs 6

Omega Rank for R : cycles: {{4, 6, 7}, {2, 3, 8}} order: 3

See Matrix

[0, y₁, y₃, y₂, 0, y₁, y₃, y₂]

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

Omega Rank for B : cycles: {{1, 4, 6}, {2, 5, 8}} order: 3

See Matrix

[y₁, y₂, 0, y₃, y₁, y₂, 0, y₃]

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

M \ ; N

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

τ= 32 , r'= 1/2

R: [3, 3, 8, 6, 7, 7, 4, 2]
B: [6, 8, 1, 1, 2, 4, 5, 5]

Ranges

Action of R on ranges, [[3], [3], [4], [2]]
Action of B on ranges, [[2], [4], [1], [1]]

Cycles: R , {{4, 6, 7}, {2, 3, 8}}, B , {{1, 4, 6}, {2, 5, 8}}

β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

Partitions

Action of R on partitions, [[2], [3], [1], [2]]
Action of B on partitions, [[2], [4], [2], [1]]

α([{1, 2, 3, 4}, {5, 6, 7, 8}]) = 1/5
α([{3, 4, 5, 6}, {1, 2, 7, 8}]) = 2/5
α([{1, 2, 4, 7}, {3, 5, 6, 8}]) = 1/5
α([{1, 6, 7, 8}, {2, 3, 4, 5}]) = 1/5

b1 = {1, 6, 7, 8} ` , ` b2 = {2, 3, 4, 5} ` , ` b3 = {3, 4, 5, 6} ` , ` b4 = {1, 2, 3, 4} ` , ` b5 = {5, 6, 7, 8} ` , ` b6 = {1, 2, 7, 8} ` , ` b7 = {1, 2, 4, 7} ` , ` b8 = {3, 5, 6, 8}

Action of R and B on the blocks of the partitions: = [3, 6, 7, 6, 3, 8, 5, 4] [4, 5, 1, 3, 6, 2, 3, 6]
with invariant measure [1, 1, 2, 1, 1, 2, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {3, 4, 7, 8}

Rank 2

R,B [3, 3, 8, 6, 7, 7, 4, 2], [6, 8, 1, 1, 2, 4, 5, 5]

π₂ [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, 0, 0, 0]

u₂ [1, 4, 3, 5, 4, 1, 2, 3, 2, 4, 5, 2, 3, 1, 1, 2, 5, 4, 2, 3, 4, 5, 1, 4, 3, 3, 2, 1] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

91 . Coloring, {3, 5, 6, 7}

R: [3, 3, 8, 1, 2, 4, 4, 5]
B: [6, 8, 1, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 6 vs 6 3 vs 6

Omega Rank for R : cycles: {{2, 3, 5, 8}} order: 4

See Matrix

[y₄, y₃, y₂, y₁, y₆, 0, 0, y₅]

Omega Rank for B : cycles: {{2, 8}, {5, 7}} order: 4

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵ p = - s ³ + s ⁶

92 . Coloring, {3, 5, 6, 8}

R: [3, 3, 8, 1, 2, 4, 5, 2]
B: [6, 8, 1, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R : cycles: {{2, 3, 8}} order: 3

See Matrix

[y₁, y₃, y₂, y₅, y₅, 0, 0, y₄]

p = s ³ - s ⁶

Omega Rank for B : cycles: {{4, 6, 7}} order: 3

See Matrix

[y₄, 0, 0, y₂, y₁, y₅, y₃, y₄]

p = - s ³ + s ⁶

93 . Coloring, {3, 5, 7, 8}

R: [3, 3, 8, 1, 2, 7, 4, 2]
B: [6, 8, 1, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 6 vs 6 2 vs 6

Omega Rank for R : cycles: {{2, 3, 8}} order: 6

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, y₅, y₆]

Omega Rank for B : cycles: {{4, 6}, {5, 7}} order: 2

See Matrix

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

p' = s ³ - s ⁵ p' = s ⁴ - s ⁵ p' = s ² - s ⁵ p = s ² - s ⁶

94 . Coloring, {3, 6, 7, 8}

R: [3, 3, 8, 1, 7, 4, 4, 2]
B: [6, 8, 1, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 6 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 3, 8}} order: 6

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, y₅, y₆]

Omega Rank for B : cycles: {{2, 5, 8}} order: 6

See Matrix

[y₆, y₅, 0, 0, y₄, y₃, y₂, y₁]

95 . Coloring, {4, 5, 6, 7}

R: [3, 3, 1, 6, 2, 4, 4, 5]
B: [6, 8, 8, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 4 vs 6 4 vs 6

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

See Matrix

[y₂, y₁, y₄, 3 y₂ + 3 y₁ - 4 y₃, 4 y₂ + 4 y₁ - y₄ - 5 y₃, y₃, 0, 0]

p' = s ³ - s ⁵ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 8}, {5, 7}} order: 4

See Matrix

[-5 y₁ + 4 y₃ + 4 y₂ - y₄, y₁, 0, 0, y₃, y₂, y₄, -4 y₁ + 3 y₃ + 3 y₂]

p = s ³ - s ⁵ p' = - s ³ + s ⁵

96 . Coloring, {4, 5, 6, 8}

R: [3, 3, 1, 6, 2, 4, 5, 2]
B: [6, 8, 8, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 6 6 vs 6

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

See Matrix

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

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

Omega Rank for B : cycles: {{1, 4, 6, 7}} order: 4

See Matrix

[y₁, 0, 0, y₄, y₅, y₆, y₂, y₃]

97 . Coloring, {4, 5, 7, 8}

R: [3, 3, 1, 6, 2, 7, 4, 2]
B: [6, 8, 8, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 6 3 vs 6

Omega Rank for R : cycles: {{1, 3}, {4, 6, 7}} order: 6

See Matrix

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

p = s ² - s ⁶ p' = s ³ - s ⁵ p' = s ² - s ⁴

Omega Rank for B : cycles: {{1, 4, 6}, {5, 7}} order: 6

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴ p = - s ² + s ⁶

98 . Coloring, {4, 6, 7, 8}

R: [3, 3, 1, 6, 7, 4, 4, 2]
B: [6, 8, 8, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 6 6 vs 6

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

See Matrix

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

p' = s ³ - s ⁵ p' = s ⁴ - s ⁵ p = s ² - s ⁶ p' = s ² - s ⁵

Omega Rank for B : cycles: {{2, 5, 8}} order: 6

See Matrix

[y₃, y₂, 0, 0, y₁, y₅, y₆, y₄]

99 . Coloring, {5, 6, 7, 8}

Ωp(Δ)=0: p = s ³ p' = s ³ p' = s ⁴ p' = s ⁵

R: [3, 3, 1, 1, 2, 4, 4, 2]
B: [6, 8, 8, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 6 2 vs 6 2 vs 6 2 vs 4 2 vs 4

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

See Matrix

[y₂, y₁, y₂, y₁, 0, 0, 0, 0]

p = - s ² + s ⁴ p = - s ² + s ³

Omega Rank for B : cycles: {{5, 7}} order: 2

See Matrix

[0, 0, 0, 0, y₂, y₁, y₂, y₁]

p' = s ² - s ³ p = s ² - s ⁴

M \ ; N

τ= 32 , r'= 1/2

R: [3, 3, 1, 1, 2, 4, 4, 2]
B: [6, 8, 8, 6, 7, 7, 5, 5]

Ranges

Action of R on ranges, [[1], [1], [2], [2]]
Action of B on ranges, [[4], [4], [3], [3]]

Cycles: R , {{1, 3}}, B , {{5, 7}}

β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

Partitions

Action of R on partitions, [[8], [8], [6], [8], [6], [6], [8], [6]]
Action of B on partitions, [[2], [7], [7], [2], [7], [2], [7], [2]]

α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 0/1
α([{1, 4, 5, 6}, {2, 3, 7, 8}]) = 1/4
α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 0/1
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 0/1
α([{3, 4, 5, 6}, {1, 2, 7, 8}]) = 0/1
α([{1, 2, 6, 7}, {3, 4, 5, 8}]) = 1/4
α([{2, 3, 5, 6}, {1, 4, 7, 8}]) = 1/4
α([{3, 4, 6, 7}, {1, 2, 5, 8}]) = 1/4

b1 = {1, 4, 5, 8} ` , ` b2 = {3, 4, 5, 6} ` , ` b3 = {1, 2, 6, 7} ` , ` b4 = {1, 4, 6, 7} ` , ` b5 = {1, 2, 5, 6} ` , ` b6 = {3, 4, 5, 8} ` , ` b7 = {2, 3, 5, 6} ` , ` b8 = {1, 4, 7, 8} ` , ` b9 = {3, 4, 6, 7} ` , ` b10 = {1, 2, 5, 8} ` , ` b11 = {2, 3, 6, 7} ` , ` b12 = {1, 2, 7, 8} ` , ` b13 = {3, 4, 7, 8} ` , ` b14 = {2, 3, 5, 8} ` , ` b15 = {1, 4, 5, 6} ` , ` b16 = {2, 3, 7, 8}

Action of R and B on the blocks of the partitions: = [9, 3, 6, 9, 6, 3, A, 9, 3, 6, A, 6, 3, A, 9, A] [10, 8, F, F, 8, 10, 8, 7, F, 10, F, 7, 7, 10, 8, 7]
with invariant measure [0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {5, 6, 7, 8}

Rank 2

R,B [3, 3, 1, 1, 2, 4, 4, 2], [6, 8, 8, 6, 7, 7, 5, 5]

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

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

wpp [4, 4, 4, 4, 4, 4, 4, 4]

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

R: [3, 8, 8, 6, 2, 4, 5, 5]
B: [6, 3, 1, 1, 7, 7, 4, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 2 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 5, 8}, {4, 6}} order: 6

See Matrix

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

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

Omega Rank for B : cycles: {{1, 4, 6, 7}} order: 4

See Matrix

[y₃, y₄, y₁, y₂, 0, y₆, y₅, 0]

101 . Coloring, {2, 3, 4, 5, 7}

R: [3, 8, 8, 6, 2, 7, 4, 5]
B: [6, 3, 1, 1, 7, 4, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 4 vs 7 3 vs 7

Omega Rank for R : cycles: {{2, 5, 8}, {4, 6, 7}} order: 3

See Matrix

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

p' = s ² - s ⁵ p = - s ² + s ⁵ p' = - s ³ + s ⁶

Omega Rank for B : cycles: {{1, 4, 6}, {5, 7}} order: 6

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁶ p = - s ³ + s ⁷ p = - s ³ + s ⁵

102 . Coloring, {2, 3, 4, 5, 8}

R: [3, 8, 8, 6, 2, 7, 5, 2]
B: [6, 3, 1, 1, 7, 4, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R : cycles: {{2, 8}} order: 4

See Matrix

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

p = - s ⁴ + s ⁵ p = - s ⁴ + s ⁶

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

See Matrix

[y₁, 0, y₅, y₄, y₅, y₂, y₃, 0]

p = - s ³ + s ⁶

103 . Coloring, {2, 3, 4, 6, 7}

R: [3, 8, 8, 6, 7, 4, 4, 5]
B: [6, 3, 1, 1, 2, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

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

See Matrix

[0, 0, y₃, y₄, y₅, y₃ + y₄ + y₅ - y₁ - y₂, y₁, y₂]

p = s ⁵ - s ⁶

Omega Rank for B : cycles: {{1, 2, 3, 5, 6, 7}} order: 6

See Matrix

[y₄, y₃, y₂, 0, y₁, y₅, y₆, 0]

104 . Coloring, {2, 3, 4, 6, 8}

R: [3, 8, 8, 6, 7, 4, 5, 2]
B: [6, 3, 1, 1, 2, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R : cycles: {{2, 8}, {4, 6}, {5, 7}} order: 2

See Matrix

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

p = - s ² + s ⁴ p = - s ² + s ³ p = - s ² + s ⁵ p = - s ² + s ⁶ p = - s ² + s ⁷

Omega Rank for B : cycles: {{1, 4, 6, 7}} order: 4

See Matrix

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

p = s ⁴ - s ⁷ p' = s ⁴ - s ⁶ p' = s ⁵ - s ⁶

105 . Coloring, {2, 3, 4, 7, 8}

R: [3, 8, 8, 6, 7, 7, 4, 2]
B: [6, 3, 1, 1, 2, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 8}, {4, 6, 7}} order: 6

See Matrix

[0, -y₁ + y₄, y₁, 2 y₄ - y₂ - y₃, 0, y₂, y₃, y₄]

p = s ² - s ⁵ p' = - s ² + s ⁵

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

See Matrix

[y₁, y₂, y₃, y₅, y₆, y₄, 0, 0]

106 . Coloring, {2, 3, 5, 6, 7}

R: [3, 8, 8, 1, 2, 4, 4, 5]
B: [6, 3, 1, 6, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 6 vs 6 5 vs 6

Omega Rank for R : cycles: {{2, 5, 8}} order: 6

See Matrix

[y₅, y₆, y₄, y₃, y₂, 0, 0, y₁]

Omega Rank for B : cycles: {{5, 7}} order: 6

See Matrix

[y₂, y₁, y₂ + y₁ - y₅ - y₄ + y₃, 0, y₅, y₄, y₃, 0]

p = s ⁵ - s ⁶

107 . Coloring, {2, 3, 5, 6, 8}

R: [3, 8, 8, 1, 2, 4, 5, 2]
B: [6, 3, 1, 6, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R : cycles: {{2, 8}} order: 4

See Matrix

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

p' = - s ⁴ + s ⁵ p = s ⁴ - s ⁵

Omega Rank for B : cycles: {{4, 6, 7}} order: 3

See Matrix

[y₁, 0, y₂, y₃, y₂, y₅, y₄, 0]

p = - s ³ + s ⁶

108 . Coloring, {2, 3, 5, 7, 8}

R: [3, 8, 8, 1, 2, 7, 4, 2]
B: [6, 3, 1, 6, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 4 vs 6

Omega Rank for R : cycles: {{2, 8}} order: 6

See Matrix

[y₂ + y₁ + y₃ - y₅ - y₄, y₂, y₁, y₃, 0, 0, y₅, y₄]

p = s ⁵ - s ⁶

Omega Rank for B : cycles: {{4, 6}, {5, 7}} order: 4

See Matrix

[y₂, 0, 4 y₂ + 4 y₁ - y₃ - 5 y₄, y₁, 3 y₂ + 3 y₁ - 4 y₄, y₃, y₄, 0]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

109 . Coloring, {2, 3, 6, 7, 8}

R: [3, 8, 8, 1, 7, 4, 4, 2]
B: [6, 3, 1, 6, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R : cycles: {{2, 8}} order: 6

See Matrix

[y₂, y₃, y₁, y₂ - y₃ - y₁ + y₅ + y₄, 0, 0, y₅, y₄]

p = - s ⁵ + s ⁶

Omega Rank for B : cycles: {{1, 2, 3, 5, 6, 7}} order: 6

See Matrix

[y₁, y₂, y₆, 0, y₅, y₄, y₃, 0]

110 . Coloring, {2, 4, 5, 6, 7}

R: [3, 8, 1, 6, 2, 4, 4, 5]
B: [6, 3, 8, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R : cycles: {{4, 6}, {1, 3}, {2, 5, 8}} order: 6

See Matrix

[y₂, y₂, y₂, 3 y₂ - y₁, y₂, y₁, 0, y₂]

p = s - s ³ p' = - s + s ⁵ p' = - s ² + s ⁶ p' = - s + s ³ p' = - s ² + s ⁴

Omega Rank for B : cycles: {{2, 3, 8}, {5, 7}} order: 6

See Matrix

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

p = - s ³ + s ⁵ p' = - s ³ + s ⁵ p = - s ³ + s ⁷

111 . Coloring, {2, 4, 5, 6, 8}

R: [3, 8, 1, 6, 2, 4, 5, 2]
B: [6, 3, 8, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R : cycles: {{2, 8}, {4, 6}, {1, 3}} order: 2

See Matrix

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

p' = s ² - s ³ p = s ² - s ⁴ p' = - s ³ + s ⁴ p' = - s ³ + s ⁶ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{1, 4, 6, 7}} order: 4

See Matrix

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

p = - s ⁴ + s ⁶ p = - s ⁴ + s ⁷ p = - s ⁴ + s ⁵

112 . Coloring, {2, 4, 5, 7, 8}

R: [3, 8, 1, 6, 2, 7, 4, 2]
B: [6, 3, 8, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R : cycles: {{2, 8}, {1, 3}, {4, 6, 7}} order: 6

See Matrix

[y₂, 3 y₂ - y₁, y₂, y₂, 0, y₂, y₂, y₁]

p' = s ² - s ⁴ p' = - s + s ³ p' = - s ⁴ + s ⁶ p' = - s + s ⁵ p = s - s ⁵

Omega Rank for B : cycles: {{1, 4, 6}, {5, 7}} order: 6

See Matrix

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

p = s ³ - s ⁷ p' = s ³ - s ⁵ p' = s ⁴ - s ⁶

113 . Coloring, {2, 4, 6, 7, 8}

R: [3, 8, 1, 6, 7, 4, 4, 2]
B: [6, 3, 8, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 2 vs 7 4 vs 7

Omega Rank for R : cycles: {{2, 8}, {4, 6}, {1, 3}} order: 2

See Matrix

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

p' = - s ³ + s ⁶ p' = - s ³ + s ⁵ p = s ² - s ⁴ p' = - s ³ + s ⁴ p' = s ² - s ³

Omega Rank for B : cycles: {{2, 3, 5, 8}} order: 4

See Matrix

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

p = - s ⁴ + s ⁷ p = - s ⁴ + s ⁶ p = - s ⁴ + s ⁵

114 . Coloring, {2, 5, 6, 7, 8}

R: [3, 8, 1, 1, 2, 4, 4, 2]
B: [6, 3, 8, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 4 vs 5

Omega Rank for R : cycles: {{2, 8}, {1, 3}} order: 2

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴

Omega Rank for B : cycles: {{5, 7}} order: 4

See Matrix

[0, 0, y₁, 0, y₂, 2 y₁, y₃, y₄]

p = s ³ - s ⁵

115 . Coloring, {3, 4, 5, 6, 7}

R: [3, 3, 8, 6, 2, 4, 4, 5]
B: [6, 8, 1, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 4 vs 6

Omega Rank for R : cycles: {{4, 6}, {2, 3, 5, 8}} order: 4

See Matrix

[0, y₄, y₃, y₁, y₂, 3 y₃ - 4 y₁ + 3 y₂, 0, -y₄ + 4 y₃ - 5 y₁ + 4 y₂]

p = s - s ⁵ p' = s - s ⁵

Omega Rank for B : cycles: {{2, 8}, {5, 7}} order: 4

See Matrix

[y₃, y₄, 0, 0, -y₃ + 6 y₄ - y₁ - y₂, y₁, y₂, y₄]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

116 . Coloring, {3, 4, 5, 6, 8}

R: [3, 3, 8, 6, 2, 4, 5, 2]
B: [6, 8, 1, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 2 vs 6 6 vs 6

Omega Rank for R : cycles: {{4, 6}, {2, 3, 8}} order: 6

See Matrix

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

p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁵ p = - s ² + s ⁶

Omega Rank for B : cycles: {{1, 4, 6, 7}} order: 4

See Matrix

[y₁, 0, 0, y₂, y₃, y₄, y₅, y₆]

117 . Coloring, {3, 4, 5, 7, 8}

R: [3, 3, 8, 6, 2, 7, 4, 2]
B: [6, 8, 1, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 6 4 vs 6

Omega Rank for R : cycles: {{4, 6, 7}, {2, 3, 8}} order: 3

See Matrix

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

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

Omega Rank for B : cycles: {{5, 7}, {1, 4, 6}} order: 6

See Matrix

[y₄, 0, 0, y₃, y₂, -y₄ - y₃ + 2 y₂, y₂ - y₁, y₁]

p = - s ² + s ⁵ p' = - s ² + s ⁵

118 . Coloring, {3, 4, 6, 7, 8}

R: [3, 3, 8, 6, 7, 4, 4, 2]
B: [6, 8, 1, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 4 vs 6 6 vs 6

Omega Rank for R : cycles: {{4, 6}, {2, 3, 8}} order: 6

See Matrix

[0, y₁, -y₁ + 2 y₄ + 2 y₃ - y₂, y₄ + y₃, 0, y₄, y₃, y₂]

p' = s ² - s ⁵ p = s ² - s ⁵

Omega Rank for B : cycles: {{2, 5, 8}} order: 6

See Matrix

[y₁, y₂, 0, 0, y₄, y₅, y₆, y₃]

119 . Coloring, {3, 5, 6, 7, 8}

R: [3, 3, 8, 1, 2, 4, 4, 2]
B: [6, 8, 1, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 5 vs 5 3 vs 5

Omega Rank for R : cycles: {{2, 3, 8}} order: 3

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, 0, y₅]

Omega Rank for B : cycles: {{5, 7}} order: 4

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

120 . Coloring, {4, 5, 6, 7, 8}

R: [3, 3, 1, 6, 2, 4, 4, 2]
B: [6, 8, 8, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 4 vs 5

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

See Matrix

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

p' = s ² - s ⁴ p = - s ² + s ⁴

Omega Rank for B : cycles: {{5, 7}} order: 4

See Matrix

[y₄, 0, 0, 0, y₃, y₂, y₁, 2 y₄]

p = - s ³ + s ⁵

121 . Coloring, {2, 3, 4, 5, 6, 7}

R: [3, 8, 8, 6, 2, 4, 4, 5]
B: [6, 3, 1, 1, 7, 7, 5, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 5 vs 6 6 vs 6

Omega Rank for R : cycles: {{4, 6}, {2, 5, 8}} order: 6

See Matrix

[0, -3 y₁ + 5 y₂ - 3 y₅ + 5 y₃ - 3 y₄, 3 y₁, 3 y₂, 3 y₅, 3 y₃, 0, 3 y₄]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{5, 7}} order: 6

See Matrix

[y₄, y₁, y₃, 0, y₂, y₅, y₆, 0]

122 . Coloring, {2, 3, 4, 5, 6, 8}

Ωp(Δ)=0: p' = s + 4s ⁴ - 8s ⁵ p' = s ² - 2s ³ + 4s ⁴ - 4s ⁵ p = s + 4s ⁴ - 8s ⁵

R: [3, 8, 8, 6, 2, 4, 5, 2]
B: [6, 3, 1, 1, 7, 7, 4, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 2 vs 6 2 vs 6

Omega Rank for R : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

[0, y₂, 3 y₁ - y₂, y₁, 3 y₁ - y₂, y₁, 0, y₂]

p = s ² - s ⁶ p' = s ² - s ⁵ p' = s ⁴ - s ⁵ p' = s ³ - s ⁵

Omega Rank for B : cycles: {{1, 4, 6, 7}} order: 4

See Matrix

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

p = - s ² + s ³ p = - s ² + s ⁴ p = - s ² + s ⁶ p = - s ² + s ⁵

M \ ; N

$ [ [0, 0, 0, 0, 0, 0, 1, 0] , [0, 0, 0, 0, 0, 0, 0, 1] , [0, 0, 0, 0, 1, 0, 0, 0] , [0, 0, 0, 0, 0, 1, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 0, 0, 0, 0, 0, 0] ] $ $ [ [0, 5, 4, 4, 3, 3, 7, 2] , [5, 0, 2, 4, 5, 3, 2, 7] , [4, 2, 0, 2, 7, 5, 3, 5] , [4, 4, 2, 0, 5, 7, 3, 3] , [3, 5, 7, 5, 0, 2, 4, 2] , [3, 3, 5, 7, 2, 0, 4, 4] , [7, 2, 3, 3, 4, 4, 0, 5] , [2, 7, 5, 3, 2, 4, 5, 0] ] $

τ= 32 , r'= 1/2

R: [3, 8, 8, 6, 2, 4, 5, 2]
B: [6, 3, 1, 1, 7, 7, 4, 5]

Ranges

Action of R on ranges, [[3], [2], [2], [4]]
Action of B on ranges, [[4], [3], [1], [1]]

Cycles: R , {{2, 8}, {4, 6}}, B , {{1, 4, 6, 7}}

β({1, 7}) = 1/4
β({2, 8}) = 1/4
β({3, 5}) = 1/4
β({4, 6}) = 1/4

Partitions

Action of R on partitions, [[4], [6], [6], [4], [5], [5]]
Action of B on partitions, [[4], [3], [4], [3], [2], [1]]

α([{5, 6, 7, 8}, {1, 2, 3, 4}]) = 1/14
α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 1/14
α([{1, 3, 4, 8}, {2, 5, 6, 7}]) = 3/14
α([{2, 3, 4, 7}, {1, 5, 6, 8}]) = 5/14
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/7
α([{4, 5, 7, 8}, {1, 2, 3, 6}]) = 1/7

b1 = {4, 5, 7, 8} ` , ` b2 = {1, 4, 5, 8} ` , ` b3 = {1, 2, 3, 6} ` , ` b4 = {1, 2, 5, 6} ` , ` b5 = {2, 3, 6, 7} ` , ` b6 = {1, 3, 4, 8} ` , ` b7 = {2, 5, 6, 7} ` , ` b8 = {5, 6, 7, 8} ` , ` b9 = {2, 3, 4, 7} ` , ` b10 = {3, 4, 7, 8} ` , ` b11 = {1, 5, 6, 8} ` , ` b12 = {1, 2, 3, 4}

Action of R and B on the blocks of the partitions: = [5, 5, 2, 1, 2, 3, 1, 9, B, 3, 9, B] [8, A, C, 6, 4, 9, B, B, 7, 7, 6, 9]
with invariant measure [2, 2, 2, 1, 2, 3, 3, 1, 5, 1, 5, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 3, 4, 5, 6, 8}

Rank 2

R,B [3, 8, 8, 6, 2, 4, 5, 2], [6, 3, 1, 1, 7, 7, 4, 5]

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

u₂ [5, 4, 4, 3, 3, 7, 2, 2, 4, 5, 3, 2, 7, 2, 7, 5, 3, 5, 5, 7, 3, 3, 2, 4, 2, 4, 4, 5] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

123 . Coloring, {2, 3, 4, 5, 7, 8}

R: [3, 8, 8, 6, 2, 7, 4, 2]
B: [6, 3, 1, 1, 7, 4, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 8 vs 8 8 vs 8 3 vs 6 5 vs 6

Omega Rank for R : cycles: {{2, 8}, {4, 6, 7}} order: 6

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴ p = - s ² + s ⁶

Omega Rank for B : cycles: {{1, 4, 6}, {5, 7}} order: 6

See Matrix

[-3 y₁ - 3 y₂ + 5 y₅ - 3 y₃ + 5 y₄, 0, 3 y₁, 3 y₂, 3 y₅, 3 y₃, 3 y₄, 0]

p = - s ² - s ³ + s ⁵ + s ⁶

124 . Coloring, {2, 3, 4, 6, 7, 8}

Ωp(Δ)=0: p = s + 4s ⁴ p' = s + 4s ⁴ p' = s ² + 4s ⁵

R: [3, 8, 8, 6, 7, 4, 4, 2]
B: [6, 3, 1, 1, 2, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 6 4 vs 8 4 vs 8 2 vs 6 3 vs 6

Omega Rank for R : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

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

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

Omega Rank for B : cycles: {{1, 2, 3, 5, 6, 7}} order: 6

See Matrix

[y₁, y₂, y₃, 0, y₁, y₂, y₃, 0]

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

M \ ; N

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

τ= 32 , r'= 1/2

R: [3, 8, 8, 6, 7, 4, 4, 2]
B: [6, 3, 1, 1, 2, 7, 5, 5]

Ranges

Action of R on ranges, [[3], [4], [4], [2]]
Action of B on ranges, [[2], [3], [1], [1]]

Cycles: R , {{2, 8}, {4, 6}}, B , {{1, 2, 3, 5, 6, 7}}

β({1, 5}) = 1/4
β({2, 6}) = 1/4
β({3, 7}) = 1/4
β({4, 8}) = 1/4

Partitions

Action of R on partitions, [[3], [5], [4], [4], [5]]
Action of B on partitions, [[2], [5], [1], [5], [1]]

α([{2, 5, 7, 8}, {1, 3, 4, 6}]) = 1/4
α([{5, 6, 7, 8}, {1, 2, 3, 4}]) = 1/8
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 1/8
α([{1, 2, 3, 8}, {4, 5, 6, 7}]) = 1/8
α([{2, 3, 4, 5}, {1, 6, 7, 8}]) = 3/8

b1 = {2, 5, 7, 8} ` , ` b2 = {1, 3, 4, 6} ` , ` b3 = {1, 4, 6, 7} ` , ` b4 = {1, 2, 3, 8} ` , ` b5 = {4, 5, 6, 7} ` , ` b6 = {5, 6, 7, 8} ` , ` b7 = {2, 3, 4, 5} ` , ` b8 = {1, 6, 7, 8} ` , ` b9 = {2, 3, 5, 8} ` , ` b10 = {1, 2, 3, 4}

Action of R and B on the blocks of the partitions: = [9, 3, 5, 4, 5, 7, 8, 7, 4, 8] [6, A, 2, 7, 8, 8, 1, 2, 1, 7]
with invariant measure [2, 2, 1, 1, 1, 1, 3, 3, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 3, 4, 6, 7, 8}

Rank 2

R,B [3, 8, 8, 6, 7, 4, 4, 2], [6, 3, 1, 1, 2, 7, 5, 5]

π₂ [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, 0, 0, 0]

u₂ [3, 2, 2, 4, 1, 2, 2, 1, 2, 1, 4, 3, 2, 1, 2, 3, 4, 3, 2, 2, 3, 4, 3, 2, 2, 1, 2, 1] (dim 1)

wpp [4, 4, 4, 4, 4, 4, 4, 4]

125 . Coloring, {2, 3, 5, 6, 7, 8}

R: [3, 8, 8, 1, 2, 4, 4, 2]
B: [6, 3, 1, 6, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 5 vs 5 5 vs 5

Omega Rank for R : cycles: {{2, 8}} order: 4

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, 0, y₅]

Omega Rank for B : cycles: {{5, 7}} order: 4

See Matrix

[y₁, 0, y₅, 0, y₄, y₂, y₃, 0]

126 . Coloring, {2, 4, 5, 6, 7, 8}

Ωp(Δ)=0: p' = s - 4s ⁵ p = s - 4s ⁵ p' = s ² - 2s ⁴ p' = s ³ - 2s ⁵

R: [3, 8, 1, 6, 2, 4, 4, 2]
B: [6, 3, 8, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 6 3 vs 7 3 vs 7 2 vs 6 3 vs 6

Omega Rank for R : cycles: {{2, 8}, {4, 6}, {1, 3}} order: 2

See Matrix

[y₁, 3 y₁ - y₂, y₁, 3 y₁ - y₂, 0, y₂, 0, y₂]

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

Omega Rank for B : cycles: {{5, 7}} order: 4

See Matrix

[y₁, 0, y₁, 0, y₃, y₂, y₃, y₂]

p' = s ⁴ - s ⁵ p = s ³ - s ⁶ p' = s ³ - s ⁵

M \ ; N

τ= 32 , r'= 1/2

R: [3, 8, 1, 6, 2, 4, 4, 2]
B: [6, 3, 8, 1, 7, 7, 5, 5]

Ranges

Action of R on ranges, [[1], [4], [2], [2]]
Action of B on ranges, [[4], [1], [3], [3]]

Cycles: R , {{2, 8}, {4, 6}, {1, 3}}, B , {{5, 7}}

β({1, 3}) = 1/4
β({2, 4}) = 1/4
β({5, 7}) = 1/4
β({6, 8}) = 1/4

Partitions

Action of R on partitions, [[7], [1], [3], [6], [7], [6], [1], [3]]
Action of B on partitions, [[8], [4], [2], [5], [5], [8], [2], [4]]

α([{3, 4, 6, 7}, {1, 2, 5, 8}]) = 1/8
α([{1, 4, 5, 6}, {2, 3, 7, 8}]) = 1/8
α([{1, 2, 6, 7}, {3, 4, 5, 8}]) = 1/8
α([{2, 3, 5, 6}, {1, 4, 7, 8}]) = 1/8
α([{1, 2, 7, 8}, {3, 4, 5, 6}]) = 1/8
α([{1, 4, 5, 8}, {2, 3, 6, 7}]) = 1/8
α([{1, 4, 6, 7}, {2, 3, 5, 8}]) = 1/8
α([{1, 2, 5, 6}, {3, 4, 7, 8}]) = 1/8

b1 = {3, 4, 6, 7} ` , ` b2 = {1, 2, 5, 8} ` , ` b3 = {1, 2, 7, 8} ` , ` b4 = {1, 4, 5, 6} ` , ` b5 = {3, 4, 5, 6} ` , ` b6 = {2, 3, 7, 8} ` , ` b7 = {1, 4, 5, 8} ` , ` b8 = {1, 4, 6, 7} ` , ` b9 = {2, 3, 6, 7} ` , ` b10 = {1, 2, 6, 7} ` , ` b11 = {3, 4, 5, 8} ` , ` b12 = {2, 3, 5, 6} ` , ` b13 = {1, 4, 7, 8} ` , ` b14 = {1, 2, 5, 6} ` , ` b15 = {3, 4, 7, 8} ` , ` b16 = {2, 3, 5, 8}

Action of R and B on the blocks of the partitions: = [8, 10, 10, 1, 8, 2, 9, 1, 7, B, A, 7, 9, B, A, 2] [E, F, 5, D, 3, C, F, 4, E, 4, 6, 3, 5, D, C, 6]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 4, 5, 6, 7, 8}

Rank 2

R,B [3, 8, 1, 6, 2, 4, 4, 2], [6, 3, 8, 1, 7, 7, 5, 5]

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

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

wpp [4, 4, 4, 4, 4, 4, 4, 4]

127 . Coloring, {3, 4, 5, 6, 7, 8}

R: [3, 3, 8, 6, 2, 4, 4, 2]
B: [6, 8, 1, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 6 vs 6 6 vs 6 4 vs 5 4 vs 5

Omega Rank for R : cycles: {{4, 6}, {2, 3, 8}} order: 6

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{5, 7}} order: 4

See Matrix

[2 y₄, 0, 0, 0, y₁, y₂, y₃, y₄]

p = - s ³ + s ⁵

128 . Coloring, {2, 3, 4, 5, 6, 7, 8}

R: [3, 8, 8, 6, 2, 4, 4, 2]
B: [6, 3, 1, 1, 7, 7, 5, 5]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 6 7 vs 7 7 vs 7 3 vs 5 4 vs 5

Omega Rank for R : cycles: {{2, 8}, {4, 6}} order: 2

See Matrix

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

p' = s ² - s ⁴ p = - s ² + s ⁴

Omega Rank for B : cycles: {{5, 7}} order: 4

See Matrix

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

p = s ⁴ - s ⁵

SUMMARY

Graph Type
CC

ν(A)
2

ν(Δ)
2

π
[1, 1, 1, 1, 1, 1, 1, 1]

Dbly Stoch
true

SANDWICH
Total 18

No . Coloring Rank

1 {2, 6} 2

2 {3, 7} 2

3 {3, 4, 5, 6} 2

4 {3, 4, 7, 8} 2

5 {} 4

6 {2, 8} 2

7 {2, 3, 6, 7} 2

8 {2, 3, 4, 5, 6, 8} 2

9 {3, 5} 2

10 {2, 3, 4, 6, 7, 8} 2

11 {4, 8} 2

12 {5, 6, 7, 8} 2

13 {2, 3, 5, 8} 2

14 {2, 4} 2

15 {6, 8} 2

16 {5, 7} 2

17 {2, 4, 5, 6, 7, 8} 2

18 {4, 6} 2

RT GROUPS
Total 2

No . Coloring Rank Solv

1 {2, 4, 5, 7} 8 ["group", Not Solvable]

2 {2, 4, 6, 8} 8 ["group", Not Solvable]

CC Colorings
Total 2

No . Coloring Sandwich,Rank

1 {} true, 4

2 {5, 6, 7, 8} true, 2

Δ-RANK'D SC'D !RK'D τ-RANK'D R/B RANK'D NOT SYNC'D Total Runs 2^n-1
96 0 108 , 108 24 , 32 20 128 128

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

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

Right Group
Coloring	{2, 4, 5, 7}
Rank	8
R,B	[3, 8, 1, 6, 2, 7, 4, 5], [6, 3, 8, 1, 7, 4, 5, 2]
π₂	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 2)
wpp	[1, 1, 1, 1, 1, 1, 1, 1]
π₈	[1]
u₈	[1]

Right Group
Coloring	{2, 4, 6, 8}
Rank	8
R,B	[3, 8, 1, 6, 7, 4, 5, 2], [6, 3, 8, 1, 2, 7, 4, 5]
π₂	[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 4)
wpp	[1, 1, 1, 1, 1, 1, 1, 1]
π₈	[1]
u₈	[1]

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

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

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

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

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

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

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

No .	Coloring	Rank
SANDWICH		Total 18
1	{2, 6}	2
2	{3, 7}	2
3	{3, 4, 5, 6}	2
4	{3, 4, 7, 8}	2
5	{}	4
6	{2, 8}	2
7	{2, 3, 6, 7}	2
8	{2, 3, 4, 5, 6, 8}	2
9	{3, 5}	2
10	{2, 3, 4, 6, 7, 8}	2
11	{4, 8}	2
12	{5, 6, 7, 8}	2
13	{2, 3, 5, 8}	2
14	{2, 4}	2
15	{6, 8}	2
16	{5, 7}	2
17	{2, 4, 5, 6, 7, 8}	2
18	{4, 6}	2

No .	Coloring	Rank	Solv
RT GROUPS		Total 2
1	{2, 4, 5, 7}	8	["group", Not Solvable]
2	{2, 4, 6, 8}	8	["group", Not Solvable]