New Graph

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

π = [1, 2, 2, 3, 2, 2]

POSSIBLE RANKS

1 x 12
2 x 6
3 x 4

BASE DETERMINANT 231/2048, .1127929688

NullSpace of Δ

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

Nullspace of A

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

1 . Coloring, {}

Ωp(Δ)=0: p = s ³ + 2s ⁴

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 4 3 vs 5 3 vs 5 1 vs 4 3 vs 5

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

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

See Matrix

[0, 2 y₁, 0, 2 y₁, y₁, y₁]

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

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

See Matrix

[y₃ - y₂, 0, y₃, y₂, y₁, y₃ - y₁]

p = - s ³ + s ⁵ p = - s ³ + s ⁴

` See 3-level graph

M N

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

τ= 12 , r'= 2/3

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

Ranges

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

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

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {}

Rank 3

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

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

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

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

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

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

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

2 . Coloring, {2}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 4 vs 5 5 vs 5 3 vs 4 4 vs 4

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

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

See Matrix

[0, y₂, 0, y₂ + y₁ - y₃, y₁, y₃]

p = - s ³ + s ⁴

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

See Matrix

[y₁, 0, y₄, y₂, y₃, 0]

3 . Coloring, {3}

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 4 2 vs 4 2 vs 4 2 vs 4 2 vs 4

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

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

See Matrix

[0, y₁, 0, y₂, y₁, y₂]

p = s - s ³ p' = s - s ³

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

See Matrix

[y₁, 0, y₂, y₂, 0, y₁]

p' = - s ² + s ³ p = s ² - s ³

M N

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

τ= 18 , r'= 1/2

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

Ranges

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

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

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

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

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

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

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

` ` See level-2 partition graph.

Right Group

Coloring {3}

Rank 2

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

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

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

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

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

4 . Coloring, {4}

Ωp(Δ)=0: p = s + 2s ³ + 4s ⁴

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 4 4 vs 6 4 vs 6 2 vs 5 3 vs 6

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

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

See Matrix

[0, -y₁ + 2 y₂, y₁, 2 y₂, y₂, y₂]

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

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

See Matrix

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

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

` See 3-level graph

M N

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

τ= 12 , r'= 2/3

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

Ranges

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

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

β({1, 2, 5}) = 1/10
β({1, 2, 6}) = 1/10
β({1, 3, 6}) = 1/20
β({2, 4, 5}) = 1/5
β({2, 4, 6}) = 1/10
β({3, 4, 5}) = 1/5
β({3, 4, 6}) = 1/4

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {4}

Rank 3

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

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

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

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

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

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

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

5 . Coloring, {5}

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 4 3 vs 5 3 vs 5 3 vs 4 2 vs 4

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

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

See Matrix

[y₁, y₁ + y₂ - y₃, 0, y₂, y₃, 0]

p = s ³ - s ⁴

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

See Matrix

[0, 0, y₂, y₁, y₁, y₂]

p = s - s ³ p' = s - s ³

M N

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

τ= 18 , r'= 1/2

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

Ranges

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

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

Sandwich

Coloring {5}

Rank 2

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

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

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

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

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

6 . Coloring, {6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 3 vs 4 4 vs 4 2 vs 3 4 vs 4

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

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

See Matrix

[0, y₁ - y₂, 0, y₁, 0, y₂]

p = - s ² + s ³

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

See Matrix

[y₁, 0, y₄, 0, y₃, y₂]

7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[0, y₂, 0, 0, y₁, y₃]

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

See Matrix

[y₃, 0, y₁, y₂, 0, 0]

8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p = - s ² + s ⁴ p' = s ² - s ⁴

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

See Matrix

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

p = s + s ² - s ⁴ - s ⁵

9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 5 vs 5 5 vs 5 4 vs 5 3 vs 4

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

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

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

See Matrix

[0, 0, y₁ + y₂ - y₃, y₁, y₂, y₃]

p = s - s ² + s ³ - s ⁴

10 . Coloring, {2, 6}

Ωp(Δ)=0: p = s ³ p' = s ³

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 4 2 vs 4 2 vs 4 1 vs 3 2 vs 4

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

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

See Matrix

[0, y₁, 0, y₁, 0, y₁]

p = - s + s ² p = - s + s ³

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

See Matrix

[y₂ - y₁, 0, y₂, y₁, y₂, 0]

p = - s ² + s ⁴ p = - s ² + s ³

` See 3-level graph

M N

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

τ= 12 , r'= 2/3

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

Ranges

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

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

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {2, 6}

Rank 3

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

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

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

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

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

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

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

11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p = - s ⁴ + s ⁵

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

See Matrix

[y₁, y₃, y₄, y₅, 0, y₂]

12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, y₂, 0, y₃, y₄, 0]

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

See Matrix

[0, 0, y₃, y₂, 0, y₁]

13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 5 vs 5 5 vs 5 3 vs 4 4 vs 5

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

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

See Matrix

[0, y₁ + y₂ - y₃, 0, y₁, y₂, y₃]

p = - s ³ + s ⁴

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

See Matrix

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

p = s ⁴ - s ⁵

14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p = - s ⁴ + s ⁵

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

See Matrix

[0, y₅, y₄, y₃, y₂, y₁]

15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 5 vs 5 5 vs 5 2 vs 4 5 vs 5

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

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

See Matrix

[0, y₂, y₁, 3 y₂ + y₁, 0, 2 y₂]

p = - s ² + s ³ p = - s ² + s ⁴

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

See Matrix

[y₂, y₁, y₅, 0, y₃, y₄]

16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, y₃, 0, y₂, 0, 0]

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

See Matrix

[0, 0, y₁, 0, y₃, y₂]

17 . Coloring, {2, 3, 4}

Ωp(Δ)=0: p = s - 2s ³ - 4s ⁴

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 4 5 vs 5 5 vs 5 3 vs 4 4 vs 4

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

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

See Matrix

[0, y₃, 3 y₃, 0, y₁, y₂]

p = - s ² + s ⁴

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

See Matrix

[y₂, y₁, y₃, y₄, 0, 0]

18 . Coloring, {2, 3, 5}

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 4 2 vs 4 2 vs 4 2 vs 4 2 vs 3

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

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

See Matrix

[y₂, y₁, 0, 0, y₁, y₂]

p = - s + s ³ p' = - s + s ³

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

See Matrix

[0, 0, y₂, y₁, 0, -y₂ + y₁]

p = - s ² + s ³

M N

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

τ= 18 , r'= 1/2

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

Ranges

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

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

Sandwich

Coloring {2, 3, 5}

Rank 2

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

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

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

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

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

19 . Coloring, {2, 3, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 5 vs 5 4 vs 5 4 vs 4 3 vs 4

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

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

See Matrix

[0, y₄, 0, y₃, y₁, y₂]

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

See Matrix

[y₁ - y₂ + y₃, 0, y₁, y₂, y₃, 0]

p = - s ³ + s ⁴

20 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p' = - 1 + s ⁴ p' = - s + s ⁵

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

See Matrix

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

p = s ⁴ - s ⁵

21 . Coloring, {2, 4, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[0, y₁, y₄, y₃, 0, y₂]

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

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, y₂, 0, y₃, 0, y₄]

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

See Matrix

[0, 0, y₁ + y₂ - y₃, y₁, y₂, y₃]

p = - s ³ + s ⁴

23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₃, y₄, y₅, y₂, y₁, 0]

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

See Matrix

[0, y₁, y₂, y₃, 0, y₄]

24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

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

See Matrix

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

p' = 1 - s + s ² - s ³ + s ⁴ - s ⁵

25 . Coloring, {3, 5, 6}

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 4 3 vs 5 3 vs 5 3 vs 4 2 vs 4

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

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

See Matrix

[y₂, y₁ + y₂ - y₃, 0, y₁, y₃, 0]

p = - s ³ + s ⁴

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

See Matrix

[0, 0, y₂, y₁, y₁, y₂]

p = s - s ³ p' = s - s ³

M N

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

τ= 18 , r'= 1/2

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

Ranges

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

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

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

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

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

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

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

` ` See level-2 partition graph.

Right Group

Coloring {3, 5, 6}

Rank 2

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

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

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

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

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

26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 5 vs 5 5 vs 5 4 vs 4 3 vs 4

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

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

See Matrix

[y₁, y₂, y₃, y₄, 0, 0]

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

See Matrix

[0, 3 y₁, y₁, 0, y₂, y₃]

p = s ² - s ⁴

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 5 vs 5 5 vs 5 5 vs 5 2 vs 4

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

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

See Matrix

[y₃, y₄, y₅, 0, y₁, y₂]

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

See Matrix

[0, -3 y₂ + y₁, y₂, y₁, 0, 2 y₂]

p = - s ² + s ³ p = - s ² + s ⁴

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[0, y₁, y₂, y₃, y₄, y₅]

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

See Matrix

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

p = - s ⁴ + s ⁵

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

Ωp(Δ)=0: p = s ³ - 2s ⁴

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 4 3 vs 5 3 vs 5 3 vs 5 1 vs 4

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

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

See Matrix

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

p = - s ³ + s ⁵ p = - s ³ + s ⁴

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

See Matrix

[0, 0, 2 y₁, 2 y₁, y₁, y₁]

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

` See 3-level graph

M N

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

τ= 12 , r'= 2/3

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

Ranges

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

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

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {2, 3, 5, 6}

Rank 3

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

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

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

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

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

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

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

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

4 vs 4 5 vs 5 5 vs 5 5 vs 5 3 vs 5

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

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

See Matrix

[y₄, y₁, y₂, y₃, 0, y₅]

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

See Matrix

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

p = - s ² + s ⁴ p' = - s ² + s ⁴

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

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₃, y₂, y₁, y₅, y₄, 0]

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

See Matrix

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

p = s ⁴ - s ⁵

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

Ωp(Δ)=0: p = s - 2s ³ + 4s ⁴

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 4 4 vs 6 4 vs 6 4 vs 6 2 vs 5

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

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

See Matrix

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

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

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

See Matrix

[0, y₁, -y₁ + 2 y₂, 2 y₂, y₂, y₂]

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

` See 3-level graph

M N

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

τ= 12 , r'= 2/3

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

Ranges

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

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

β({1, 2, 5}) = 1/27
β({1, 2, 6}) = 1/54
β({1, 3, 5}) = 2/27
β({1, 3, 6}) = 13/108
β({2, 4, 5}) = 13/54
β({2, 4, 6}) = 11/54
β({3, 4, 5}) = 4/27
β({3, 4, 6}) = 17/108

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

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

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

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

` ` See level-3 partition graph.

Right Group

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

Rank 3

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

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

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

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

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

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

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

SUMMARY

Graph Type
CC

ν(A)
2

ν(Δ)
2

π
[1, 2, 2, 3, 2, 2]

Dbly Stoch
false

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

SANDWICH
Total 2

No . Coloring Rank

1 {5} 2

2 {2, 3, 5} 2

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

RT GROUPS
Total 7

No . Coloring Rank Solv

1 {3} 2 Solvable

2 {} 3 Not Solvable

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

4 {2, 6} 3 Solvable

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

6 {3, 5, 6} 2 Solvable

7 {4} 3 Not Solvable

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

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