New Graph

          [4, 4, 4, 7, 7, 7, 1, 1, 1], [2, 9, 5, 8, 3, 8, 5, 6, 2]

π = [3, 2, 1, 3, 2, 1, 3, 2, 1]

POSSIBLE RANKS

1 x 18
2 x 9
3 x 6

BASE DETERMINANT 2151937075/68719476736, .3131480589e-1

NullSpace of Δ

{1, 2, 4, 5, 6, 9}, {3, 7, 8}

Nullspace of A

[{2, 5, 6, 9},{1, 4}] `,` [{3, 8},{7}]

1 . Coloring, {}

Ωp(Δ)=0: p' = s ⁴ - 4s ⁶ p' = s ³ - 8s ⁶ p = s ² - 32s ⁷ p' = s ² - 16s ⁶ p' = s ⁵ - 2s ⁶

R: [4, 4, 4, 7, 7, 7, 1, 1, 1]
B: [2, 9, 5, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 7 2 vs 7 2 vs 7 1 vs 3 2 vs 6

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

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

See Matrix

[y₁, 0, 0, y₁, 0, 0, y₁, 0, 0]

p = - s + s ³ p = - s + s ²

Omega Rank for B : cycles: {{2, 9}, {3, 5}, {6, 8}} order: 2

See Matrix

[0, y₂, y₁, 0, y₂, y₁, 0, y₂, y₁]

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

` See 3-level graph

M \ ; N

$ [ [0, 0, 0, 3, 0, 0, 3, 0, 0] , [0, 0, 0, 0, 2, 0, 0, 2, 0] , [0, 0, 0, 0, 0, 1, 0, 0, 1] , [3, 0, 0, 0, 0, 0, 3, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 2, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 1] , [3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 2, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 1, 0, 0, 0] ] $ $ [ [0, 3, 3, 6, 4, 6, 6, 5, 3] , [3, 0, 2, 4, 6, 4, 5, 6, 6] , [3, 2, 0, 6, 6, 6, 3, 4, 6] , [6, 4, 6, 0, 2, 0, 6, 6, 6] , [4, 6, 6, 2, 0, 2, 6, 6, 4] , [6, 4, 6, 0, 2, 0, 6, 6, 6] , [6, 5, 3, 6, 6, 6, 0, 1, 3] , [5, 6, 4, 6, 6, 6, 1, 0, 2] , [3, 6, 6, 6, 4, 6, 3, 2, 0] ] $

τ= 27 , r'= 2/3

R: [4, 4, 4, 7, 7, 7, 1, 1, 1]
B: [2, 9, 5, 8, 3, 8, 5, 6, 2]

Ranges

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

Cycles: R , {{1, 4, 7}}, B , {{2, 9}, {3, 5}, {6, 8}}

β({1, 4, 7}) = 1/2
β({2, 5, 8}) = 1/3
β({3, 6, 9}) = 1/6

Partitions

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

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

b1 = {1, 2, 3} ` , ` b2 = {1, 5, 9} ` , ` b3 = {1, 8, 9} ` , ` b4 = {2, 3, 7} ` , ` b5 = {2, 4, 6} ` , ` b6 = {3, 7, 8} ` , ` b7 = {4, 5, 6} ` , ` b8 = {7, 8, 9}

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

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

` ` See level-3 partition graph.

Sandwich

Coloring {}

Rank 3

R,B [4, 4, 4, 7, 7, 7, 1, 1, 1], [2, 9, 5, 8, 3, 8, 5, 6, 2]

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

u₂ [3, 3, 6, 4, 6, 6, 5, 3, 2, 4, 6, 4, 5, 6, 6, 6, 6, 6, 3, 4, 6, 2, 0, 6, 6, 6, 2, 6, 6, 4, 6, 6, 6, 1, 3, 2] (dim 1)

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

π₃ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 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, 2, 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]

u₃ [2, 1, 1, 1, 2, 2, 0, 3, 1, 3, 0, 0, 0, 0, 0, 6, 5, 3, 0, 4, 3, 3, 6, 5, 3, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 3, 4, 4, 0, 5, 6, 4, 3, 4, 4, 0, 2, 2, 2, 0, 3, 4, 6, 2, 3, 4, 4, 3, 4, 6, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 2, 2, 2, 0, 1, 1, 0, 1, 3, 2, 0]

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

2 . Coloring, {2}

Ωp(Δ)=0: p = s ³ - s ⁴ - 8s ⁷

R: [4, 9, 4, 7, 7, 7, 1, 1, 1]
B: [2, 4, 5, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₄, 0, 0, y₃, 0, 0, y₂, 0, y₁]

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

See Matrix

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

p = - s ³ + s ⁵ p' = s ³ - s ⁵

3 . Coloring, {3}

R: [4, 4, 5, 7, 7, 7, 1, 1, 1]
B: [2, 9, 4, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 2 vs 4 5 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	2 vs 4	5 vs 7

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

See Matrix

[y₂, 0, 0, y₁, -y₁ + y₂, 0, y₂, 0, 0]

p = - s ² + s ⁴ p = - s ² + s ³

Omega Rank for B : cycles: {{6, 8}, {2, 9}} order: 4

See Matrix

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

p' = s ⁴ - s ⁶ p = s ⁴ - s ⁶

4 . Coloring, {4}

R: [4, 4, 4, 8, 7, 7, 1, 1, 1]
B: [2, 9, 5, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 7 vs 8 8 vs 8 2 vs 4 3 vs 7

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

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

See Matrix

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

p = - s ² + s ³ p = - s ² + s ⁴

Omega Rank for B : cycles: {{6, 8}, {2, 9}, {3, 5}} order: 2

See Matrix

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

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

5 . Coloring, {5}

Ωp(Δ)=0: p = - s ³ + s ⁴ + 4s ⁵ - 8s ⁷

R: [4, 4, 4, 7, 3, 7, 1, 1, 1]
B: [2, 9, 5, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₃, 0, y₁, y₂, 0, 0, y₄, 0, 0]

Omega Rank for B : cycles: {{6, 8}, {5, 7}, {2, 9}} order: 2

See Matrix

[0, y₁, 0, 0, y₁, y₂, y₂, y₁, y₂]

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

6 . Coloring, {6}

R: [4, 4, 4, 7, 7, 8, 1, 1, 1]
B: [2, 9, 5, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 7 vs 8 8 vs 8 2 vs 4 5 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	7 vs 8	8 vs 8	2 vs 4	5 vs 7

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

See Matrix

[y₂, 0, 0, y₂, 0, 0, y₂ - y₁, y₁, 0]

p = - s ² + s ⁴ p = - s ² + s ³

Omega Rank for B : cycles: {{3, 5}, {2, 9}} order: 4

See Matrix

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

p = s ⁴ - s ⁶ p' = - s ⁴ + s ⁶

7 . Coloring, {7}

Ωp(Δ)=0: p = s ³ - 16s ⁵ + 8s ⁶ + 32s ⁷ p' = s ³ - 4s ⁴ + 8s ⁶

R: [4, 4, 4, 7, 7, 7, 5, 1, 1]
B: [2, 9, 5, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 7 8 vs 8 8 vs 8 3 vs 4 3 vs 7

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

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

See Matrix

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

p = - s ³ + s ⁴

Omega Rank for B : cycles: {{3, 5}, {6, 8}, {2, 9}} order: 2

See Matrix

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

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

8 . Coloring, {8}

Ωp(Δ)=0: p = s ² - 6s ⁴ + 16s ⁷

R: [4, 4, 4, 7, 7, 7, 1, 6, 1]
B: [2, 9, 5, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, 0, 0, y₂, 0, y₃, y₄, 0, 0]

Omega Rank for B : cycles: {{3, 5}, {2, 9}} order: 4

See Matrix

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

p' = s ³ - s ⁵ p = - s ³ + s ⁵

9 . Coloring, {9}

R: [4, 4, 4, 7, 7, 7, 1, 1, 2]
B: [2, 9, 5, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[-y₁ + y₂, y₁, 0, y₂, 0, 0, y₂, 0, 0]

p = s ² - s ³ p' = - s ² + s ³

Omega Rank for B : cycles: {{3, 5}, {1, 2, 9}, {6, 8}} order: 6

See Matrix

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

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

10 . Coloring, {2, 3}

R: [4, 9, 5, 7, 7, 7, 1, 1, 1]
B: [2, 4, 4, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	5 vs 6

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

See Matrix

[y₃, 0, 0, y₂, y₁, 0, y₄, 0, 2 y₁]

p = - s ² + s ⁵

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

See Matrix

[0, 4 y₃, 3 y₁, 3 y₂, 3 y₃, 3 y₄, 0, 3 y₅, 0]

p = - s ⁴ + s ⁶

11 . Coloring, {2, 4}

R: [4, 9, 4, 8, 7, 7, 1, 1, 1]
B: [2, 4, 5, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	5 vs 7

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

See Matrix

[2 y₁, 0, 0, 2 y₂, 0, 0, 3 y₃, 2 y₄, 2 y₃]

p = - s ² + s ⁵

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

See Matrix

[0, -y₁ + 4 y₅ - y₃ + y₄, y₁, -y₂ + y₅ + 4 y₄, y₂, y₅, y₃, y₄, 0]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

12 . Coloring, {2, 5}

Ωp(Δ)=0: p = s ² - 2s ⁴ - 16s ⁷

R: [4, 9, 4, 7, 3, 7, 1, 1, 1]
B: [2, 4, 5, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₂, 0, y₃, y₁, 0, 0, y₄, 0, y₃]

p = - s ² + s ⁵

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

See Matrix

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

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

13 . Coloring, {2, 6}

R: [4, 9, 4, 7, 7, 8, 1, 1, 1]
B: [2, 4, 5, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 7 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	7 vs 7

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

See Matrix

[y₁, 0, 0, y₂, 0, 0, y₄, y₃, 2 y₃]

p = - s ² + s ⁵

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

See Matrix

[0, y₂, y₁, y₄, y₃, y₇, y₆, y₅, 0]

14 . Coloring, {2, 7}

R: [4, 9, 4, 7, 7, 7, 5, 1, 1]
B: [2, 4, 5, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	5 vs 7

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

See Matrix

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

p = s ⁴ - s ⁵

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

See Matrix

[2 y₁ - y₅ + 3 y₂ - y₃, 3 y₁ + 2 y₂ - y₄, y₁, y₅, y₂, y₃, 0, y₄, 0]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

15 . Coloring, {2, 8}

Ωp(Δ)=0: p = - s ³ - s ⁴ + 4s ⁵ + 8s ⁷

R: [4, 9, 4, 7, 7, 7, 1, 6, 1]
B: [2, 4, 5, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 7 7 vs 7 7 vs 7 4 vs 5 2 vs 6

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

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

See Matrix

[y₂, 0, 0, y₁, 0, y₄, y₃, 0, y₄]

p = - s ² + s ⁵

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

See Matrix

[y₂, y₁, y₂, y₂, y₁, 0, 0, y₁, 0]

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

16 . Coloring, {2, 9}

R: [4, 9, 4, 7, 7, 7, 1, 1, 2]
B: [2, 4, 5, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	5 vs 7

Omega Rank for R : cycles: {{1, 4, 7}, {2, 9}} order: 6

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

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

See Matrix

[y₄, 3 y₄ - 4 y₃ + 3 y₂ + 3 y₁ - y₅, y₃, y₂, 2 y₄ - 3 y₃ + 2 y₂ + 2 y₁, y₁, 0, y₅, 0]

p' = - s ⁴ + s ⁶ p = - s ⁴ + s ⁶

17 . Coloring, {3, 4}

R: [4, 4, 5, 8, 7, 7, 1, 1, 1]
B: [2, 9, 4, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 3 vs 5 4 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	3 vs 5	4 vs 8

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

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{6, 8}, {3, 4, 5, 7}, {2, 9}} order: 4

See Matrix

[0, 2 y₂ + 2 y₃ - 4 y₄, y₁, y₂, y₃, y₂ + y₃ - 2 y₄, -y₁ + 2 y₂ + 2 y₃ - 3 y₄, y₄, 2 y₄]

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

18 . Coloring, {3, 5}

R: [4, 4, 5, 7, 3, 7, 1, 1, 1]
B: [2, 9, 4, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 3 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	3 vs 7

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

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{6, 8}, {5, 7}, {2, 9}} order: 2

See Matrix

[0, 8 y₃ + 8 y₁ - 10 y₂, 0, y₃, 5 y₃ + 5 y₁ - 6 y₂, y₁, y₂, 6 y₃ + 6 y₁ - 7 y₂, -2 y₃ - 2 y₁ + 4 y₂]

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

19 . Coloring, {3, 6}

R: [4, 4, 5, 7, 7, 8, 1, 1, 1]
B: [2, 9, 4, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 2 vs 5 4 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	2 vs 5	4 vs 8

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

See Matrix

[y₁ + y₂, 0, 0, y₁, y₂, 0, y₁, y₂, 0]

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

Omega Rank for B : cycles: {{3, 4, 5, 6, 7, 8}, {2, 9}} order: 6

See Matrix

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

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

20 . Coloring, {3, 7}

Ωp(Δ)=0: p = - s ³ + s ⁴ - 4s ⁵ + 8s ⁷

R: [4, 4, 5, 7, 7, 7, 5, 1, 1]
B: [2, 9, 4, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p = - s ³ + s ⁴

Omega Rank for B : cycles: {{2, 9}, {6, 8}} order: 4

See Matrix

[9 y₁ - 6 y₂ - 6 y₄ - 3 y₃, 2 y₁, 6 y₁ - 4 y₂ - 4 y₄ - 2 y₃, 2 y₂, 0, 2 y₄, 0, 2 y₃, -5 y₁ + 4 y₂ + 4 y₄ + 3 y₃]

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

21 . Coloring, {3, 8}

R: [4, 4, 5, 7, 7, 7, 1, 6, 1]
B: [2, 9, 4, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 7 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	7 vs 7

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

See Matrix

[y₁, 0, 0, y₂, y₃, 2 y₃, y₄, 0, 0]

p = s ² - s ⁵

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₁, y₂, y₃, y₄, y₅, 0, 0, y₆, y₇]

22 . Coloring, {3, 9}

R: [4, 4, 5, 7, 7, 7, 1, 1, 2]
B: [2, 9, 4, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 2 vs 5 6 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	2 vs 5	6 vs 8

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

See Matrix

[y₁, y₂, 0, y₁, y₂, 0, y₁ + y₂, 0, 0]

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

Omega Rank for B : cycles: {{1, 2, 9}, {6, 8}} order: 6

See Matrix

[-y₁ + y₂ + y₃ + y₄ - y₆, y₁, -y₅ + y₂ + y₃ + y₄, y₂, y₃, y₄, 0, y₅, y₆]

p' = - s ⁴ + s ⁷ p = - s ⁴ + s ⁷

23 . Coloring, {4, 5}

R: [4, 4, 4, 8, 3, 7, 1, 1, 1]
B: [2, 9, 5, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 2 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	2 vs 6

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

See Matrix

[y₁, 0, 2 y₄, y₂, 0, 0, y₄, y₃, 0]

p = - s ² + s ⁵

Omega Rank for B : cycles: {{6, 8}, {2, 9}, {5, 7}} order: 2

See Matrix

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

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

24 . Coloring, {4, 6}

Ωp(Δ)=0: p = s ² + 2s ⁴ - 16s ⁷

R: [4, 4, 4, 8, 7, 8, 1, 1, 1]
B: [2, 9, 5, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₂, 0, 0, y₂, 0, 0, y₁, y₂ - y₁, 0]

p = s ² - s ⁴ p' = s ² - s ³

Omega Rank for B : cycles: {{3, 5}, {2, 9}} order: 4

See Matrix

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

p' = s ³ - s ⁵ p = s ³ - s ⁵

25 . Coloring, {4, 7}

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

R: [4, 4, 4, 8, 7, 7, 5, 1, 1]
B: [2, 9, 5, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 7 9 vs 9 9 vs 9 3 vs 5 4 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
5 vs 7	9 vs 9	9 vs 9	3 vs 5	4 vs 8

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

See Matrix

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

p = s - s ⁴ p' = - s + s ⁴

Omega Rank for B : cycles: {{3, 5}, {2, 9}, {6, 8}} order: 4

See Matrix

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

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

26 . Coloring, {4, 8}

R: [4, 4, 4, 8, 7, 7, 1, 6, 1]
B: [2, 9, 5, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 5 4 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	5 vs 5	4 vs 7

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

See Matrix

[y₁, 0, 0, y₂, 0, y₃, y₄, y₅, 0]

Omega Rank for B : cycles: {{3, 5}, {2, 9}} order: 4

See Matrix

[y₄ - y₂, y₃ + 2 y₁, y₃, 0, y₄, 0, 3 y₁, y₁, y₂]

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

27 . Coloring, {4, 9}

R: [4, 4, 4, 8, 7, 7, 1, 1, 2]
B: [2, 9, 5, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 2 vs 5 5 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	2 vs 5	5 vs 8

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

See Matrix

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

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

Omega Rank for B : cycles: {{3, 5}, {1, 2, 9}, {6, 8}} order: 6

See Matrix

[y₁, -y₁ + 2 y₅ + 2 y₃ - y₂, 2 y₅ + y₃ - y₄, 0, y₅ + 2 y₃, y₅, y₄, y₃, y₂]

p = - s ² + s ⁸ p = s ² - s ⁴ - s ⁵ + s ⁷ p = - s ² - s ³ + s ⁵ + s ⁶

28 . Coloring, {5, 6}

R: [4, 4, 4, 7, 3, 8, 1, 1, 1]
B: [2, 9, 5, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	4 vs 6

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

See Matrix

[y₁, 0, 2 y₄, y₂, 0, 0, y₃, y₄, 0]

p = - s ² + s ⁵

Omega Rank for B : cycles: {{2, 9}, {5, 7}} order: 4

See Matrix

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

p' = s ³ - s ⁵ p = - s ³ + s ⁵

29 . Coloring, {5, 7}

R: [4, 4, 4, 7, 3, 7, 5, 1, 1]
B: [2, 9, 5, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	5 vs 7

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 9}, {6, 8}} order: 4

See Matrix

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

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

30 . Coloring, {5, 8}

Ωp(Δ)=0: p = s ³ - 3s ⁴ + 8s ⁷

R: [4, 4, 4, 7, 3, 7, 1, 6, 1]
B: [2, 9, 5, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, 0, y₃, y₂, 0, y₃, y₄, 0, 0]

p = - s ² + s ⁵

Omega Rank for B : cycles: {{2, 9}, {5, 7}} order: 4

See Matrix

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

p = s ³ - s ⁵ p' = s ³ - s ⁵

31 . Coloring, {5, 9}

R: [4, 4, 4, 7, 3, 7, 1, 1, 2]
B: [2, 9, 5, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	4 vs 7

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

See Matrix

[y₃, y₄, 2 y₄, y₁, 0, 0, y₂, 0, 0]

p = - s ² + s ⁵

Omega Rank for B : cycles: {{1, 2, 9}, {5, 7}, {6, 8}} order: 6

See Matrix

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

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

32 . Coloring, {6, 7}

R: [4, 4, 4, 7, 7, 8, 5, 1, 1]
B: [2, 9, 5, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 5 6 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	5 vs 5	6 vs 8

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

See Matrix

[y₄, 0, 0, y₃, y₂, 0, y₁, y₅, 0]

Omega Rank for B : cycles: {{2, 9}, {3, 5}} order: 6

See Matrix

[y₄, y₃, y₂, 0, y₁, -y₄ + 2 y₂ + 3 y₁ - y₆, -y₃ + 3 y₂ + 2 y₁ - y₅, y₅, y₆]

p' = s ⁵ - s ⁷ p = s ⁵ - s ⁷

33 . Coloring, {6, 8}

R: [4, 4, 4, 7, 7, 8, 1, 6, 1]
B: [2, 9, 5, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	4 vs 7

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

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 9}, {3, 5}} order: 4

See Matrix

[-16 y₃ + 33 y₂ - 48 y₁ - 5 y₄, 5 y₃, -7 y₃ + 16 y₂ - 26 y₁, 0, 5 y₂, 0, 5 y₁, 15 y₁, 5 y₄]

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

34 . Coloring, {6, 9}

R: [4, 4, 4, 7, 7, 8, 1, 1, 2]
B: [2, 9, 5, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 2 vs 5 6 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	2 vs 5	6 vs 8

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

See Matrix

[y₁, y₂, 0, y₂ + y₁, 0, 0, y₁, y₂, 0]

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

Omega Rank for B : cycles: {{3, 5}, {1, 2, 9}} order: 6

See Matrix

[-y₁ + y₂ + y₆ - y₅, y₁, y₂ + y₆ - y₃ - y₄, 0, y₂, y₆, y₃, y₄, y₅]

p = - s ⁴ + s ⁷ p' = - s ⁴ + s ⁷

35 . Coloring, {7, 8}

R: [4, 4, 4, 7, 7, 7, 5, 6, 1]
B: [2, 9, 5, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	4 vs 6

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

See Matrix

[y₁, 0, 0, y₃, y₂, 2 y₁, y₄, 0, 0]

p = s ³ - s ⁵

Omega Rank for B : cycles: {{3, 5}, {2, 9}} order: 4

See Matrix

[2 y₁ + 3 y₃ - y₄, 3 y₁ + 2 y₃ - y₂, y₁, 0, y₃, 0, 0, y₂, y₄]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

36 . Coloring, {7, 9}

R: [4, 4, 4, 7, 7, 7, 5, 1, 2]
B: [2, 9, 5, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p' = - s ³ + s ⁴ p = s ³ - s ⁴

Omega Rank for B : cycles: {{3, 5}, {6, 8}, {1, 2, 9}} order: 6

See Matrix

[y₁, -y₁ + 3 y₂ + 3 y₃ - y₄, y₂, 0, y₃, 2 y₃, 0, 2 y₂, y₄]

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

37 . Coloring, {8, 9}

R: [4, 4, 4, 7, 7, 7, 1, 6, 2]
B: [2, 9, 5, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	4 vs 5	5 vs 6

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

See Matrix

[y₁, y₂, 0, y₃, 0, 2 y₂, y₄, 0, 0]

p = s ² - s ⁵

Omega Rank for B : cycles: {{3, 5}, {1, 2, 9}} order: 6

See Matrix

[-y₁ + 2 y₂ + 2 y₃ - y₄ - y₅, y₁, y₂, 0, y₃, 0, 0, y₄, y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

38 . Coloring, {2, 3, 4}

R: [4, 9, 5, 8, 7, 7, 1, 1, 1]
B: [2, 4, 4, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 5 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	5 vs 6	5 vs 7

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

See Matrix

[y₁, 0, 0, y₅, y₄, 0, y₃, y₂, 2 y₄]

p = - s ³ + s ⁶

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

See Matrix

[0, -y₁ + 4 y₃ - y₄ + y₅, y₁, -y₂ + y₃ + 4 y₅, y₂, y₃, y₄, y₅, 0]

p' = - s ² + s ⁶ p = - s ² + s ⁶

39 . Coloring, {2, 3, 5}

R: [4, 9, 5, 7, 3, 7, 1, 1, 1]
B: [2, 4, 4, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	8 vs 8	8 vs 8	5 vs 6	4 vs 6

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[0, y₄, 0, y₃, -5 y₃ - 5 y₂ + 14 y₁, y₂, y₁, -y₄ - 14 y₃ - 14 y₂ + 39 y₁, 0]

p' = s ³ - s ⁵ p = s ³ - s ⁵

40 . Coloring, {2, 3, 6}

R: [4, 9, 5, 7, 7, 8, 1, 1, 1]
B: [2, 4, 4, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 6 7 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	4 vs 6	7 vs 7

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

See Matrix

[y₁, 0, 0, y₂, y₄, 0, y₃, y₄, 2 y₄]

p' = - s ² + s ⁵ p = s ² - s ⁵

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

See Matrix

[0, y₁, y₂, y₃, y₄, y₅, y₆, y₇, 0]

41 . Coloring, {2, 3, 7}

Ωp(Δ)=0: p = s ² + 6s ⁴ + 16s ⁷

R: [4, 9, 5, 7, 7, 7, 5, 1, 1]
B: [2, 4, 4, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

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

p = - s ⁴ + s ⁵

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

See Matrix

[3 y₂, 2 y₁, 2 y₂, 2 y₅, 0, 2 y₄, 0, 2 y₃, 0]

p = - s ⁴ + s ⁶

42 . Coloring, {2, 3, 8}

R: [4, 9, 5, 7, 7, 7, 1, 6, 1]
B: [2, 4, 4, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, 0, 0, y₂, y₃, 2 y₃, y₄, 0, 2 y₃]

p = - s ² + s ⁵ p' = - s ² + s ⁵

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

See Matrix

[y₁, y₄, y₅, y₆, y₃, 0, 0, y₂, 0]

43 . Coloring, {2, 3, 9}

Ωp(Δ)=0: p = s ² + 2s ³ - 4s ⁵ - 8s ⁶ - 16s ⁷

R: [4, 9, 5, 7, 7, 7, 1, 1, 2]
B: [2, 4, 4, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 7

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 7	8 vs 8	8 vs 8	5 vs 6	4 vs 7

Omega Rank for R : cycles: {{1, 4, 7}, {2, 9}} order: 6

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[y₂, y₁, -7 y₂ + 3 y₁, -11 y₂ + 4 y₁ - y₄ + y₃, 3 y₂, y₄, 0, y₃, 0]

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

44 . Coloring, {2, 4, 5}

R: [4, 9, 4, 8, 3, 7, 1, 1, 1]
B: [2, 4, 5, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, 0, 2 y₄, y₂, 0, 0, y₄, y₃, 2 y₄]

p' = s ² - s ⁵ p = - s ² + s ⁵

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

See Matrix

[0, 4 y₃ + 4 y₄ - 15 y₁ - y₂, 0, y₃, y₄, y₃ + y₄ - 4 y₁, y₂, y₁, 0]

p' = - s ³ + s ⁵ p = - s ³ + s ⁵

45 . Coloring, {2, 4, 6}

Ωp(Δ)=0: p = s ³ + s ⁴ + 4s ⁵ + 8s ⁷

R: [4, 9, 4, 8, 7, 8, 1, 1, 1]
B: [2, 4, 5, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, 0, 0, y₂, 0, 0, y₄, y₃, y₄]

p = - s ² + s ⁵

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

See Matrix

[0, 2 y₄, y₁, y₂, y₃, y₄, y₅, 0, 0]

p = - s ⁴ + s ⁶

46 . Coloring, {2, 4, 7}

R: [4, 9, 4, 8, 7, 7, 5, 1, 1]
B: [2, 4, 5, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 6 4 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
7 vs 7	9 vs 9	9 vs 9	4 vs 6	4 vs 8

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

See Matrix

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

p' = s ² - s ⁵ p = s ² - s ⁵

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

See Matrix

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

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

47 . Coloring, {2, 4, 8}

R: [4, 9, 4, 8, 7, 7, 1, 6, 1]
B: [2, 4, 5, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 6 7 vs 7

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

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

See Matrix

[y₁, 0, 0, y₅, 0, y₂, y₃, y₄, y₆]

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

See Matrix

[y₁, y₂, y₃, y₆, y₇, 0, y₄, y₅, 0]

48 . Coloring, {2, 4, 9}

Ωp(Δ)=0: p = s ² - 2s ³ + 4s ⁵ + 8s ⁶ - 16s ⁷

R: [4, 9, 4, 8, 7, 7, 1, 1, 2]
B: [2, 4, 5, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 7 9 vs 9 9 vs 9 5 vs 6 6 vs 8

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
6 vs 7	9 vs 9	9 vs 9	5 vs 6	6 vs 8

Omega Rank for R : cycles: {{2, 9}, {1, 4, 8}} order: 6

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[-y₃ - y₄ + 2 y₂ + 3 y₅, -y₁ + 3 y₂ - y₆ + 2 y₅, y₁, y₃, y₄, y₂, y₆, y₅, 0]

p' = - s ⁵ + s ⁷ p = - s ⁵ + s ⁷

49 . Coloring, {2, 5, 6}

R: [4, 9, 4, 7, 3, 8, 1, 1, 1]
B: [2, 4, 5, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, 0, 2 y₄, y₃, 0, 0, y₂, y₄, 2 y₄]

p = - s ² + s ⁵ p' = - s ² + s ⁵

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

See Matrix

[0, y₁, 0, y₄, y₂, y₃, y₆, y₅, 0]

50 . Coloring, {2, 5, 7}

R: [4, 9, 4, 7, 3, 7, 5, 1, 1]
B: [2, 4, 5, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 7 vs 7

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

See Matrix

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

p = - s ³ + s ⁴ - s ⁵ + s ⁶

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

See Matrix

[y₁, y₂, 0, y₃, y₄, y₅, y₆, y₇, 0]

51 . Coloring, {2, 5, 8}

Ωp(Δ)=0: p' = s ⁵ - 2s ⁶ p' = s ² - 16s ⁶ p' = s ⁴ - 4s ⁶ p' = s ³ - 8s ⁶ p = s ² - 32s ⁷

R: [4, 9, 4, 7, 3, 7, 1, 6, 1]
B: [2, 4, 5, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, y₂, y₁, 0, 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, 2, 4, 8}, {5, 7}} order: 4

See Matrix

[y₁, y₂, 0, y₁, y₂, 0, y₁, y₂, 0]

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

` See 3-level graph

M \ ; N

$ [ [0, 0, 0, 3, 0, 0, 3, 0, 0] , [0, 0, 0, 0, 2, 0, 0, 2, 0] , [0, 0, 0, 0, 0, 1, 0, 0, 1] , [3, 0, 0, 0, 0, 0, 3, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 2, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 1] , [3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 2, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 1, 0, 0, 0] ] $ $ [ [0, 16, 10, 20, 16, 20, 20, 8, 10] , [16, 0, 17, 12, 20, 12, 12, 20, 11] , [10, 17, 0, 20, 15, 20, 10, 8, 20] , [20, 12, 20, 0, 14, 0, 20, 14, 20] , [16, 20, 15, 14, 0, 14, 10, 20, 11] , [20, 12, 20, 0, 14, 0, 20, 14, 20] , [20, 12, 10, 20, 10, 20, 0, 18, 10] , [8, 20, 8, 14, 20, 14, 18, 0, 18] , [10, 11, 20, 20, 11, 20, 10, 18, 0] ] $

τ= 27 , r'= 2/3

R: [4, 9, 4, 7, 3, 7, 1, 6, 1]
B: [2, 4, 5, 8, 7, 8, 5, 1, 2]

Ranges

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

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

β({1, 4, 7}) = 1/2
β({2, 5, 8}) = 1/3
β({3, 6, 9}) = 1/6

Partitions

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

α([{1, 8, 9}, {2, 4, 6}, {3, 5, 7}]) = 1/12
α([{1, 3, 8}, {2, 7, 9}, {4, 5, 6}]) = 1/4
α([{1, 2, 9}, {3, 5, 7}, {4, 6, 8}]) = 1/6
α([{1, 3, 8}, {2, 4, 6}, {5, 7, 9}]) = 1/4
α([{1, 5, 9}, {2, 3, 7}, {4, 6, 8}]) = 2/15
α([{1, 5, 9}, {2, 4, 6}, {3, 7, 8}]) = 1/15
α([{1, 8, 9}, {2, 3, 7}, {4, 5, 6}]) = 1/60
α([{1, 2, 9}, {3, 7, 8}, {4, 5, 6}]) = 1/30

b1 = {1, 2, 9} ` , ` b2 = {1, 3, 8} ` , ` b3 = {1, 5, 9} ` , ` b4 = {1, 8, 9} ` , ` b5 = {2, 3, 7} ` , ` b6 = {2, 4, 6} ` , ` b7 = {2, 7, 9} ` , ` b8 = {3, 5, 7} ` , ` b9 = {3, 7, 8} ` , ` b10 = {4, 5, 6} ` , ` b11 = {4, 6, 8} ` , ` b12 = {5, 7, 9}

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

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

` ` See level-3 partition graph.

Sandwich

Coloring {2, 5, 8}

Rank 3

R,B [4, 9, 4, 7, 3, 7, 1, 6, 1], [2, 4, 5, 8, 7, 8, 5, 1, 2]

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

u₂ [16, 10, 20, 16, 20, 20, 8, 10, 17, 12, 20, 12, 12, 20, 11, 20, 15, 20, 10, 8, 20, 14, 0, 20, 14, 20, 14, 10, 20, 11, 20, 14, 20, 18, 10, 18] (dim 1)

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

π₃ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 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, 2, 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]

u₃ [3, 8, 12, 8, 8, 4, 5, 10, 1, 10, 0, 6, 0, 10, 0, 20, 2, 10, 10, 6, 4, 5, 20, 2, 10, 6, 0, 0, 9, 12, 9, 5, 5, 8, 6, 0, 4, 6, 3, 6, 2, 20, 2, 4, 6, 3, 10, 3, 9, 9, 0, 10, 2, 20, 9, 5, 3, 6, 10, 2, 20, 0, 0, 6, 0, 4, 8, 5, 0, 0, 0, 12, 10, 12, 4, 8, 5, 8, 1, 9, 12, 10, 12, 6]

52 . Coloring, {2, 5, 9}

R: [4, 9, 4, 7, 3, 7, 1, 1, 2]
B: [2, 4, 5, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 7}} order: 6

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[3 y₁ - y₃ - 4 y₄ - y₅ + 3 y₂, y₁, 0, y₃, y₄, y₅, 2 y₁ - 3 y₄ + 2 y₂, y₂, 0]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

53 . Coloring, {2, 6, 7}

R: [4, 9, 4, 7, 7, 8, 5, 1, 1]
B: [2, 4, 5, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 6 vs 8

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

See Matrix

[y₁, 0, 0, y₂, y₃, 0, y₄, y₅, 2 y₅]

p = s ⁴ - s ⁶

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

See Matrix

[2 y₁ - y₃ + 3 y₂ - y₆, 3 y₁ + 2 y₂ - y₅ - y₄, y₁, y₃, y₂, y₆, y₅, y₄, 0]

p = - s + s ⁷ p' = - s + s ⁷

54 . Coloring, {2, 6, 8}

R: [4, 9, 4, 7, 7, 8, 1, 6, 1]
B: [2, 4, 5, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[-16 y₁ - 5 y₃ + 33 y₂ - 16 y₅, 5 y₁, -7 y₁ + 16 y₂ - 5 y₄ - 7 y₅, 5 y₃, 5 y₂, 0, 5 y₄, 5 y₅, 0]

p' = - s ² + s ⁶ p = - s ² + s ⁶

55 . Coloring, {2, 6, 9}

R: [4, 9, 4, 7, 7, 8, 1, 1, 2]
B: [2, 4, 5, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R : cycles: {{1, 4, 7}, {2, 9}} order: 6

See Matrix

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

p = s ² + s ³ - s ⁵ - s ⁶

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

See Matrix

[y₁, y₁ - y₃ + y₄ + y₂ + y₅ - y₆ - y₇, y₃, y₄, y₂, y₅, y₆, y₇, 0]

p = - s ⁷ + s ⁸

56 . Coloring, {2, 7, 8}

R: [4, 9, 4, 7, 7, 7, 5, 6, 1]
B: [2, 4, 5, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

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

See Matrix

[y₁, 0, 0, y₂, y₃, y₅, y₄, 0, y₅]

p = - s ⁴ + s ⁶

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

See Matrix

[2 y₁ - y₂ + 3 y₃, 3 y₁ + 2 y₃ - y₄, y₁, y₂, y₃, 0, 0, y₄, 0]

p = - s + s ⁵ p' = - s + s ⁵

57 . Coloring, {2, 7, 9}

R: [4, 9, 4, 7, 7, 7, 5, 1, 2]
B: [2, 4, 5, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {5, 7}} order: 4

See Matrix

[2 y₁ - y₃ + 3 y₂, y₁, 0, y₄, 3 y₁ - y₄ + 2 y₂, 0, y₃, 0, y₂]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

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

See Matrix

[y₂, 2 y₁ + 3 y₅ - y₄, y₁, -y₂ + 3 y₁ + 2 y₅ - y₃, y₅, y₃, 0, y₄, 0]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

58 . Coloring, {2, 8, 9}

R: [4, 9, 4, 7, 7, 7, 1, 6, 2]
B: [2, 4, 5, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Omega Rank for R : cycles: {{2, 9}, {1, 4, 7}} order: 6

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[y₃, 3 y₃ - 4 y₁ + 3 y₂ - y₄, y₁, y₂, 2 y₃ - 3 y₁ + 2 y₂, 0, 0, y₄, 0]

p = - s + s ⁵ p' = - s + s ⁵

59 . Coloring, {3, 4, 5}

R: [4, 4, 5, 8, 3, 7, 1, 1, 1]
B: [2, 9, 4, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 3 vs 7

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{2, 9}, {5, 7}, {6, 8}} order: 2

See Matrix

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

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

60 . Coloring, {3, 4, 6}

R: [4, 4, 5, 8, 7, 8, 1, 1, 1]
B: [2, 9, 4, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 3 vs 5 5 vs 7

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

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 9}, {3, 4, 5, 7}} order: 4

See Matrix

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

p = - s ² + s ⁶ p' = - s ² + s ⁶

61 . Coloring, {3, 4, 7}

R: [4, 4, 5, 8, 7, 7, 5, 1, 1]
B: [2, 9, 4, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 6 vs 8

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

See Matrix

[-7 y₁ + 11 y₂ + 11 y₃ - 7 y₄, 0, 0, 7 y₁, 7 y₂, 0, 7 y₃, 7 y₄, 0]

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 9}, {6, 8}} order: 6

See Matrix

[-y₂ + y₃ + 4 y₅ - y₆, -y₁ + 4 y₃ - y₄ + y₅, y₁, y₂, 0, y₃, y₄, y₅, y₆]

p = - s ⁵ + s ⁷ p' = - s ⁵ + s ⁷

62 . Coloring, {3, 4, 8}

R: [4, 4, 5, 8, 7, 7, 1, 6, 1]
B: [2, 9, 4, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 6 6 vs 8

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

See Matrix

[y₁, 0, 0, y₅, y₆, y₃, y₄, y₂, 0]

Omega Rank for B : cycles: {{2, 9}, {3, 4, 5, 7}} order: 4

See Matrix

[y₆, y₅, y₄, y₃, y₂, 0, y₁, -y₅ + y₄ + y₁, -y₆ + y₃ + y₂]

p' = s ³ - s ⁷ p = s ³ - s ⁷

63 . Coloring, {3, 4, 9}

R: [4, 4, 5, 8, 7, 7, 1, 1, 2]
B: [2, 9, 4, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 9 9 vs 9 3 vs 6 6 vs 9

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

See Matrix

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

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

Omega Rank for B : cycles: {{3, 4, 5, 7}, {1, 2, 9}, {6, 8}}

See Matrix

[-y₁ + 2 y₃ + 2 y₅ - y₆, y₁, 2 y₃ + y₅ - y₄, -y₂ + y₃ + 2 y₅, y₂, y₃, y₄, y₅, y₆]

p' = - 1 - s - s ² + s ⁴ + s ⁵ + s ⁶ p' = 1 - s ³ - s ⁴ + s ⁷ p' = s - s ⁴ - s ⁵ + s ⁸

64 . Coloring, {3, 5, 6}

R: [4, 4, 5, 7, 3, 8, 1, 1, 1]
B: [2, 9, 4, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 5 vs 7

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{2, 9}, {5, 7}} order: 4

See Matrix

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

p' = - s ⁴ + s ⁶ p = - s ⁴ + s ⁶

65 . Coloring, {3, 5, 7}

Ωp(Δ)=0: p = s ² - 2s ⁴ + 8s ⁵ - 16s ⁷

R: [4, 4, 5, 7, 3, 7, 5, 1, 1]
B: [2, 9, 4, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

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

p = s ⁴ - s ⁵

Omega Rank for B : cycles: {{6, 8}, {2, 9}} order: 4

See Matrix

[2 y₁, 9 y₁ - 15 y₃ - 11 y₂ + 9 y₄, 0, 2 y₃, 0, 2 y₂, 4 y₃, 7 y₁ - 9 y₃ - 9 y₂ + 7 y₄, 2 y₄]

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

66 . Coloring, {3, 5, 8}

R: [4, 4, 5, 7, 3, 7, 1, 6, 1]
B: [2, 9, 4, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[-14 y₄ - y₂ + 39 y₃ - 14 y₁ - y₅, y₄, 0, y₂, y₃, 0, -5 y₄ + 14 y₃ - 5 y₁, y₁, y₅]

p = - s ⁴ + s ⁶ p' = s ⁴ - s ⁶

67 . Coloring, {3, 5, 9}

R: [4, 4, 5, 7, 3, 7, 1, 1, 2]
B: [2, 9, 4, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 8

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

See Matrix

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

p = s ² + s ³ - s ⁵ - s ⁶

Omega Rank for B : cycles: {{6, 8}, {5, 7}, {1, 2, 9}} order: 6

See Matrix

[-y₁ + 6 y₂ + 6 y₃ - 6 y₄ - y₅, y₁, 0, y₂, 5 y₂ + 5 y₃ - 6 y₄, y₃, y₄, 6 y₂ + 6 y₃ - 7 y₄, y₅]

p = - s ² - s ³ + s ⁵ + s ⁶ p = - s ² + s ⁸ p = s ² - s ⁴ - s ⁵ + s ⁷

68 . Coloring, {3, 6, 7}

R: [4, 4, 5, 7, 7, 8, 5, 1, 1]
B: [2, 9, 4, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 5 8 vs 8

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

See Matrix

[y₁, 0, 0, y₂, y₃, 0, y₄, y₅, 0]

Omega Rank for B : cycles: {{2, 9}} order: 8

See Matrix

[y₁, y₂, y₃, y₄, 0, y₅, y₆, y₇, y₈]

69 . Coloring, {3, 6, 8}

R: [4, 4, 5, 7, 7, 8, 1, 6, 1]
B: [2, 9, 4, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 8 vs 8

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{2, 9}} order: 8

See Matrix

[y₂, y₁, y₅, y₄, y₃, 0, y₆, y₇, y₈]

70 . Coloring, {3, 6, 9}

Ωp(Δ)=0: p = s - 5s ⁵ - 12s ⁷ p' = s + s ⁴ - 4s ⁶ p' = s ² + 3s ⁴ + 4s ⁶ p' = s ³ + s ⁴ + 4s ⁶ p' = - s ⁴ + s ⁵ - 2s ⁶

R: [4, 4, 5, 7, 7, 8, 1, 1, 2]
B: [2, 9, 4, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

2 vs 7 2 vs 9 3 vs 9 2 vs 6 3 vs 9

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

See Matrix

[y₁, y₂, 0, y₁, y₂, 0, y₁, y₂, 0]

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, 2, 9}, {3, 4, 5, 6, 7, 8}} order: 6

See Matrix

[y₁, y₂, y₃, y₁, y₂, y₃, y₁, y₂, y₃]

p' = - s ² + s ⁵ p' = - 1 + s ⁶ p' = - s ² + s ⁸ p' = - s + s ⁷ p' = - 1 + s ³ p' = - s + s ⁴

` See 3-level graph

M \ ; N

$ [ [0, 0, 0, 3, 0, 0, 3, 0, 0] , [0, 0, 0, 0, 2, 0, 0, 2, 0] , [0, 0, 0, 0, 0, 1, 0, 0, 1] , [3, 0, 0, 0, 0, 0, 3, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 2, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 1] , [3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 2, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 1, 0, 0, 0] ] $ $ [ [0, 186, 280, 535, 414, 480, 535, 470, 310] , [186, 0, 293, 434, 535, 405, 450, 535, 372] , [280, 293, 0, 520, 525, 535, 270, 252, 535] , [535, 434, 520, 0, 126, 90, 535, 510, 460] , [414, 535, 525, 126, 0, 180, 530, 535, 365] , [480, 405, 535, 90, 180, 0, 500, 485, 535] , [535, 450, 270, 535, 530, 500, 0, 90, 300] , [470, 535, 252, 510, 535, 485, 90, 0, 333] , [310, 372, 535, 460, 365, 535, 300, 333, 0] ] $

τ= 27 , r'= 2/3

R: [4, 4, 5, 7, 7, 8, 1, 1, 2]
B: [2, 9, 4, 8, 3, 7, 5, 6, 1]

Ranges

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

Cycles: R , {{1, 4, 7}}, B , {{1, 2, 9}, {3, 4, 5, 6, 7, 8}}

β({1, 4, 7}) = 1/2
β({2, 5, 8}) = 1/3
β({3, 6, 9}) = 1/6

Partitions

Too many possibilities to consider

Sandwich

Coloring {3, 6, 9}

Rank 3

R,B [4, 4, 5, 7, 7, 8, 1, 1, 2], [2, 9, 4, 8, 3, 7, 5, 6, 1]

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

u₂ [186, 280, 535, 414, 480, 535, 470, 310, 293, 434, 535, 405, 450, 535, 372, 520, 525, 535, 270, 252, 535, 126, 90, 535, 510, 460, 180, 530, 535, 365, 500, 485, 535, 90, 300, 333] (dim 1)

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

π₃ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 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, 2, 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]

u₃ [323, 255, 195, 235, 303, 363, 0, 795, 447, 675, 45, 70, 165, 15, 105, 1605, 1335, 705, 110, 1227, 1047, 685, 1335, 1095, 765, 75, 225, 245, 531, 849, 489, 25, 30, 390, 75, 125, 1047, 1227, 646, 150, 1335, 1605, 606, 855, 1065, 726, 15, 470, 510, 343, 225, 765, 686, 1335, 510, 785, 726, 1065, 705, 606, 1605, 0, 105, 150, 0, 363, 303, 35, 165, 145, 45, 195, 675, 699, 430, 390, 30, 255, 375, 489, 215, 795, 849, 55]

71 . Coloring, {3, 7, 8}

Ωp(Δ)=0: p = s ³ + s ⁴ - 8s ⁷

R: [4, 4, 5, 7, 7, 7, 5, 6, 1]
B: [2, 9, 4, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, 0, y₂, y₃, 2 y₁, y₄, 0, 0]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, 0, y₅, y₆]

72 . Coloring, {3, 7, 9}

R: [4, 4, 5, 7, 7, 7, 5, 1, 2]
B: [2, 9, 4, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 3 vs 5 6 vs 7

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

See Matrix

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

p = - s ³ + s ⁴ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{6, 8}, {1, 2, 9}} order: 6

See Matrix

[y₃, y₄, y₃ + y₄ - y₁ - y₂ - y₆ + y₅, y₁, 0, y₂, 0, y₆, y₅]

p = - s ³ - s ⁴ + s ⁶ + s ⁷

73 . Coloring, {3, 8, 9}

R: [4, 4, 5, 7, 7, 7, 1, 6, 2]
B: [2, 9, 4, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, y₃, 0, y₂, y₃, 2 y₃, y₄, 0, 0]

p' = s ² - s ⁵ p = s ² - s ⁵

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₂, y₁, y₇, y₆, y₅, 0, 0, y₄, y₃]

74 . Coloring, {4, 5, 6}

Ωp(Δ)=0: p = s ³ + s ⁴ - 8s ⁷

R: [4, 4, 4, 8, 3, 8, 1, 1, 1]
B: [2, 9, 5, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₂, 0, y₁, y₄, 0, 0, 0, y₃, 0]

Omega Rank for B : cycles: {{2, 9}, {5, 7}} order: 2

See Matrix

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

p = s ² - s ⁴ p' = - s ² + s ⁴

75 . Coloring, {4, 5, 7}

R: [4, 4, 4, 8, 3, 7, 5, 1, 1]
B: [2, 9, 5, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 6 5 vs 7

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

See Matrix

[y₆, 0, y₅, y₃, y₄, 0, y₁, y₂, 0]

Omega Rank for B : cycles: {{2, 9}, {6, 8}} order: 4

See Matrix

[y₂ + 4 y₄ - y₁ - y₅, 4 y₂ - y₃ + y₄, 0, 0, y₁, y₂, y₃, y₄, y₅]

p = s ⁴ - s ⁶ p' = - s ⁴ + s ⁶

76 . Coloring, {4, 5, 8}

R: [4, 4, 4, 8, 3, 7, 1, 6, 1]
B: [2, 9, 5, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₃, 0, y₂, y₁, 0, y₅, y₆, y₄, 0]

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

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

p' = s ³ - s ⁵ p = s ³ - s ⁵

77 . Coloring, {4, 5, 9}

R: [4, 4, 4, 8, 3, 7, 1, 1, 2]
B: [2, 9, 5, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 6 4 vs 7

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

See Matrix

[y₁, y₂, 2 y₂, y₃, 0, 0, y₂, y₄, 0]

p = - s ² + s ⁵ p' = - s ² + s ⁵

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}, {6, 8}} order: 6

See Matrix

[y₁, -y₁ + 2 y₂ - 2 y₃ - y₄, 0, 0, y₂, y₂ - 2 y₃, 2 y₂ - 3 y₃, y₃, y₄]

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

78 . Coloring, {4, 6, 7}

R: [4, 4, 4, 8, 7, 8, 5, 1, 1]
B: [2, 9, 5, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

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

See Matrix

[-5 y₁ + 13 y₂ + 13 y₃ - 5 y₄, 0, 0, 5 y₁, 5 y₂, 0, 5 y₃, 5 y₄, 0]

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 9}, {3, 5}} order: 4

See Matrix

[2 y₅ + 3 y₃ - y₂ - y₁, y₄, y₅, 0, y₃, y₂, -y₄ + 3 y₅ + 2 y₃, 0, y₁]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

79 . Coloring, {4, 6, 8}

Ωp(Δ)=0: p = s ² - 2s ⁴ + 8s ⁵ - 16s ⁷

R: [4, 4, 4, 8, 7, 8, 1, 6, 1]
B: [2, 9, 5, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, 0, y₂, 0, y₅, y₄, y₃, 0]

Omega Rank for B : cycles: {{2, 9}, {3, 5}} order: 2

See Matrix

[-5 y₃ + 2 y₁ + 2 y₂, -4 y₃ + 2 y₁ + 2 y₂, y₁, 0, y₂, 0, -10 y₃ + 4 y₁ + 4 y₂, 0, y₃]

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

80 . Coloring, {4, 6, 9}

R: [4, 4, 4, 8, 7, 8, 1, 1, 2]
B: [2, 9, 5, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 2 vs 5 5 vs 7

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

See Matrix

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

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

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}} order: 6

See Matrix

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

p = s ³ - s ⁶ p' = s ³ - s ⁶

81 . Coloring, {4, 7, 8}

R: [4, 4, 4, 8, 7, 7, 5, 6, 1]
B: [2, 9, 5, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 6 4 vs 7

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

See Matrix

[y₄, 0, 0, y₅, y₁, y₆, y₂, y₃, 0]

Omega Rank for B : cycles: {{2, 9}, {3, 5}} order: 4

See Matrix

[2 y₁ + 3 y₂ - y₄, 3 y₁ + 2 y₂ - 4 y₃, y₁, 0, y₂, 0, 3 y₃, y₃, y₄]

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

82 . Coloring, {4, 7, 9}

R: [4, 4, 4, 8, 7, 7, 5, 1, 2]
B: [2, 9, 5, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 7 vs 8 8 vs 8 4 vs 6 5 vs 8

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

See Matrix

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

p' = - s ² + s ⁵ p = - s ² + s ⁵

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}, {6, 8}} order: 6

See Matrix

[-y₁ + 4 y₅ - y₄ + 4 y₃ - y₂, y₁, y₅, 0, y₃, y₅, y₄, y₃, y₂]

p = - s ² + s ⁸ p = - s ² - s ³ + s ⁵ + s ⁶ p' = s ² + s ³ - s ⁵ - s ⁶

83 . Coloring, {4, 8, 9}

R: [4, 4, 4, 8, 7, 7, 1, 6, 2]
B: [2, 9, 5, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 6 3 vs 7

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

See Matrix

[y₁, y₂, 0, y₃, 0, y₄, y₅, y₆, 0]

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}} order: 6

See Matrix

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

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

84 . Coloring, {5, 6, 7}

R: [4, 4, 4, 7, 3, 8, 5, 1, 1]
B: [2, 9, 5, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 6 6 vs 7

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

See Matrix

[y₂, 0, y₃, y₁, y₆, 0, y₄, y₅, 0]

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₂, y₁, 0, 0, y₄, y₃, y₅, 3 y₄, y₆]

p = s ⁵ - s ⁷

85 . Coloring, {5, 6, 8}

R: [4, 4, 4, 7, 3, 8, 1, 6, 1]
B: [2, 9, 5, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[5 y₁, -16 y₁ + 33 y₃ - 5 y₄ - 16 y₂, 0, 0, -7 y₁ + 16 y₃ - 7 y₂, 0, 5 y₃, 5 y₄, 5 y₂]

p = s ³ - s ⁵ p' = s ³ - s ⁵

86 . Coloring, {5, 6, 9}

R: [4, 4, 4, 7, 3, 8, 1, 1, 2]
B: [2, 9, 5, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 6 5 vs 7

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

See Matrix

[y₁, y₃, 2 y₃, y₄, 0, 0, y₂, y₃, 0]

p' = - s ² + s ⁵ p = - s ² + s ⁵

Omega Rank for B : cycles: {{1, 2, 9}, {5, 7}} order: 6

See Matrix

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

p = - s ³ + s ⁶ p' = - s ³ + s ⁶

87 . Coloring, {5, 7, 8}

R: [4, 4, 4, 7, 3, 7, 5, 6, 1]
B: [2, 9, 5, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

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

See Matrix

[y₄, 0, y₃, y₁, y₂, 2 y₄, y₅, 0, 0]

p = - s ² + s ⁶

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₁, y₂, 0, 0, y₃, 0, y₄, 4 y₃, y₅]

p = - s ⁴ + s ⁶

88 . Coloring, {5, 7, 9}

R: [4, 4, 4, 7, 3, 7, 5, 1, 2]
B: [2, 9, 5, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

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

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

Omega Rank for B : cycles: {{6, 8}, {1, 2, 9}} order: 6

See Matrix

[2 y₃ + 2 y₂, 2 y₃ + 2 y₂ - 2 y₁, 0, 0, 2 y₁, 2 y₄, 2 y₃, 3 y₃ + 3 y₂ - 2 y₄, 2 y₂]

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

89 . Coloring, {5, 8, 9}

R: [4, 4, 4, 7, 3, 7, 1, 6, 2]
B: [2, 9, 5, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 6

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

See Matrix

[y₁, y₂, 2 y₂, y₄, 0, 2 y₂, y₃, 0, 0]

p' = - s ² + s ⁵ p = - s ² + s ⁵

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}} order: 6

See Matrix

[-y₁ + 2 y₂ + 2 y₃ - y₄ - y₅, y₁, 0, 0, y₂, 0, y₃, y₄, y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

90 . Coloring, {6, 7, 8}

R: [4, 4, 4, 7, 7, 8, 5, 6, 1]
B: [2, 9, 5, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 6 4 vs 7

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

See Matrix

[-15 y₂ - y₁ + 4 y₄ + 4 y₃, 0, 0, y₄, y₃, y₂, y₁, y₄ + y₃ - 4 y₂, 0]

p' = - s ³ + s ⁵ p = s ³ - s ⁵

Omega Rank for B : cycles: {{2, 9}, {3, 5}} order: 4

See Matrix

[2 y₁ + 3 y₂ - y₃, 3 y₁ + 2 y₂ - 4 y₄, y₁, 0, y₂, 0, y₄, 3 y₄, y₃]

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

91 . Coloring, {6, 7, 9}

R: [4, 4, 4, 7, 7, 8, 5, 1, 2]
B: [2, 9, 5, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 7 vs 8 8 vs 8 5 vs 6 7 vs 8

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

See Matrix

[y₁, y₄, 0, y₃, y₂, 0, y₅, y₄, 0]

p = s ⁴ - s ⁶

Omega Rank for B : cycles: {{3, 5}, {1, 2, 9}} order: 6

See Matrix

[y₃, -y₃ + 5 y₂ + 5 y₁ - y₇ - y₅ - y₆ - y₄, y₂, 0, y₁, y₇, y₅, y₆, y₄]

p = - s ⁴ - s ⁵ + s ⁷ + s ⁸

92 . Coloring, {6, 8, 9}

R: [4, 4, 4, 7, 7, 8, 1, 6, 2]
B: [2, 9, 5, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 5 vs 7

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{3, 5}, {1, 2, 9}} order: 6

See Matrix

[-7 y₁ + 11 y₂ + 11 y₃ - 10 y₄ - 7 y₅, 7 y₁, 7 y₂, 0, 7 y₃, 0, 7 y₄, 21 y₄, 7 y₅]

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

93 . Coloring, {7, 8, 9}

R: [4, 4, 4, 7, 7, 7, 5, 6, 2]
B: [2, 9, 5, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[0, y₄, 0, y₂, y₃, 2 y₄, y₁, 0, 0]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{3, 5}, {1, 2, 9}} order: 6

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

R: [4, 9, 5, 8, 3, 7, 1, 1, 1]
B: [2, 4, 4, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 4 vs 6

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

See Matrix

[y₃, 0, y₂, -y₃ + 5 y₂ + 5 y₁ - 3 y₄ - y₅, y₁, 0, y₄, y₅, 2 y₄]

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

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

See Matrix

[0, y₄, 0, y₃, -y₃ - 15 y₁ + 4 y₄ + 4 y₂, y₁, y₂, y₄ - 4 y₁ + y₂, 0]

p = s ³ - s ⁵ p' = s ³ - s ⁵

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

R: [4, 9, 5, 8, 7, 8, 1, 1, 1]
B: [2, 4, 4, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

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

See Matrix

[y₁, 0, 0, y₅, y₄, 0, y₂, y₃, 2 y₄]

p = s ³ - s ⁶

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

See Matrix

[0, 2 y₅, y₂, y₁, y₄, y₅, y₃, 0, 0]

p = s ² - s ⁶

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

R: [4, 9, 5, 8, 7, 7, 5, 1, 1]
B: [2, 4, 4, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

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

See Matrix

[-7 y₃ + 11 y₄ + 11 y₁ - 7 y₅ - 7 y₂, 0, 0, 7 y₃, 7 y₄, 0, 7 y₁, 7 y₅, 7 y₂]

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[-y₅ + y₂ + 4 y₁, y₃, y₄, y₅, 0, y₂, -y₃ - y₄ + 4 y₂ + y₁, y₁, 0]

p = s ² - s ⁶ p' = s ² - s ⁶

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

R: [4, 9, 5, 8, 7, 7, 1, 6, 1]
B: [2, 4, 4, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 7 vs 7

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

See Matrix

[y₁, 0, 0, y₆, y₅, y₃, y₄, y₂, 2 y₅]

p = - s ² + s ⁷

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

See Matrix

[y₁, y₂, y₃, y₄, y₅, 0, y₆, y₇, 0]

98 . Coloring, {2, 3, 4, 9}

R: [4, 9, 5, 8, 7, 7, 1, 1, 2]
B: [2, 4, 4, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R : cycles: {{1, 4, 8}, {2, 9}} order: 6

See Matrix

[5 y₄ - y₂ - y₃ - y₅ - y₁ + 5 y₆, y₄, 0, y₂, y₃, 0, y₅, y₁, y₆]

p = - s ³ - s ⁴ + s ⁶ + s ⁷

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

See Matrix

[y₆, y₅, y₃, y₄, -y₆ - y₄ + 2 y₂ + 3 y₁, y₂, -y₅ - y₃ + 3 y₂ + 2 y₁, y₁, 0]

p' = - s ³ + s ⁷ p = - s ³ + s ⁷

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

R: [4, 9, 5, 7, 3, 8, 1, 1, 1]
B: [2, 4, 4, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 6

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

See Matrix

[5 y₁ - y₂ + 5 y₃ - y₄ - 3 y₅, 0, y₁, y₂, y₃, 0, y₄, y₅, 2 y₅]

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

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

See Matrix

[0, y₆, 0, y₅, y₄, y₃, y₂, y₁, 0]

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

Ωp(Δ)=0: p = s ³ + 3s ⁴ + 8s ⁷

R: [4, 9, 5, 7, 3, 7, 5, 1, 1]
B: [2, 4, 4, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

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

p = - s ⁵ + s ⁶

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

See Matrix

[y₁, y₂, 0, y₄, 0, y₃, y₅, y₆, 0]

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

R: [4, 9, 5, 7, 3, 7, 1, 6, 1]
B: [2, 4, 4, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

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

See Matrix

[5 y₁ - y₂ + 5 y₃ - y₄ - 2 y₅, 0, y₁, y₂, y₃, y₅, y₄, 0, y₅]

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

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

See Matrix

[-14 y₁ - y₃ + 39 y₄ - 14 y₂, y₁, 0, y₃, y₄, 0, -5 y₁ + 14 y₄ - 5 y₂, y₂, 0]

p' = - s + s ⁵ p = - s + s ⁵

102 . Coloring, {2, 3, 5, 9}

Ωp(Δ)=0: p = s ² - 8s ⁴ - 12s ⁵ + 8s ⁶ + 16s ⁷

R: [4, 9, 5, 7, 3, 7, 1, 1, 2]
B: [2, 4, 4, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 7 8 vs 8 8 vs 8 4 vs 7 5 vs 7

Omega Rank for R : cycles: {{1, 4, 7}, {3, 5}, {2, 9}} order: 6

See Matrix

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

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

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

See Matrix

[9 y₁ - 4 y₂ - 13 y₄ - 4 y₅ + 9 y₃, 4 y₁, 0, 4 y₂, 4 y₄, 4 y₅, 5 y₁ - 9 y₄ + 5 y₃, 4 y₃, 0]

p' = - s ⁴ + s ⁶ p = - s ⁴ + s ⁶

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

R: [4, 9, 5, 7, 7, 8, 5, 1, 1]
B: [2, 4, 4, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 7 vs 7

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

See Matrix

[y₃, 0, 0, y₁, y₂, 0, y₄, y₅, 2 y₅]

p = s ⁴ - s ⁶

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

See Matrix

[y₄, y₃, y₂, y₁, 0, y₇, y₆, y₅, 0]

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

R: [4, 9, 5, 7, 7, 8, 1, 6, 1]
B: [2, 4, 4, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 7 vs 7

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

See Matrix

[y₅, 0, 0, y₂, y₃, y₄, -y₅ - y₂ - 3 y₃ + 5 y₄ + 5 y₁, y₁, 2 y₃]

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

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

See Matrix

[y₄, y₁, y₂, y₃, y₆, 0, y₇, y₅, 0]

105 . Coloring, {2, 3, 6, 9}

R: [4, 9, 5, 7, 7, 8, 1, 1, 2]
B: [2, 4, 4, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R : cycles: {{2, 9}, {1, 4, 7}} order: 6

See Matrix

[y₅, y₄, 0, y₃, y₂, 0, -y₅ + 5 y₄ - y₃ - 2 y₂ + 5 y₁, y₂, y₁]

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

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

See Matrix

[y₁ + y₂ - y₃ - y₄ - y₇ + y₆ + y₅, y₁, y₂, y₃, y₄, y₇, y₆, y₅, 0]

p = s ³ - s ⁴ + s ⁵ - s ⁶ + s ⁷ - s ⁸

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

Ωp(Δ)=0: p = s ² + 2s ⁴ + 8s ⁵ + 16s ⁷

R: [4, 9, 5, 7, 7, 7, 5, 6, 1]
B: [2, 4, 4, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, 0, y₅, y₄, y₃, y₂, 0, y₃]

p = - s ⁴ + s ⁶

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

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, 0, y₅, 0]

107 . Coloring, {2, 3, 7, 9}

R: [4, 9, 5, 7, 7, 7, 5, 1, 2]
B: [2, 4, 4, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[2 y₁ - y₂ + 3 y₃, y₁, 0, 3 y₁ - y₄ + 2 y₃, y₄, 0, y₂, 0, y₃]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

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

See Matrix

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

p = - s ⁴ + s ⁵ p = - s ⁴ + s ⁶

108 . Coloring, {2, 3, 8, 9}

R: [4, 9, 5, 7, 7, 7, 1, 6, 2]
B: [2, 4, 4, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R : cycles: {{1, 4, 7}, {2, 9}} order: 6

See Matrix

[5 y₁ - y₂ - 3 y₃ - y₅ + 5 y₄, y₁, 0, y₂, y₃, 2 y₃, y₅, 0, y₄]

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

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

See Matrix

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

p = - s ³ + s ⁴ - s ⁵ + s ⁶

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

Ωp(Δ)=0: p = s ² + 2s ⁴ + 8s ⁵ + 16s ⁷

R: [4, 9, 4, 8, 3, 8, 1, 1, 1]
B: [2, 4, 5, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, y₄, y₂, 0, 0, 0, y₃, y₄]

p = s ² - s ⁵

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

See Matrix

[0, 2 y₂, 0, y₃, y₄, y₂, y₁, 0, 0]

p = s ³ - s ⁵

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

R: [4, 9, 4, 8, 3, 7, 5, 1, 1]
B: [2, 4, 5, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 5 vs 7

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

See Matrix

[y₂, 0, y₁, y₆, y₅, 0, y₄, y₃, 2 y₄]

p = - s ⁴ + s ⁷

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

See Matrix

[y₃ + 4 y₅ - y₁ - y₂, 4 y₃ - y₄ + y₅, 0, y₁, y₂, y₃, y₄, y₅, 0]

p = - s ² + s ⁶ p' = - s ² + s ⁶

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

R: [4, 9, 4, 8, 3, 7, 1, 6, 1]
B: [2, 4, 5, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 6

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

See Matrix

[y₁, 0, y₅, y₂, 0, y₃, y₆, y₄, y₅]

p = - s ² + s ⁷

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

See Matrix

[y₂, y₁, 0, y₄, y₃, 0, y₅, y₆, 0]

112 . Coloring, {2, 4, 5, 9}

Ωp(Δ)=0: p = s ² - 4s ⁵ - 8s ⁶ + 16s ⁷

R: [4, 9, 4, 8, 3, 7, 1, 1, 2]
B: [2, 4, 5, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

6 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 8}} order: 6

See Matrix

[5 y₁ - y₂ - 3 y₃ - y₄ + 5 y₅, y₁, 2 y₃, y₂, 0, 0, y₃, y₄, y₅]

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

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

See Matrix

[-y₁ - y₂ + 2 y₅ + 3 y₄, 3 y₅ - y₃ + 2 y₄, 0, y₁, y₂, y₅, y₃, y₄, 0]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

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

R: [4, 9, 4, 8, 7, 8, 5, 1, 1]
B: [2, 4, 5, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

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

See Matrix

[-5 y₁ + 13 y₂ + 13 y₃ - 5 y₄ - 5 y₅, 0, 0, 5 y₁, 5 y₂, 0, 5 y₃, 5 y₄, 5 y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[y₅, y₄, y₃, y₂, y₁, -y₅ + 2 y₃ - y₂ + 3 y₁, -y₄ + 3 y₃ + 2 y₁, 0, 0]

p = - s ² + s ⁶ p' = - s ² + s ⁶

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

Ωp(Δ)=0: p = s ³ + 3s ⁴ + 8s ⁷

R: [4, 9, 4, 8, 7, 8, 1, 6, 1]
B: [2, 4, 5, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, 0, y₂, 0, y₃, y₅, y₄, y₅]

p = - s ⁴ + s ⁶

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

See Matrix

[y₁, y₂, y₃, y₄, y₅, 0, y₆, 0, 0]

115 . Coloring, {2, 4, 6, 9}

R: [4, 9, 4, 8, 7, 8, 1, 1, 2]
B: [2, 4, 5, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R : cycles: {{1, 4, 8}, {2, 9}} order: 6

See Matrix

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

p = s ² + s ³ - s ⁵ - s ⁶

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

See Matrix

[y₃, y₂, y₁, -3 y₃ + y₂ + y₁ - y₅ + y₄, y₅, 2 y₃, y₄, 0, 0]

p' = s ⁵ - s ⁶ p = s ⁵ - s ⁷

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

R: [4, 9, 4, 8, 7, 7, 5, 6, 1]
B: [2, 4, 5, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 7 vs 7 5 vs 7

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

See Matrix

[y₃, 0, 0, y₁, y₂, y₆, y₇, y₄, y₅]

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

See Matrix

[2 y₁ - y₄ + 3 y₃, y₅, y₁, y₄, y₃, 0, y₂, -y₅ + 3 y₁ + 2 y₃ - y₂, 0]

p = - s ² + s ⁶ p' = - s ² + s ⁶

117 . Coloring, {2, 4, 7, 9}

Ωp(Δ)=0: p = s - 8s ³ - 12s ⁴ + 32s ⁶ + 32s ⁷

R: [4, 9, 4, 8, 7, 7, 5, 1, 2]
B: [2, 4, 5, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{5, 7}, {2, 9}, {1, 4, 8}} order: 6

See Matrix

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

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

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

See Matrix

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

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

M \ ; N

$ [ [0, 142, 0, 0, 104, 87, 191, 0, 46] , [142, 0, 52, 94, 0, 0, 0, 92, 0] , [0, 52, 0, 0, 0, 57, 81, 0, 0] , [0, 94, 0, 0, 162, 46, 171, 0, 97] , [104, 0, 0, 162, 0, 0, 0, 114, 0] , [87, 0, 57, 46, 0, 0, 0, 0, 0] , [191, 0, 81, 171, 0, 0, 0, 127, 0] , [0, 92, 0, 0, 114, 0, 127, 0, 47] , [46, 0, 0, 97, 0, 0, 0, 47, 0] ] $ $ [ [0, 1, 0, 0, 1, 1, 1, 0, 1] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 1, 1, 1, 0, 1] , [0, 1, 0, 0, 1, 1, 1, 0, 1] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 1, 1, 1, 0, 1] , [1, 0, 1, 1, 0, 0, 0, 1, 0] ] $

τ= 41 , r'= 1/2

R: [4, 9, 4, 8, 7, 7, 5, 1, 2]
B: [2, 4, 5, 7, 3, 8, 1, 6, 1]

Ranges

Action of R on ranges, [[14], [13], [13], [11], [7], [14], [17], [5], [13], [11], [16], [16], [15], [8], [4], [2], [1]]
Action of B on ranges, [[7], [6], [8], [1], [1], [11], [13], [12], [15], [2], [10], [16], [4], [4], [9], [3], [3]]

Cycles: R , {{5, 7}, {2, 9}, {1, 4, 8}}, B , {{3, 5}, {6, 8}, {1, 2, 4, 7}}

β({1, 2}) = 71/855
β({1, 5}) = 52/855
β({1, 6}) = 29/570
β({1, 7}) = 191/1710
β({1, 9}) = 23/855
β({2, 3}) = 26/855
β({2, 4}) = 47/855
β({2, 8}) = 46/855
β({3, 6}) = 1/30
β({3, 7}) = 9/190
β({4, 5}) = 9/95
β({4, 6}) = 23/855
β({4, 7}) = 1/10
β({4, 9}) = 97/1710
β({5, 8}) = 1/15
β({7, 8}) = 127/1710
β({8, 9}) = 47/1710

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

b1 = {1, 3, 4, 8} ` , ` b2 = {2, 5, 6, 7, 9}

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

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

` ` See level-2 partition graph.

Right Group

Coloring {2, 4, 7, 9}

Rank 2

R,B [4, 9, 4, 8, 7, 7, 5, 1, 2], [2, 4, 5, 7, 3, 8, 1, 6, 1]

π₂ [142, 0, 0, 104, 87, 191, 0, 46, 52, 94, 0, 0, 0, 92, 0, 0, 0, 57, 81, 0, 0, 162, 46, 171, 0, 97, 0, 0, 114, 0, 0, 0, 0, 127, 0, 47]

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

wpp [4, 5, 4, 4, 5, 5, 5, 4, 5]

118 . Coloring, {2, 4, 8, 9}

R: [4, 9, 4, 8, 7, 7, 1, 6, 2]
B: [2, 4, 5, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 6 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 6, 7, 8}}

See Matrix

[y₆, y₅, 0, y₄, 0, y₃, y₂, -y₆ + 5 y₅ - y₄ - y₃ - y₂ + 5 y₁, y₁]

p = - s - s ² + s ⁶ + s ⁷

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

See Matrix

[y₂, y₁, y₂ - y₁ + y₅ + y₄ - y₃ - y₆, y₅, y₄, 0, y₃, y₆, 0]

p = s ⁶ - s ⁷

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

R: [4, 9, 4, 7, 3, 8, 5, 1, 1]
B: [2, 4, 5, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 7 vs 7

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

See Matrix

[y₅, 0, y₄, y₃, y₂, 0, y₁, y₆, 2 y₆]

p = s ³ - s ⁷

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

See Matrix

[y₁, y₂, 0, y₃, y₇, y₄, y₅, y₆, 0]

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

R: [4, 9, 4, 7, 3, 8, 1, 6, 1]
B: [2, 4, 5, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

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

See Matrix

[-2 y₃ - y₁ + 5 y₅ - y₄ + 5 y₂, 0, y₃, y₁, 0, y₅, y₄, y₂, y₃]

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

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

See Matrix

[-16 y₄ - 5 y₃ + 33 y₂ - 16 y₁, 5 y₄, 0, 5 y₃, 5 y₂, 0, -7 y₄ + 16 y₂ - 7 y₁, 5 y₁, 0]

p' = - s + s ⁵ p = - s + s ⁵

121 . Coloring, {2, 5, 6, 9}

R: [4, 9, 4, 7, 3, 8, 1, 1, 2]
B: [2, 4, 5, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 7}} order: 6

See Matrix

[5 y₁ - y₂ - y₃ - 3 y₄ + 5 y₅, y₁, 2 y₄, y₂, 0, 0, y₃, y₄, y₅]

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

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

See Matrix

[y₆, y₅, 0, y₄, y₃, y₂, y₁, y₆ - y₅ + y₄ + y₃ + y₂ - y₁, 0]

p = s ⁶ - s ⁷

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

Ωp(Δ)=0: p = s ³ - 16s ⁵ + 8s ⁶ - 32s ⁷ p' = s ³ + 4s ⁴ + 8s ⁶

R: [4, 9, 4, 7, 3, 7, 5, 6, 1]
B: [2, 4, 5, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 6

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

See Matrix

[y₁, 0, y₂, y₃, y₆, y₅, y₄, 0, y₅]

p = - s ³ + s ⁷

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

See Matrix

[y₁, y₂, 0, y₃, y₄, 0, y₅, y₆, 0]

123 . Coloring, {2, 5, 7, 9}

R: [4, 9, 4, 7, 3, 7, 5, 1, 2]
B: [2, 4, 5, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {3, 4, 5, 7}} order: 4

See Matrix

[2 y₁ - y₂ - y₄ + 3 y₅, y₁, y₂, 3 y₁ - y₃ + 2 y₅, y₃, 0, y₄, 0, y₅]

p = - s ² + s ⁶ p' = - s ² + s ⁶

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

See Matrix

[y₄ - y₁ - y₂ - y₃ + y₆ + y₅, y₄, 0, y₁, y₂, y₃, y₆, y₅, 0]

p = - s ⁶ + s ⁷

124 . Coloring, {2, 5, 8, 9}

R: [4, 9, 4, 7, 3, 7, 1, 6, 2]
B: [2, 4, 5, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

Omega Rank for R : cycles: {{1, 4, 7}, {2, 9}} order: 6

See Matrix

[y₅, y₁, y₂, -y₅ + 5 y₁ - 2 y₂ - y₃ + 5 y₄, 0, y₂, y₃, 0, y₄]

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

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

See Matrix

[3 y₁ - y₂ - 4 y₃ + 3 y₄, y₁, 0, y₂, y₃, 0, 2 y₁ - 3 y₃ + 2 y₄, y₄, 0]

p = - s + s ⁵ p' = - s + s ⁵

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

R: [4, 9, 4, 7, 7, 8, 5, 6, 1]
B: [2, 4, 5, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 7

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

See Matrix

[y₂ - y₃ + 4 y₄, 0, 0, -y₁ + 4 y₂ + y₄ - y₅, y₁, y₂, y₃, y₄, y₅]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

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

See Matrix

[2 y₁ - y₂ + 3 y₃, 3 y₁ + 2 y₃ - y₄ - y₅, y₁, y₂, y₃, 0, y₄, y₅, 0]

p' = - s ² + s ⁶ p = - s ² + s ⁶

126 . Coloring, {2, 6, 7, 9}

R: [4, 9, 4, 7, 7, 8, 5, 1, 2]
B: [2, 4, 5, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 8

Omega Rank for R : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[3 y₅ - y₂ + 2 y₁, y₅, 0, y₄, y₃, 0, y₂, 2 y₅ - y₄ - y₃ + 3 y₁, y₁]

p = s ⁴ - s ⁶ p' = s ⁴ - s ⁶

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

See Matrix

[3 y₁ - y₃ + 2 y₂ - y₄, 2 y₁ + 3 y₂ - y₅ - y₆, y₁, y₃, y₂, y₄, y₅, y₆, 0]

p = - s + s ⁷ p' = - s + s ⁷

127 . Coloring, {2, 6, 8, 9}

R: [4, 9, 4, 7, 7, 8, 1, 6, 2]
B: [2, 4, 5, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 7 5 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 7}, {6, 8}} order: 6

See Matrix

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

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

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

See Matrix

[9 y₁ - 2 y₂ - 11 y₃ + 9 y₅, 2 y₁, 7 y₁ - 9 y₃ - 2 y₄ + 7 y₅, 2 y₂, 2 y₃, 0, 2 y₄, 2 y₅, 0]

p = - s ² + s ⁶ p' = - s ² + s ⁶

128 . Coloring, {2, 7, 8, 9}

R: [4, 9, 4, 7, 7, 7, 5, 6, 2]
B: [2, 4, 5, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{5, 7}, {2, 9}} order: 2

See Matrix

[0, y₁ + 3 y₂ - 4 y₃, 0, 2 y₂, y₁, y₂, 4 y₁ + 12 y₂ - 15 y₃, 0, y₃]

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

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

See Matrix

[3 y₁ - y₂ + 2 y₃, 2 y₁ + 3 y₃ - y₄, y₁, y₂, y₃, 0, 0, y₄, 0]

p = - s + s ⁵ p' = - s + s ⁵

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

R: [4, 4, 5, 8, 3, 8, 1, 1, 1]
B: [2, 9, 4, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 3 vs 6

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

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 2

See Matrix

[0, y₃ - y₂, 0, y₁, -3 y₁ + y₃, 2 y₁, y₃, 0, y₂]

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

130 . Coloring, {3, 4, 5, 7}

R: [4, 4, 5, 8, 3, 7, 5, 1, 1]
B: [2, 9, 4, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

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

See Matrix

[11 y₁ - 7 y₂ + 11 y₃ + 11 y₄ - 7 y₅, 0, 7 y₁, 7 y₂, 7 y₃, 0, 7 y₄, 7 y₅, 0]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{2, 9}, {6, 8}} order: 4

See Matrix

[y₁, 4 y₂ - y₃ + y₄, 0, -y₁ + y₂ + 4 y₄ - y₅, 0, y₂, y₃, y₄, y₅]

p' = s ⁴ - s ⁶ p = s ⁴ - s ⁶

131 . Coloring, {3, 4, 5, 8}

R: [4, 4, 5, 8, 3, 7, 1, 6, 1]
B: [2, 9, 4, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 4 vs 7

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

See Matrix

[5 y₁ - y₂ + 5 y₄ - y₃ - y₅ - y₆, 0, y₁, y₂, y₄, y₃, y₅, y₆, 0]

p = - s - s ² + s ⁶ + s ⁷

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

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

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

132 . Coloring, {3, 4, 5, 9}

R: [4, 4, 5, 8, 3, 7, 1, 1, 2]
B: [2, 9, 4, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 8

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

See Matrix

[-2 y₄ + 5 y₁ - y₂ + 5 y₃ - y₅, y₄, y₁, y₂, y₃, 0, y₄, y₅, 0]

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

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}, {6, 8}} order: 6

See Matrix

[-y₁ + 2 y₃ + 2 y₄ - y₅, y₁, 0, -y₂ + y₃ + 2 y₄, y₂, y₃, 2 y₃ + y₄, y₄, y₅]

p = - s ² + s ⁸ p = - s ² + s ⁴ + s ⁵ - s ⁷ p = s ² + s ³ - s ⁵ - s ⁶

133 . Coloring, {3, 4, 6, 7}

Ωp(Δ)=0: p = s ³ - 3s ⁴ + 8s ⁷

R: [4, 4, 5, 8, 7, 8, 5, 1, 1]
B: [2, 9, 4, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₁, y₂, y₄, y₃, 0, y₄, y₅, 0, y₆]

p = s ⁵ - s ⁷

134 . Coloring, {3, 4, 6, 8}

R: [4, 4, 5, 8, 7, 8, 1, 6, 1]
B: [2, 9, 4, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

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

See Matrix

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

p = - s ⁵ + s ⁶

Omega Rank for B : cycles: {{3, 4, 5, 7}, {2, 9}} order: 4

See Matrix

[2 y₄, 2 y₄ + 2 y₅, 2 y₃, 2 y₂, 2 y₁, 0, 5 y₄ - 2 y₃ - 2 y₂ - 2 y₁ + 5 y₅, 0, 2 y₅]

p' = s ² - s ⁶ p = s ² - s ⁶

135 . Coloring, {3, 4, 6, 9}

R: [4, 4, 5, 8, 7, 8, 1, 1, 2]
B: [2, 9, 4, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

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

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

Omega Rank for B : cycles: {{3, 4, 5, 7}, {1, 2, 9}}

See Matrix

[-y₄ + y₁ + y₂ + y₃ - y₅, y₄, -y₆ + y₁ + y₂ + y₃, y₁, y₂, y₃, y₆, 0, y₅]

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

136 . Coloring, {3, 4, 7, 8}

R: [4, 4, 5, 8, 7, 7, 5, 6, 1]
B: [2, 9, 4, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 6 6 vs 7

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

See Matrix

[y₁, 0, 0, y₂, y₃, y₄, y₅, y₆, 0]

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₂, y₃, 2 y₆, y₁, 0, 0, y₄, y₆, y₅]

p = s ⁵ - s ⁷

137 . Coloring, {3, 4, 7, 9}

R: [4, 4, 5, 8, 7, 7, 5, 1, 2]
B: [2, 9, 4, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 7 vs 8

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

See Matrix

[-7 y₁ - 7 y₂ + 11 y₃ + 11 y₄ - 7 y₅, 7 y₁, 0, 7 y₂, 7 y₃, 0, 7 y₄, 7 y₅, 0]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}, {6, 8}} order: 6

See Matrix

[-y₁ - y₂ - y₃ + 5 y₄ - y₅ + 5 y₆ - y₇, y₁, y₂, y₃, 0, y₄, y₅, y₆, y₇]

p = - s ⁴ - s ⁵ + s ⁷ + s ⁸

138 . Coloring, {3, 4, 8, 9}

R: [4, 4, 5, 8, 7, 7, 1, 6, 2]
B: [2, 9, 4, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 5 vs 8

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

See Matrix

[y₁, y₅, 0, y₂, y₅, y₆, y₃, y₄, 0]

p = - s ² + s ⁷

Omega Rank for B : cycles: {{1, 2, 9}, {3, 4, 5, 7}}

See Matrix

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

p = - s ² + s ⁶ p' = - s ³ + s ⁷ p' = - s ² + s ⁶

139 . Coloring, {3, 5, 6, 7}

R: [4, 4, 5, 7, 3, 8, 5, 1, 1]
B: [2, 9, 4, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 6 7 vs 7

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

See Matrix

[y₆, 0, y₄, y₅, y₂, 0, y₃, y₁, 0]

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₁, y₃, 0, y₂, 0, y₆, y₅, y₄, y₇]

140 . Coloring, {3, 5, 6, 8}

R: [4, 4, 5, 7, 3, 8, 1, 6, 1]
B: [2, 9, 4, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 7 5 vs 7

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

See Matrix

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

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

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[y₁, y₂, 0, -y₁ - y₂ + 4 y₃ - y₄ - y₅, y₃, 0, y₃, y₄, y₅]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

141 . Coloring, {3, 5, 6, 9}

R: [4, 4, 5, 7, 3, 8, 1, 1, 2]
B: [2, 9, 4, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 8

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

See Matrix

[-2 y₂ + 5 y₁ - y₄ + 5 y₅ - y₃, y₂, y₁, y₄, y₅, 0, y₃, y₂, 0]

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

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}} order: 6

See Matrix

[-y₁ + y₄ + y₅ + y₂ - y₆, y₁, 0, y₄, y₅, y₂, y₃, y₄ + y₅ + y₂ - y₃, y₆]

p' = - s ⁴ + s ⁷ p = - s ⁴ + s ⁷

142 . Coloring, {3, 5, 7, 8}

Ωp(Δ)=0: p = s ² + 2s ⁴ - 16s ⁷

R: [4, 4, 5, 7, 3, 7, 5, 6, 1]
B: [2, 9, 4, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, y₅, y₄, y₃, 2 y₁, y₂, 0, 0]

p = s ⁴ - s ⁶

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₁, y₂, 0, y₃, 0, 0, 2 y₃, y₅, y₄]

p = - s ⁴ + s ⁶

143 . Coloring, {3, 5, 7, 9}

R: [4, 4, 5, 7, 3, 7, 5, 1, 2]
B: [2, 9, 4, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 7

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

See Matrix

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

p = s ⁴ - s ⁶ p' = s ⁴ - s ⁵

Omega Rank for B : cycles: {{6, 8}, {1, 2, 9}} order: 6

See Matrix

[3 y₃, 3 y₂, 0, -7 y₃ - 7 y₂ + 11 y₁ + 11 y₅ - 7 y₄, 0, 3 y₁, -14 y₃ - 14 y₂ + 22 y₁ + 22 y₅ - 14 y₄, 3 y₅, 3 y₄]

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

144 . Coloring, {3, 5, 8, 9}

R: [4, 4, 5, 7, 3, 7, 1, 6, 2]
B: [2, 9, 4, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 7

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

See Matrix

[y₂, y₃, y₄, y₅, y₁, 2 y₃, -y₂ - 3 y₃ + 5 y₄ - y₅ + 5 y₁, 0, 0]

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

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}} order: 6

See Matrix

[-5 y₁ - 5 y₂ + 13 y₃ + 13 y₄ - 5 y₅ - 5 y₆, 5 y₁, 0, 5 y₂, 5 y₃, 0, 5 y₄, 5 y₅, 5 y₆]

p = - s ³ - s ⁴ + s ⁶ + s ⁷

145 . Coloring, {3, 6, 7, 8}

R: [4, 4, 5, 7, 7, 8, 5, 6, 1]
B: [2, 9, 4, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

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

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₅, y₄, 2 y₂, y₃, 0, 0, y₂, y₁, y₆]

p = s ⁵ - s ⁷

146 . Coloring, {3, 6, 7, 9}

R: [4, 4, 5, 7, 7, 8, 5, 1, 2]
B: [2, 9, 4, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 6 8 vs 8

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

See Matrix

[y₁, y₅, 0, y₂, y₃, 0, y₄, y₅, 0]

p = - s ⁴ + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₃, y₄, y₅, y₂, 0, y₈, y₁, y₇, y₆]

147 . Coloring, {3, 6, 8, 9}

R: [4, 4, 5, 7, 7, 8, 1, 6, 2]
B: [2, 9, 4, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 8 vs 8

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

See Matrix

[-2 y₄ - y₁ + 5 y₃ - y₂ + 5 y₅, y₄, 0, y₁, y₄, y₃, y₂, y₅, 0]

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

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₁, y₅, y₄, y₃, y₂, 0, y₈, y₇, y₆]

148 . Coloring, {3, 7, 8, 9}

R: [4, 4, 5, 7, 7, 7, 5, 6, 2]
B: [2, 9, 4, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 6 vs 6

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

See Matrix

[0, y₄, 0, y₂, y₃, 2 y₄, y₁, 0, 0]

p = s ³ - s ⁵

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₁, y₂, y₄, y₃, 0, 0, 0, y₅, y₆]

149 . Coloring, {4, 5, 6, 7}

R: [4, 4, 4, 8, 3, 8, 5, 1, 1]
B: [2, 9, 5, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 5 5 vs 6

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

See Matrix

[y₄, 0, y₂, y₃, y₁, 0, 0, y₅, 0]

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₁, y₂, 0, 0, y₃, 2 y₃, y₄, 0, y₅]

p = - s ⁴ + s ⁶

150 . Coloring, {4, 5, 6, 8}

Ωp(Δ)=0: p = s ³ - s ⁴ + 4s ⁵ - 8s ⁷

R: [4, 4, 4, 8, 3, 8, 1, 6, 1]
B: [2, 9, 5, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[2 y₁, 0, y₁, y₃, 0, y₂, 0, y₄, 0]

p = - s ³ + s ⁵

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 2

See Matrix

[2 y₁ - 2 y₃, 2 y₁, 0, 0, 5 y₁ - 2 y₂, 0, 2 y₂, 0, 2 y₃]

p = - s ² + s ⁴ p' = - s ² + s ⁴

151 . Coloring, {4, 5, 6, 9}

R: [4, 4, 4, 8, 3, 8, 1, 1, 2]
B: [2, 9, 5, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 6

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

See Matrix

[y₁, y₂, 2 y₂, y₄, 0, 0, 0, y₃, 0]

p = - s ² + s ⁵

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}} order: 6

See Matrix

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

p = s ² - s ⁵ p' = - s ² + s ⁵

152 . Coloring, {4, 5, 7, 8}

R: [4, 4, 4, 8, 3, 7, 5, 6, 1]
B: [2, 9, 5, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 7 vs 7 5 vs 6

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

See Matrix

[y₁, 0, y₂, y₃, y₄, y₅, y₆, y₇, 0]

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₂, y₁, 0, 0, y₃, 0, y₅, y₃, y₄]

p = - s ⁴ + s ⁶

153 . Coloring, {4, 5, 7, 9}

R: [4, 4, 4, 8, 3, 7, 5, 1, 2]
B: [2, 9, 5, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 7

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

See Matrix

[y₁, y₂, y₄, y₅, y₆, 0, y₂, y₃, 0]

p = - s ⁴ + s ⁷

Omega Rank for B : cycles: {{6, 8}, {1, 2, 9}} order: 6

See Matrix

[y₁, -y₁ - y₃ + 5 y₂ - y₆ + 5 y₅ - y₄, 0, 0, y₃, y₂, y₆, y₅, y₄]

p = - s ³ - s ⁴ + s ⁶ + s ⁷

154 . Coloring, {4, 5, 8, 9}

R: [4, 4, 4, 8, 3, 7, 1, 6, 2]
B: [2, 9, 5, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 3 vs 6

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

See Matrix

[y₅, y₆, 2 y₆, y₄, 0, y₃, y₁, y₂, 0]

p = s ² - s ⁷

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}} order: 6

See Matrix

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

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

155 . Coloring, {4, 6, 7, 8}

R: [4, 4, 4, 8, 7, 8, 5, 6, 1]
B: [2, 9, 5, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[-14 y₁ + 39 y₂ - 14 y₃ - y₄, 0, 0, y₁, y₂, y₃, -5 y₁ + 14 y₂ - 5 y₃, y₄, 0]

p = s ³ - s ⁵ p' = s ³ - s ⁵

Omega Rank for B : cycles: {{2, 9}, {3, 5}} order: 4

See Matrix

[y₄, y₃, y₂, 0, y₁, 0, -y₃ + 3 y₂ + 2 y₁, 0, -y₄ + 2 y₂ + 3 y₁]

p = - s ³ + s ⁵ p' = s ³ - s ⁵

156 . Coloring, {4, 6, 7, 9}

R: [4, 4, 4, 8, 7, 8, 5, 1, 2]
B: [2, 9, 5, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[-5 y₁ - 5 y₂ + 13 y₃ + 13 y₄ - 5 y₅, 5 y₁, 0, 5 y₂, 5 y₃, 0, 5 y₄, 5 y₅, 0]

p = s ² + s ³ - s ⁵ - s ⁶

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}} order: 6

See Matrix

[-y₁ + 5 y₂ + 5 y₃ - y₄ - y₅ - y₆, y₁, y₂, 0, y₃, y₄, y₅, 0, y₆]

p = s ³ + s ⁴ - s ⁶ - s ⁷

157 . Coloring, {4, 6, 8, 9}

R: [4, 4, 4, 8, 7, 8, 1, 6, 2]
B: [2, 9, 5, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

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

See Matrix

[y₄, y₅, 0, y₃, 0, y₂, 2 y₅, y₁, 0]

p = s ⁴ - s ⁶

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}} order: 6

See Matrix

[4 y₅, 4 y₄, 4 y₃, 0, 4 y₂, 0, 5 y₅ + 5 y₄ - 4 y₃ - 4 y₂ + 5 y₁, 0, 4 y₁]

p = - s ² - s ³ + s ⁵ + s ⁶

158 . Coloring, {4, 7, 8, 9}

R: [4, 4, 4, 8, 7, 7, 5, 6, 2]
B: [2, 9, 5, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 6 5 vs 7

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

See Matrix

[0, y₅, 0, y₄, y₃, y₁, y₂, y₆, 0]

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}} order: 6

See Matrix

[-y₅ + 5 y₄ + 5 y₃ - 4 y₂ - y₁, y₅, y₄, 0, y₃, 0, 3 y₂, y₂, y₁]

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

159 . Coloring, {5, 6, 7, 8}

R: [4, 4, 4, 7, 3, 8, 5, 6, 1]
B: [2, 9, 5, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 6

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

See Matrix

[-y₁ + y₃ + 4 y₅ - y₄, 0, y₁, -y₂ + 4 y₃ + y₅, y₂, y₃, y₄, y₅, 0]

p' = - s ² + s ⁶ p = - s ² + s ⁶

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₂, y₁, 0, 0, y₃, 0, y₄, 3 y₃, y₅]

p = - s ⁴ + s ⁶

160 . Coloring, {5, 6, 7, 9}

R: [4, 4, 4, 7, 3, 8, 5, 1, 2]
B: [2, 9, 5, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 7

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

See Matrix

[y₁, y₃, y₆, y₄, y₅, 0, y₂, y₃, 0]

p = - s ³ + s ⁷

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₂, y₁, 0, 0, y₄, y₃, y₆, 3 y₄, y₅]

p = - s ⁴ + s ⁷

161 . Coloring, {5, 6, 8, 9}

R: [4, 4, 4, 7, 3, 8, 1, 6, 2]
B: [2, 9, 5, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 6

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

See Matrix

[-3 y₁ - y₂ + 5 y₃ - y₄ + 5 y₅, y₁, 2 y₁, y₂, 0, y₃, y₄, y₅, 0]

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

Omega Rank for B : cycles: {{1, 2, 9}, {5, 7}} order: 6

See Matrix

[-7 y₁ + 11 y₂ + 11 y₃ - 7 y₄ - 7 y₅, 7 y₁, 0, 0, 7 y₂, 0, 7 y₃, 7 y₄, 7 y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

162 . Coloring, {5, 7, 8, 9}

R: [4, 4, 4, 7, 3, 7, 5, 6, 2]
B: [2, 9, 5, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[0, y₁, y₂, y₅, y₄, 2 y₁, y₃, 0, 0]

p = - s ² + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₁, y₂, 0, 0, y₃, 0, y₅, 4 y₃, y₄]

p = - s ³ + s ⁶

163 . Coloring, {6, 7, 8, 9}

R: [4, 4, 4, 7, 7, 8, 5, 6, 2]
B: [2, 9, 5, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 7

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

See Matrix

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

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}} order: 6

See Matrix

[y₂, y₃, y₄, 0, y₅, 0, y₁, 3 y₁, -y₂ - y₃ + 5 y₄ + 5 y₅ - 4 y₁]

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

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

R: [4, 9, 5, 8, 3, 8, 1, 1, 1]
B: [2, 4, 4, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 5

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

See Matrix

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

p = s ² + s ³ - s ⁵ - s ⁶

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

See Matrix

[0, 2 y₃, 0, y₁, y₂, y₃, y₄, 0, 0]

p = - s ³ + s ⁵

165 . Coloring, {2, 3, 4, 5, 7}

R: [4, 9, 5, 8, 3, 7, 5, 1, 1]
B: [2, 4, 4, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

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

See Matrix

[3 y₂, 0, 3 y₁, 3 y₄, 3 y₅, 0, -7 y₂ + 11 y₁ - 7 y₄ + 11 y₅ - 7 y₃, 3 y₃, -14 y₂ + 22 y₁ - 14 y₄ + 22 y₅ - 14 y₃]

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

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

See Matrix

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

p = - s + s ⁵ p' = - s + s ⁵

166 . Coloring, {2, 3, 4, 5, 8}

R: [4, 9, 5, 8, 3, 7, 1, 6, 1]
B: [2, 4, 4, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 7 vs 8 6 vs 6

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

See Matrix

[5 y₅ - y₆ + 5 y₄ - y₂ - y₃ - y₁ - y₇, 0, y₅, y₆, y₄, y₂, y₃, y₁, y₇]

p = - s ² - s ³ + s ⁷ + s ⁸

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

See Matrix

[y₁, y₂, 0, y₃, y₄, 0, y₅, y₆, 0]

167 . Coloring, {2, 3, 4, 5, 9}

R: [4, 9, 5, 8, 3, 7, 1, 1, 2]
B: [2, 4, 4, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 8 5 vs 7

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {1, 4, 8}} order: 6

See Matrix

[4 y₁ + 4 y₄ - y₅ - y₃ - y₂, y₁, y₄, y₅, y₁, 0, y₃, y₂, y₄]

p = s ² + s ³ - s ⁵ - s ⁶ p = s ² - s ⁴ - s ⁵ + s ⁷ p = - s ² + s ⁸

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

See Matrix

[y₃, y₄, 0, -y₃ - y₁ + 2 y₂ + 3 y₅, y₁, y₂, -y₄ + 3 y₂ + 2 y₅, y₅, 0]

p' = - s ⁴ + s ⁶ p = - s ⁴ + s ⁶

168 . Coloring, {2, 3, 4, 6, 7}

Ωp(Δ)=0: p = s ² + 2s ⁴ - 8s ⁵ - 16s ⁷

R: [4, 9, 5, 8, 7, 8, 5, 1, 1]
B: [2, 4, 4, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[-y₂ + 2 y₃ + 2 y₁ - y₄ - y₅, 0, 0, y₂, y₃, 0, y₁, y₄, y₅]

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[y₁, y₂, y₃, y₄, 0, y₃, y₅, 0, 0]

p = - s ² + s ⁶

169 . Coloring, {2, 3, 4, 6, 8}

R: [4, 9, 5, 8, 7, 8, 1, 6, 1]
B: [2, 4, 4, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 6

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

See Matrix

[y₂ + y₁ + y₃ + y₄ - y₅, 0, 0, y₂, y₁, y₃, y₄, y₅, 2 y₁]

p = - s ⁵ + s ⁷ p = - s ⁵ + s ⁶

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

See Matrix

[y₁, y₂, y₃, y₄, y₆, 0, y₅, 0, 0]

170 . Coloring, {2, 3, 4, 6, 9}

R: [4, 9, 5, 8, 7, 8, 1, 1, 2]
B: [2, 4, 4, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 7 5 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 8}} order: 6

See Matrix

[5 y₁ - y₆ - y₅ - y₄ - y₃ + 5 y₂, y₁, 0, y₆, y₅, 0, y₄, y₃, y₂]

p = - s ³ - s ⁴ + s ⁶ + s ⁷

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

See Matrix

[y₁, 3 y₁ - y₂ + y₃ + y₄ - y₅, y₂, y₃, y₄, 2 y₁, y₅, 0, 0 ]

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

171 . Coloring, {2, 3, 4, 7, 8}

R: [4, 9, 5, 8, 7, 7, 5, 6, 1]
B: [2, 4, 4, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 7 vs 7 5 vs 6

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

See Matrix

[y₁, 0, 0, y₂, y₃, y₄, y₅, y₆, y₇]

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

See Matrix

[y₂, y₃, 2 y₅, y₁, 0, 0, y₄, y₅, 0]

p = - s ² + s ⁶

172 . Coloring, {2, 3, 4, 7, 9}

R: [4, 9, 5, 8, 7, 7, 5, 1, 2]
B: [2, 4, 4, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 7 5 vs 7

Omega Rank for R : cycles: {{5, 7}, {2, 9}, {1, 4, 8}} order: 6

See Matrix

[-2 y₁ + 8 y₃ - 2 y₂ - 8 y₄, 3 y₃ - 5 y₄, 0, 2 y₁, 2 y₃, 0, 5 y₃ - 7 y₄, 2 y₂, 2 y₄]

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

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

See Matrix

[-y₂ + 2 y₃ + 3 y₅, -y₁ + 3 y₃ - y₄ + 2 y₅, y₁, y₂, 0, y₃, y₄, y₅, 0]

p' = - s ² + s ⁶ p = - s ² + s ⁶

173 . Coloring, {2, 3, 4, 8, 9}

R: [4, 9, 5, 8, 7, 7, 1, 6, 2]
B: [2, 4, 4, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 3 vs 8 6 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 6, 7, 8}}

See Matrix

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

p' = - s ² + s ⁶ p = - s ² + s ⁴ p' = - s ² + s ⁴ p = - s ² + s ⁸ p = - s ² + s ⁶

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

See Matrix

[y₂, y₁, y₂ - y₁ + y₆ + y₅ - y₄ - y₃, y₆, y₅, 0, y₄, y₃, 0]

p = - s ⁴ + s ⁵ - s ⁶ + s ⁷

174 . Coloring, {2, 3, 5, 6, 7}

R: [4, 9, 5, 7, 3, 8, 5, 1, 1]
B: [2, 4, 4, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, y₂, y₃, y₄, 0, y₅, y₆, 2 y₆]

p = - s ⁵ + s ⁷

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

See Matrix

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

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

175 . Coloring, {2, 3, 5, 6, 8}

R: [4, 9, 5, 7, 3, 8, 1, 6, 1]
B: [2, 4, 4, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 8 4 vs 6

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

See Matrix

[y₄, 0, y₃, y₁, y₂, y₃, -y₄ + 4 y₃ - y₁ + 4 y₂ - y₅, y₂, y₅]

p = - s ² - s ³ + s ⁵ + s ⁶ p' = - s ² - s ³ + s ⁵ + s ⁶ p = - s ² + s ⁸

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

See Matrix

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

p = s - s ⁵ p' = s - s ⁵

176 . Coloring, {2, 3, 5, 6, 9}

R: [4, 9, 5, 7, 3, 8, 1, 1, 2]
B: [2, 4, 4, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 8 6 vs 7

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {1, 4, 7}} order: 6

See Matrix

[4 y₄ - y₃ + 4 y₂ - y₁ - y₅, y₂, y₄, y₃, y₂, 0, y₁, y₅, y₄]

p = - s ² - s ³ + s ⁵ + s ⁶ p = s ² - s ⁴ - s ⁵ + s ⁷ p = - s ² + s ⁸

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

See Matrix

[y₁, y₁ + y₆ + y₄ + y₅ - y₃ - y₂, 0, y₆, y₄, y₅, y₃, y₂, 0]

p = - s ⁶ + s ⁷

177 . Coloring, {2, 3, 5, 7, 8}

Ωp(Δ)=0: p = s ³ + s ⁴ + 4s ⁵ + 8s ⁷

R: [4, 9, 5, 7, 3, 7, 5, 6, 1]
B: [2, 4, 4, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, y₂, y₃, y₄, y₆, y₅, 0, y₆]

p = s ⁵ - s ⁷

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

See Matrix

[y₁, y₂, 0, y₃, 0, 0, y₄, y₅, 0]

178 . Coloring, {2, 3, 5, 7, 9}

R: [4, 9, 5, 7, 3, 7, 5, 1, 2]
B: [2, 4, 4, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R : cycles: {{2, 9}, {3, 5}} order: 4

See Matrix

[y₂, y₁, -y₂ + 2 y₁ - y₃ + 3 y₅, 3 y₁ - y₄ + 2 y₅, y₄, 0, y₃, 0, y₅]

p' = - s ⁴ + s ⁶ p = s ⁴ - s ⁶

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

See Matrix

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

p = - s ⁵ + s ⁶

179 . Coloring, {2, 3, 5, 8, 9}

R: [4, 9, 5, 7, 3, 7, 1, 6, 2]
B: [2, 4, 4, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 8 4 vs 6

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {1, 4, 7}} order: 6

See Matrix

[4 y₅ - y₁ + 4 y₂ - y₃ - y₄, y₂, y₅, y₁, y₂, y₃, y₄, 0, y₅]

p = - s ² + s ⁸ p = - s ² - s ³ + s ⁵ + s ⁶ p = s ² - s ⁴ - s ⁵ + s ⁷

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

See Matrix

[9 y₁ - 4 y₃ - 13 y₂ + 9 y₄, 4 y₁, 0, 4 y₃, 4 y₂, 0, 5 y₁ - 9 y₂ + 5 y₄, 4 y₄, 0]

p' = - s + s ⁵ p = - s + s ⁵

180 . Coloring, {2, 3, 6, 7, 8}

R: [4, 9, 5, 7, 7, 8, 5, 6, 1]
B: [2, 4, 4, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

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

See Matrix

[y₅, 0, 0, y₃, y₄, y₂, -y₅ + y₂ + 4 y₁, y₁, -y₃ - y₄ + 4 y₂ + y₁]

p' = s ⁴ - s ⁶ p = s ⁴ - s ⁶

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

See Matrix

[y₁, y₂, 2 y₄, y₃, 0, 0, y₄, y₅, 0]

p = - s ² + s ⁶

181 . Coloring, {2, 3, 6, 7, 9}

R: [4, 9, 5, 7, 7, 8, 5, 1, 2]
B: [2, 4, 4, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[y₅, y₃, 0, y₄, y₂, 0, -y₅ + 3 y₃ + 2 y₁, 2 y₃ - y₄ - y₂ + 3 y₁, y₁]

p' = s ⁴ - s ⁶ p = - s ⁴ + s ⁶

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

See Matrix

[y₂ + y₃ - y₄ - y₁ + y₅ + y₆, y₂, y₃, y₄, 0, y₁, y₅, y₆, 0]

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

182 . Coloring, {2, 3, 6, 8, 9}

R: [4, 9, 5, 7, 7, 8, 1, 6, 2]
B: [2, 4, 4, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 8 6 vs 7

Omega Rank for R : cycles: {{1, 4, 7}, {2, 9}, {6, 8}} order: 6

See Matrix

[4 y₅ - y₁ - y₂ - y₃ + 4 y₄, y₅, 0, y₁, y₂, y₄, y₃, y₅, y₄]

p = - s ² - s ³ + s ⁵ + s ⁶ p = s ² - s ⁴ - s ⁵ + s ⁷ p = - s ² + s ⁸

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

See Matrix

[y₁ + y₂ - y₃ - y₄ + y₅ + y₆, y₁, y₂, y₃, y₄, 0, y₅, y₆, 0]

p = s ⁴ - s ⁵ + s ⁶ - s ⁷

183 . Coloring, {2, 3, 7, 8, 9}

R: [4, 9, 5, 7, 7, 7, 5, 6, 2]
B: [2, 4, 4, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 3 vs 6 4 vs 5

Omega Rank for R : cycles: {{5, 7}, {2, 9}} order: 2

See Matrix

[0, 2 y₁ + 5 y₂ - 8 y₃, 0, 3 y₂, 2 y₁, 2 y₂, 8 y₁ + 20 y₂ - 30 y₃, 0, 2 y₃]

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

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

184 . Coloring, {2, 4, 5, 6, 7}

R: [4, 9, 4, 8, 3, 8, 5, 1, 1]
B: [2, 4, 5, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

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

See Matrix

[2 y₃, 0, 2 y₁, 2 y₂, 3 y₅, 0, 0, 2 y₄, 2 y₅]

p = - s ³ + s ⁶

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

See Matrix

[y₁, y₂, 0, y₃, y₄, 2 y₄, y₅, 0, 0]

p = - s ² + s ⁶

185 . Coloring, {2, 4, 5, 6, 8}

Ωp(Δ)=0: p = s ² + 6s ⁴ + 16s ⁷

R: [4, 9, 4, 8, 3, 8, 1, 6, 1]
B: [2, 4, 5, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, y₅, y₂, 0, y₃, 0, y₄, y₅]

p = - s ⁴ + s ⁶

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

See Matrix

[y₄, y₃, 0, y₂, y₁, 0, y₅, 0, 0]

186 . Coloring, {2, 4, 5, 6, 9}

R: [4, 9, 4, 8, 3, 8, 1, 1, 2]
B: [2, 4, 5, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Omega Rank for R : cycles: {{1, 4, 8}, {2, 9}} order: 6

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[y₁, 3 y₁ + y₂ + y₄ - y₃, 0, y₂, y₄, 2 y₁, y₃, 0, 0]

p = - s ⁴ + s ⁵ p = - s ⁴ + s ⁶

187 . Coloring, {2, 4, 5, 7, 8}

R: [4, 9, 4, 8, 3, 7, 5, 6, 1]
B: [2, 4, 5, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 8 vs 8 5 vs 6

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

See Matrix

[y₅, 0, y₄, y₁, y₂, y₃, y₈, y₇, y₆]

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

See Matrix

[y₁, y₂, 0, y₅, y₄, 0, y₃, y₄, 0]

p = s ² - s ⁶

188 . Coloring, {2, 4, 5, 7, 9}

R: [4, 9, 4, 8, 3, 7, 5, 1, 2]
B: [2, 4, 5, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 7 vs 8 5 vs 7

Omega Rank for R : cycles: {{2, 9}, {1, 4, 8}} order: 6

See Matrix

[5 y₁ - y₂ - y₃ - y₄ - y₅ - y₆ + 5 y₇, y₁, y₂, y₃, y₄, 0, y₅, y₆, y₇]

p = - s ⁴ - s ⁵ + s ⁷ + s ⁸

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

See Matrix

[y₄, y₅, 0, -y₄ - y₃ + 2 y₂ + 3 y₁, y₃, y₂, -y₅ + 3 y₂ + 2 y₁, y₁, 0]

p = - s ² + s ⁶ p' = - s ² + s ⁶

189 . Coloring, {2, 4, 5, 8, 9}

R: [4, 9, 4, 8, 3, 7, 1, 6, 2]
B: [2, 4, 5, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 7 vs 8 5 vs 6

Omega Rank for R : cycles: {{1, 4, 6, 7, 8}, {2, 9}}

See Matrix

[y₄, y₂, y₃, y₁, 0, -y₄ + 5 y₂ - y₃ - y₁ - y₅ - y₆ + 5 y₇, y₅, y₆, y₇]

p = - s ² - s ³ + s ⁷ + s ⁸

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

See Matrix

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

p = - s ⁵ + s ⁶

190 . Coloring, {2, 4, 6, 7, 8}

R: [4, 9, 4, 8, 7, 8, 5, 6, 1]
B: [2, 4, 5, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

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

See Matrix

[y₁, 0, 0, -14 y₁ - y₂ + 39 y₃ - 14 y₄ - y₅, -5 y₁ + 14 y₃ - 5 y₄, y₂, y₃, y₄, y₅]

p' = s ⁴ - s ⁶ p = - s ⁴ + s ⁶

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

See Matrix

[y₂, 3 y₁ + 2 y₄ - y₃, y₁, -y₂ + 2 y₁ + 3 y₄, y₄, 0, y₃, 0, 0]

p = s - s ⁵ p' = - s + s ⁵

191 . Coloring, {2, 4, 6, 7, 9}

R: [4, 9, 4, 8, 7, 8, 5, 1, 2]
B: [2, 4, 5, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{5, 7}, {2, 9}, {1, 4, 8}} order: 6

See Matrix

[y₂, 3 y₁ - 4 y₃, 0, -y₂ + 10 y₁ - y₄ - 10 y₃, y₁, 0, 4 y₁ - 5 y₃, y₄, y₃]

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

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

See Matrix

[y₄, y₅, y₃, y₂, y₁, -y₄ + 3 y₃ - y₂ + 2 y₁, -y₅ + 2 y₃ + 3 y₁, 0, 0]

p = - s ² + s ⁶ p' = s ² - s ⁶

192 . Coloring, {2, 4, 6, 8, 9}

R: [4, 9, 4, 8, 7, 8, 1, 6, 2]
B: [2, 4, 5, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R : cycles: {{2, 9}, {6, 8}} order: 4

See Matrix

[3 y₁ - y₂ + 2 y₃, y₁, 0, 2 y₁ - y₄ - y₅ + 3 y₃, 0, y₄, y₅, y₂, y₃]

p' = s ⁴ - s ⁶ p = - s ⁴ + s ⁶

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

See Matrix

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

p = s ⁵ - s ⁶

193 . Coloring, {2, 4, 7, 8, 9}

R: [4, 9, 4, 8, 7, 7, 5, 6, 2]
B: [2, 4, 5, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 7

Omega Rank for R : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[0, y₁ + y₂ + y₃ - 4 y₅, 0, y₁, y₂, y₃, 4 y₁ + 4 y₂ + 4 y₃ - 15 y₅ - y₄, y₄, y₅]

p' = s ⁴ - s ⁶ p = - s ⁴ + s ⁶

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

See Matrix

[y₃, y₄, y₅, -y₃ + 3 y₅ + 2 y₂, y₂, 0, -y₄ + 2 y₅ + 3 y₂ - y₁, y₁, 0]

p = - s ² + s ⁶ p' = - s ² + s ⁶

194 . Coloring, {2, 5, 6, 7, 8}

R: [4, 9, 4, 7, 3, 8, 5, 6, 1]
B: [2, 4, 5, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 8 6 vs 6

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

See Matrix

[-y₂ + y₅ - y₃ + 4 y₄, 0, y₂, y₁, -y₁ + 4 y₅ + y₄ - y₆, y₅, y₃, y₄, y₆]

p = - s ³ + s ⁷ p' = - s ³ + s ⁷

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

See Matrix

[y₁, y₂, 0, y₃, y₅, 0, y₄, y₆, 0]

195 . Coloring, {2, 5, 6, 7, 9}

R: [4, 9, 4, 7, 3, 8, 5, 1, 2]
B: [2, 4, 5, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 8 6 vs 7

Omega Rank for R : cycles: {{2, 9}, {3, 4, 5, 7}} order: 4

See Matrix

[y₄, y₅, y₂, y₃, 2 y₅ - y₃ - y₆ + 3 y₁, 0, -y₄ + 3 y₅ - y₂ + 2 y₁, y₆, y₁]

p' = - s ³ + s ⁷ p = - s ³ + s ⁷

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

See Matrix

[y₁, y₁ + y₆ + y₅ + y₃ - y₄ - y₂, 0, y₆, y₅, y₃, y₄, y₂, 0]

p = s ² - s ³ + s ⁴ - s ⁵ + s ⁶ - s ⁷

196 . Coloring, {2, 5, 6, 8, 9}

R: [4, 9, 4, 7, 3, 8, 1, 6, 2]
B: [2, 4, 5, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 8 4 vs 6

Omega Rank for R : cycles: {{2, 9}, {6, 8}, {1, 4, 7}} order: 6

See Matrix

[y₄, y₃, y₂, y₁, 0, y₅, -y₄ + 4 y₃ - y₂ - y₁ + 4 y₅, y₃, y₅]

p = - s ² - s ³ + s ⁵ + s ⁶ p = - s ² + s ⁸ p' = - s ² - s ³ + s ⁵ + s ⁶

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

See Matrix

[2 y₃, 9 y₃ + 9 y₂ - 11 y₁ - 2 y₄, 0, 2 y₂, 7 y₃ + 7 y₂ - 9 y₁, 0, 2 y₁, 2 y₄, 0]

p' = - s + s ⁵ p = - s + s ⁵

197 . Coloring, {2, 5, 7, 8, 9}

R: [4, 9, 4, 7, 3, 7, 5, 6, 2]
B: [2, 4, 5, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{3, 4, 5, 7}, {2, 9}} order: 4

See Matrix

[0, -4 y₅ + y₁ + y₂ + y₃, -15 y₅ + 4 y₁ + 4 y₂ + 4 y₃ - y₄, y₁, y₂, y₃, y₄, 0, y₅]

p' = - s ² + s ⁶ p = - s ² + s ⁶

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

See Matrix

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

p = - s ³ + s ⁴ - s ⁵ + s ⁶

198 . Coloring, {2, 6, 7, 8, 9}

R: [4, 9, 4, 7, 7, 8, 5, 6, 2]
B: [2, 4, 5, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 3 vs 7 5 vs 7

Omega Rank for R : cycles: {{5, 7}, {2, 9}, {6, 8}} order: 2

See Matrix

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

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

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

See Matrix

[3 y₁ - y₂ + 2 y₃, 2 y₁ + 3 y₃ - y₄ - y₅, y₁, y₂, y₃, 0, y₄, y₅, 0]

p = s ² - s ⁶ p' = - s ² + s ⁶

199 . Coloring, {3, 4, 5, 6, 7}

Ωp(Δ)=0: p = - s ² + 6s ⁴ - 16s ⁷

R: [4, 4, 5, 8, 3, 8, 5, 1, 1]
B: [2, 9, 4, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₁, y₃, 0, y₂, 0, 2 y₂, y₅, 0, y₄]

p = - s ⁴ + s ⁶

200 . Coloring, {3, 4, 5, 6, 8}

R: [4, 4, 5, 8, 3, 8, 1, 6, 1]
B: [2, 9, 4, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[3 y₁ + 2 y₂ - y₄, 0, y₁, 2 y₁ + 3 y₂ - y₃, y₂, y₃, 0, y₄, 0]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{5, 7}, {2, 9}} order: 2

See Matrix

[4 y₂, 2 y₃, 0, 2 y₂, 2 y₁, 0, -2 y₂ - 2 y₁ + 5 y₃, 0, 2 y₃ - 4 y₂]

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

201 . Coloring, {3, 4, 5, 6, 9}

R: [4, 4, 5, 8, 3, 8, 1, 1, 2]
B: [2, 9, 4, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 7

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

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}, {5, 7}} order: 6

See Matrix

[y₂, -y₂ + y₃ - y₄, 0, y₁, -3 y₁ + y₃, 2 y₁, y₃, 0, y₄]

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

202 . Coloring, {3, 4, 5, 7, 8}

R: [4, 4, 5, 8, 3, 7, 5, 6, 1]
B: [2, 9, 4, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 7 vs 7 5 vs 6

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

See Matrix

[y₁, 0, y₄, y₂, y₃, y₇, y₅, y₆, 0]

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₅, y₄, 0, y₃, 0, 0, y₂, y₃, y₁]

p = - s ⁴ + s ⁶

203 . Coloring, {3, 4, 5, 7, 9}

R: [4, 4, 5, 8, 3, 7, 5, 1, 2]
B: [2, 9, 4, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 7

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

See Matrix

[4 y₄, 7 y₄ - 11 y₃ + 7 y₁ - 11 y₂ + 7 y₅, 4 y₃, 4 y₁, 4 y₂, 0, 7 y₄ - 11 y₃ + 7 y₁ - 11 y₂ + 7 y₅, 4 y₅, 0]

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

Omega Rank for B : cycles: {{6, 8}, {1, 2, 9}} order: 6

See Matrix

[-y₁ - y₂ + 5 y₃ - y₄ + 5 y₅ - y₆, y₁, 0, y₂, 0, y₃, y₄, y₅, y₆]

p = s ³ + s ⁴ - s ⁶ - s ⁷

204 . Coloring, {3, 4, 5, 8, 9}

R: [4, 4, 5, 8, 3, 7, 1, 6, 2]
B: [2, 9, 4, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 7 vs 8 3 vs 7

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

See Matrix

[-y₁ + 5 y₂ - y₃ + 5 y₄ - y₅ - y₆ - y₇, y₁, y₂, y₃, y₄, y₅, y₆, y₇, 0]

p = - s ² - s ³ + s ⁷ + s ⁸

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}} order: 6

See Matrix

[y₁ + y₃, y₁ + y₃, 0, y₁, 2 y₁ + 3 y₃ - y₂, 0, y₂, y₁, y₃ ]

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

205 . Coloring, {3, 4, 6, 7, 8}

Ωp(Δ)=0: p = - s ³ + s ⁴ + 4s ⁵ - 8s ⁷

R: [4, 4, 5, 8, 7, 8, 5, 6, 1]
B: [2, 9, 4, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[3 y₁ - 4 y₄ + 3 y₂ - y₃, 0, 0, y₁, y₄, y₂, 2 y₁ - 3 y₄ + 2 y₂, y₃, 0]

p' = s ³ - s ⁵ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 9}} order: 6

See Matrix

[y₄, y₁, y₂, y₃, 0, 0, y₅, 0, y₆]

206 . Coloring, {3, 4, 6, 7, 9}

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

R: [4, 4, 5, 8, 7, 8, 5, 1, 2]
B: [2, 9, 4, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

5 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 7

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

See Matrix

[y₃, y₂, 0, -y₃ - y₂ + 2 y₁ + 2 y₅ - y₄, y₁, 0, y₅, y₄, 0]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

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

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

207 . Coloring, {3, 4, 6, 8, 9}

R: [4, 4, 5, 8, 7, 8, 1, 6, 2]
B: [2, 9, 4, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 7

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

See Matrix

[y₅ - 2 y₄ + y₃ + y₂ - y₁, y₄, 0, y₅, y₄, y₃, y₂, y₁, 0]

p = - s ⁵ + s ⁶ p = - s ⁵ + s ⁷

Omega Rank for B : cycles: {{1, 2, 9}, {3, 4, 5, 7}}

See Matrix

[4 y₂, 4 y₃, 5 y₂ + 5 y₃ - 4 y₁ - 4 y₄ - 4 y₅ + 5 y₆, 4 y₁, 4 y₄, 0, 4 y₅, 0, 4 y₆]

p = - s - s ² - s ³ + s ⁵ + s ⁶ + s ⁷

208 . Coloring, {3, 4, 7, 8, 9}

R: [4, 4, 5, 8, 7, 7, 5, 6, 2]
B: [2, 9, 4, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 6 6 vs 7

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

See Matrix

[0, y₁, 0, y₃, y₄, y₂, y₅, y₆, 0]

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₁, y₂, 2 y₆, y₃, 0, 0, y₄, y₆, y₅]

p = - s ⁴ + s ⁷

209 . Coloring, {3, 5, 6, 7, 8}

R: [4, 4, 5, 7, 3, 8, 5, 6, 1]
B: [2, 9, 4, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

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

See Matrix

[y₁, 0, -y₁ + y₃ - y₂ + 4 y₅, y₄, 4 y₃ + y₅ - y₄, y₃, y₂, y₅, 0]

p = s ⁴ - s ⁶ p' = s ⁴ - s ⁶

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₃, y₂, 0, y₁, 0, 0, 3 y₁, y₅, y₄]

p = s ⁴ - s ⁶

210 . Coloring, {3, 5, 6, 7, 9}

R: [4, 4, 5, 7, 3, 8, 5, 1, 2]
B: [2, 9, 4, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 7 7 vs 7

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

See Matrix

[y₄, y₅, y₆, y₃, y₁, 0, y₂, y₅, 0]

p = - s ⁵ + s ⁷

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₆, y₇, 0, y₄, 0, y₅, y₁, y₂, y₃]

211 . Coloring, {3, 5, 6, 8, 9}

R: [4, 4, 5, 7, 3, 8, 1, 6, 2]
B: [2, 9, 4, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 9 9 vs 9 5 vs 8 5 vs 7

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

See Matrix

[-y₁ + 4 y₃ - y₂ + 4 y₅ - y₄, y₁, y₃, y₂, y₅, y₃, y₄, y₅, 0]

p = - s ² - s ³ + s ⁵ + s ⁶ p = s ² - s ⁴ - s ⁵ + s ⁷ p = - s ² + s ⁸

Omega Rank for B : cycles: {{1, 2, 9}, {5, 7}} order: 6

See Matrix

[-y₁ - y₂ + 4 y₃ - y₄ - y₅, y₁, 0, y₂, y₃, 0, y₃, y₄, y₅]

p = - s ³ + s ⁶ p' = - s ³ + s ⁶

212 . Coloring, {3, 5, 7, 8, 9}

R: [4, 4, 5, 7, 3, 7, 5, 6, 2]
B: [2, 9, 4, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

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

See Matrix

[0, y₄, y₅, y₁, y₂, 2 y₄, y₃, 0, 0]

p = - s ⁴ + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₃, y₂, 0, y₁, 0, 0, 2 y₁, y₅, y₄]

p = s ³ - s ⁶

213 . Coloring, {3, 6, 7, 8, 9}

R: [4, 4, 5, 7, 7, 8, 5, 6, 2]
B: [2, 9, 4, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 6 6 vs 7

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

See Matrix

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

p' = s ³ - s ⁵ p = - s ³ + s ⁵

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₃, y₁, 2 y₆, y₂, 0, 0, y₆, y₅, y₄]

p = - s ⁴ + s ⁷

214 . Coloring, {4, 5, 6, 7, 8}

R: [4, 4, 4, 8, 3, 8, 5, 6, 1]
B: [2, 9, 5, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 5

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

See Matrix

[y₁, 0, y₂, y₃, 3 y₁, y₄, 0, y₅, 0]

p = - s ⁴ + s ⁶

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₁, y₃, 0, 0, y₂, 0, y₄, 0, y₅]

215 . Coloring, {4, 5, 6, 7, 9}

R: [4, 4, 4, 8, 3, 8, 5, 1, 2]
B: [2, 9, 5, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, y₂, y₃, y₄, 3 y₂, 0, 0, y₅, 0]

p = - s ³ + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₁, y₂, 0, 0, y₃, 2 y₃, y₄, 0, y₅]

p = - s ³ + s ⁶

216 . Coloring, {4, 5, 6, 8, 9}

R: [4, 4, 4, 8, 3, 8, 1, 6, 2]
B: [2, 9, 5, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 6 4 vs 5

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

See Matrix

[3 y₂, y₂, 2 y₂, y₁, 0, y₄, 0, y₃, 0]

p' = s ³ - s ⁵ p = s ³ - s ⁵

Omega Rank for B : cycles: {{5, 7}, {1, 2, 9}} order: 6

See Matrix

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

p = - s - s ² + s ⁴ + s ⁵

217 . Coloring, {4, 5, 7, 8, 9}

R: [4, 4, 4, 8, 3, 7, 5, 6, 2]
B: [2, 9, 5, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 7 vs 7 5 vs 6

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

See Matrix

[0, y₁, y₇, y₅, y₆, y₄, y₂, y₃, 0]

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₁, y₂, 0, 0, y₄, 0, y₃, y₄, y₅]

p = - s ³ + s ⁶

218 . Coloring, {4, 6, 7, 8, 9}

R: [4, 4, 4, 8, 7, 8, 5, 6, 2]
B: [2, 9, 5, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[0, y₁, 0, -14 y₁ - y₂ + 39 y₃ - 14 y₄, -5 y₁ + 14 y₃ - 5 y₄, y₂, y₃, y₄, 0]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{1, 2, 9}, {3, 5}} order: 6

See Matrix

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

p = - s ² - s ³ + s ⁵ + s ⁶

219 . Coloring, {5, 6, 7, 8, 9}

R: [4, 4, 4, 7, 3, 8, 5, 6, 2]
B: [2, 9, 5, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

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

See Matrix

[0, y₂, y₃, y₁, -y₁ + 4 y₂ + 4 y₃ + 4 y₄ - 15 y₅, y₂ + y₃ + y₄ - 4 y₅, y₄, y₅, 0]

p = - s ² + s ⁶ p' = - s ² + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₁, y₂, 0, 0, y₃, 0, y₄, 3 y₃, y₅]

p = - s ³ + s ⁶

220 . Coloring, {2, 3, 4, 5, 6, 7}

Ωp(Δ)=0: p = s ³ + s ⁴ - 4s ⁵ - 8s ⁷

R: [4, 9, 5, 8, 3, 8, 5, 1, 1]
B: [2, 4, 4, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[2 y₁ - y₂ + 2 y₃ - y₄ - y₅, 0, y₁, y₂, y₃, 0, 0, y₄, y₅]

p = s ² + s ³ - s ⁵ - s ⁶

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

See Matrix

[y₁, y₂, 0, y₅, 0, y₃, y₄, 0, 0]

221 . Coloring, {2, 3, 4, 5, 6, 8}

R: [4, 9, 5, 8, 3, 8, 1, 6, 1]
B: [2, 4, 4, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 5

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

See Matrix

[y₃, 0, y₄, y₅, y₂, 2 y₄ - y₅ + 3 y₂ - y₁, 0, -y₃ + 3 y₄ + 2 y₂, y₁]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

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

See Matrix

[y₁, y₂, 0, y₃, y₄, 0, y₅, 0, 0]

222 . Coloring, {2, 3, 4, 5, 6, 9}

R: [4, 9, 5, 8, 3, 8, 1, 1, 2]
B: [2, 4, 4, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {1, 4, 8}} order: 6

See Matrix

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

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

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

See Matrix

[y₄, y₃, 0, y₂, y₁, 2 y₄, 3 y₄ - y₃ + y₂ + y₁, 0, 0]

p = - s ⁴ + s ⁵ p = - s ⁴ + s ⁶

223 . Coloring, {2, 3, 4, 5, 7, 8}

R: [4, 9, 5, 8, 3, 7, 5, 6, 1]
B: [2, 4, 4, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 8 vs 8 5 vs 5

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

See Matrix

[y₁, 0, y₂, y₃, y₄, y₅, y₆, y₇, y₈]

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

See Matrix

[y₄, y₃, 0, y₂, 0, 0, y₁, y₅, 0]

224 . Coloring, {2, 3, 4, 5, 7, 9}

R: [4, 9, 5, 8, 3, 7, 5, 1, 2]
B: [2, 4, 4, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 8 4 vs 6

Omega Rank for R : cycles: {{2, 9}, {1, 4, 8}, {3, 5}} order: 6

See Matrix

[2 y₅, 2 y₁, 2 y₃, 2 y₄, -7 y₁ + 5 y₃ + 5 y₂, 0, 2 y₂, -2 y₅ - 8 y₁ + 8 y₃ - 2 y₄ + 8 y₂, -5 y₁ + 3 y₃ + 3 y₂]

p = - s ² - s ³ + s ⁵ + s ⁶ p = s ² - s ⁴ - s ⁵ + s ⁷ p = - s ² + s ⁸

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

See Matrix

[y₃, y₂, 0, -y₃ + 2 y₁ + 3 y₄, 0, y₁, -y₂ + 3 y₁ + 2 y₄, y₄, 0]

p' = - s + s ⁵ p = - s + s ⁵

225 . Coloring, {2, 3, 4, 5, 8, 9}

R: [4, 9, 5, 8, 3, 7, 1, 6, 2]
B: [2, 4, 4, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 8 vs 9 6 vs 9 5 vs 6

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {1, 4, 6, 7, 8}}

See Matrix

[4 y₂ + 4 y₆ - y₁ - y₃ - y₄ - y₅, y₂, y₆, y₁, y₂, y₃, y₄, y₅, y₆]

p' = s ² + s ³ - s ⁷ - s ⁸ p' = s + s ² - s ⁶ - s ⁷ p' = 1 - s ² - s ⁵ + s ⁷

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

See Matrix

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

p = s ⁵ - s ⁶

226 . Coloring, {2, 3, 4, 6, 7, 8}

Ωp(Δ)=0: p = s ² - 2s ⁴ - 16s ⁷

R: [4, 9, 5, 8, 7, 8, 5, 6, 1]
B: [2, 4, 4, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₁, 0, 0, 3 y₁ - y₂ - 4 y₃ + 3 y₄ - y₅, 2 y₁ - 3 y₃ + 2 y₄, y₂, y₃, y₄, y₅]

p = - s ⁴ + s ⁶ p' = s ⁴ - s ⁶

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

See Matrix

[y₁, y₂, y₃, y₄, 0, 0, y₅, 0, 0]

227 . Coloring, {2, 3, 4, 6, 7, 9}

R: [4, 9, 5, 8, 7, 8, 5, 1, 2]
B: [2, 4, 4, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{5, 7}, {1, 4, 8}, {2, 9}} order: 6

See Matrix

[y₄, y₃, 0, y₂, 2 y₁, 0, 2 y₃, -y₄ + 3 y₃ - y₂ + 3 y₁, y₁]

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

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

See Matrix

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

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

228 . Coloring, {2, 3, 4, 6, 8, 9}

R: [4, 9, 5, 8, 7, 8, 1, 6, 2]
B: [2, 4, 4, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 8 5 vs 6

Omega Rank for R : cycles: {{2, 9}, {6, 8}} order: 6

See Matrix

[2 y₁ - y₂ - y₅ + 3 y₆, y₁, 0, 3 y₁ - y₃ - y₄ + 2 y₆, y₂, y₃, y₄, y₅, y₆]

p = s ⁵ - s ⁷ p' = - s ⁵ + s ⁷

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

See Matrix

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

p = - s ³ + s ⁴ - s ⁵ + s ⁶

229 . Coloring, {2, 3, 4, 7, 8, 9}

R: [4, 9, 5, 8, 7, 7, 5, 6, 2]
B: [2, 4, 4, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 4 vs 6

Omega Rank for R : cycles: {{5, 7}, {2, 9}} order: 4

See Matrix

[0, y₅, 0, y₃, y₄, -15 y₅ - y₃ - y₄ + 4 y₁ + 4 y₂, y₁, y₂, -4 y₅ + y₁ + y₂]

p = s ⁴ - s ⁶ p' = s ⁴ - s ⁶

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

See Matrix

[y₄, y₃, 2 y₁, y₂, 0, 0, y₄ - y₃ + y₂ - 3 y₁, y₁, 0]

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

230 . Coloring, {2, 3, 5, 6, 7, 8}

R: [4, 9, 5, 7, 3, 8, 5, 6, 1]
B: [2, 4, 4, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 6 vs 8 5 vs 5

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

See Matrix

[-y₁ + y₃ - y₄ + 4 y₅, 0, y₁, -y₂ + 4 y₃ + y₅ - y₆, y₂, y₃, y₄, y₅, y₆]

p = - s ⁵ + s ⁷ p' = - s ⁵ + s ⁷

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

See Matrix

[y₁, y₂, 0, y₃, 0, 0, y₄, y₅, 0]

231 . Coloring, {2, 3, 5, 6, 7, 9}

R: [4, 9, 5, 7, 3, 8, 5, 1, 2]
B: [2, 4, 4, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 8 5 vs 6

Omega Rank for R : cycles: {{2, 9}, {3, 5}} order: 6

See Matrix

[y₂, y₁, -y₂ + 3 y₁ - y₅ + 2 y₄, 2 y₁ - y₆ - y₃ + 3 y₄, y₆, 0, y₅, y₃, y₄]

p' = s ⁵ - s ⁷ p = s ⁵ - s ⁷

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

See Matrix

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

p = - s + s ² - s ³ + s ⁴ - s ⁵ + s ⁶

232 . Coloring, {2, 3, 5, 6, 8, 9}

Ωp(Δ)=0: p = s - 64s ⁷ p' = s + 32s ⁶ p' = s ² - 16s ⁶ p' = s ³ + 8s ⁶ p' = s ⁴ - 4s ⁶ p' = s ⁵ + 2s ⁶

R: [4, 9, 5, 7, 3, 8, 1, 6, 2]
B: [2, 4, 4, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

1 vs 7 2 vs 9 2 vs 9 2 vs 9 1 vs 6

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {6, 8}, {1, 4, 7}} order: 6

See Matrix

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

p' = - 1 + s ⁸ p' = - 1 + s ² p' = - s + s ³ p' = - 1 + s ⁴ p' = - s + s ⁵ p' = - 1 + s ⁶ p' = - s + s ⁷

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

See Matrix

[y₁, y₁, 0, y₁, y₁, 0, y₁, y₁, 0]

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

` See 6-level graph

M \ ; N

$ [ [0, 8, 4, 15, 8, 4, 15, 8, 4] , [8, 0, 0, 8, 10, 0, 8, 10, 0] , [4, 0, 0, 4, 0, 5, 4, 0, 5] , [15, 8, 4, 0, 8, 4, 15, 8, 4] , [8, 10, 0, 8, 0, 0, 8, 10, 0] , [4, 0, 5, 4, 0, 0, 4, 0, 5] , [15, 8, 4, 15, 8, 4, 0, 8, 4] , [8, 10, 0, 8, 10, 0, 8, 0, 0] , [4, 0, 5, 4, 0, 5, 4, 0, 0] ] $ $ [ [0, 3, 3, 3, 3, 3, 3, 3, 3] , [3, 0, 1, 3, 3, 2, 3, 3, 3] , [3, 1, 0, 3, 3, 3, 3, 2, 3] , [3, 3, 3, 0, 3, 3, 3, 3, 3] , [3, 3, 3, 3, 0, 1, 3, 3, 2] , [3, 2, 3, 3, 1, 0, 3, 3, 3] , [3, 3, 3, 3, 3, 3, 0, 3, 3] , [3, 3, 2, 3, 3, 3, 3, 0, 1] , [3, 3, 3, 3, 2, 3, 3, 1, 0] ] $

τ= 15 , r'= 5/6

R: [4, 9, 5, 7, 3, 8, 1, 6, 2]
B: [2, 4, 4, 8, 7, 7, 5, 1, 1]

Ranges

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

Cycles: R , {{2, 9}, {3, 5}, {6, 8}, {1, 4, 7}}, B , {{1, 2, 4, 8}, {5, 7}}

β({1, 2, 4, 5, 7, 8}) = 2/3
β({1, 3, 4, 6, 7, 9}) = 1/3

Partitions

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

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

b1 = {1} ` , ` b2 = {2, 6} ` , ` b3 = {4} ` , ` b4 = {8, 9} ` , ` b5 = {2, 3} ` , ` b6 = {7} ` , ` b7 = {5, 6} ` , ` b8 = {3, 8} ` , ` b9 = {5, 9}

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

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

` ` See level-6 partition graph.

Sandwich

Coloring {2, 3, 5, 6, 8, 9}

Rank 6

R,B [4, 9, 5, 7, 3, 8, 1, 6, 2], [2, 4, 4, 8, 7, 7, 5, 1, 1]

π₂ [8, 4, 15, 8, 4, 15, 8, 4, 0, 8, 10, 0, 8, 10, 0, 4, 0, 5, 4, 0, 5, 8, 4, 15, 8, 4, 0, 8, 10, 0, 4, 0, 5, 8, 4, 0]

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

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

π₆ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

u₆ [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0]

233 . Coloring, {2, 3, 5, 7, 8, 9}

R: [4, 9, 5, 7, 3, 7, 5, 6, 2]
B: [2, 4, 4, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 5

Omega Rank for R : cycles: {{2, 9}, {3, 5}} order: 4

See Matrix

[0, 2 y₄, 2 y₃, 3 y₂, -30 y₄ - 5 y₂ + 8 y₃ + 8 y₁, 2 y₂, 2 y₁, 0, -8 y₄ + 2 y₃ + 2 y₁]

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

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

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

234 . Coloring, {2, 3, 6, 7, 8, 9}

R: [4, 9, 5, 7, 7, 8, 5, 6, 2]
B: [2, 4, 4, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 3 vs 7 4 vs 6

Omega Rank for R : cycles: {{2, 9}, {5, 7}, {6, 8}} order: 2

See Matrix

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

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

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

See Matrix

[y₁ - y₂ + 3 y₃ + y₄, y₁, 2 y₃, y₂, 0, 0, y₃, y₄, 0]

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

235 . Coloring, {2, 4, 5, 6, 7, 8}

R: [4, 9, 4, 8, 3, 8, 5, 6, 1]
B: [2, 4, 5, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 5

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

See Matrix

[y₂, 0, y₁, y₅, -9 y₂ + 6 y₁, y₄, 0, y₃, -6 y₂ + 4 y₁]

p = - s ⁴ + s ⁶ p' = - s ⁴ + s ⁶

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

See Matrix

[y₁, y₂, 0, y₃, y₄, 0, y₅, 0, 0]

236 . Coloring, {2, 4, 5, 6, 7, 9}

R: [4, 9, 4, 8, 3, 8, 5, 1, 2]
B: [2, 4, 5, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {1, 4, 8}} order: 6

See Matrix

[y₃, y₂, y₁, -y₃ + 5 y₂ - y₁ - y₆ - y₄ + 5 y₅, y₆, 0, 0, y₄, y₅]

p = - s ³ - s ⁴ + s ⁶ + s ⁷

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

See Matrix

[y₁ - y₂ - 3 y₃ + y₄, y₁, 0, y₂, y₃, 2 y₃, y₄, 0, 0]

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

237 . Coloring, {2, 4, 5, 6, 8, 9}

R: [4, 9, 4, 8, 3, 8, 1, 6, 2]
B: [2, 4, 5, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 5

Omega Rank for R : cycles: {{2, 9}, {6, 8}} order: 4

See Matrix

[3 y₁, 5 y₁ + 2 y₄ - 8 y₃, 2 y₁, 2 y₂, 0, -2 y₂ - 30 y₃ + 20 y₁ + 8 y₄, 0, 2 y₄, 2 y₃]

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

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

See Matrix

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

p = - s ⁴ + s ⁵

238 . Coloring, {2, 4, 5, 7, 8, 9}

R: [4, 9, 4, 8, 3, 7, 5, 6, 2]
B: [2, 4, 5, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {3, 4, 5, 6, 7, 8}} order: 6

See Matrix

[0, y₁, 4 y₁ - y₄ - y₃ + y₂, y₁ - y₆ - y₅ + 4 y₂, y₆, y₅, y₄, y₃, y₂]

p' = - s + s ⁷ p = - s + s ⁷

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

See Matrix

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

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

239 . Coloring, {2, 4, 6, 7, 8, 9}

R: [4, 9, 4, 8, 7, 8, 5, 6, 2]
B: [2, 4, 5, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {5, 7}, {6, 8}} order: 2

See Matrix

[0, y₃, 0, y₂, -5 y₃ + 4 y₁, -10 y₃ - y₂ + 8 y₁, y₁, 2 y₁, -4 y₃ + 3 y₁]

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

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

See Matrix

[3 y₃ - y₄ + 2 y₂, 2 y₃ + 3 y₂ - y₁, y₃, y₄, y₂, 0, y₁, 0, 0]

p = s - s ⁵ p' = s - s ⁵

240 . Coloring, {2, 5, 6, 7, 8, 9}

R: [4, 9, 4, 7, 3, 8, 5, 6, 2]
B: [2, 4, 5, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 7 vs 8 4 vs 8 5 vs 6

Omega Rank for R : cycles: {{3, 4, 5, 7}, {2, 9}, {6, 8}} order: 4

See Matrix

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

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

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

See Matrix

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

p = - s ³ + s ⁴ - s ⁵ + s ⁶

241 . Coloring, {3, 4, 5, 6, 7, 8}

Ωp(Δ)=0: p = s ² - 2s ⁴ - 8s ⁵ + 16s ⁷

R: [4, 4, 5, 8, 3, 8, 5, 6, 1]
B: [2, 9, 4, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[3 y₁ - 4 y₃ + 3 y₂ - y₄, 0, 2 y₁ - 3 y₃ + 2 y₂, y₁, y₃, y₂, 0, y₄, 0]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{2, 9}} order: 4

See Matrix

[y₅, y₄, 0, y₃, 0, 0, y₂, 0, y₁]

242 . Coloring, {3, 4, 5, 6, 7, 9}

R: [4, 4, 5, 8, 3, 8, 5, 1, 2]
B: [2, 9, 4, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

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

See Matrix

[-y₂ + 2 y₁ - y₄ + 2 y₅ - y₃, y₂, y₁, y₄, y₅, 0, 0, y₃, 0]

p = - s ² - s ³ + s ⁵ + s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₂, y₁, 0, y₃, 0, 2 y₃, y₄, 0, y₅]

p = s ³ - s ⁶

243 . Coloring, {3, 4, 5, 6, 8, 9}

R: [4, 4, 5, 8, 3, 8, 1, 6, 2]
B: [2, 9, 4, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 7 vs 8 4 vs 7 5 vs 6

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

See Matrix

[3 y₄, y₄, y₃, y₂, y₁, 2 y₃ - y₂ + 3 y₁, 0, -4 y₄ + 3 y₃ + 2 y₁, 0]

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

Omega Rank for B : cycles: {{1, 2, 9}, {5, 7}} order: 6

See Matrix

[4 y₁, 4 y₂, 0, 4 y₅, 5 y₁ + 5 y₂ - 4 y₅ - 4 y₄ + 5 y₃, 0, 4 y₄, 0, 4 y₃]

p = - s ² - s ³ + s ⁵ + s ⁶

244 . Coloring, {3, 4, 5, 7, 8, 9}

R: [4, 4, 5, 8, 3, 7, 5, 6, 2]
B: [2, 9, 4, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 7 vs 7 5 vs 6

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

See Matrix

[0, y₃, y₂, y₁, y₄, y₇, y₆, y₅, 0]

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₁, y₂, 0, y₃, 0, 0, y₅, y₃, y₄]

p = - s ³ + s ⁶

245 . Coloring, {3, 4, 6, 7, 8, 9}

R: [4, 4, 5, 8, 7, 8, 5, 6, 2]
B: [2, 9, 4, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[0, 3 y₁ - 4 y₂ + 3 y₃ - y₄, 0, y₁, y₂, y₃, 2 y₁ - 3 y₂ + 2 y₃, y₄, 0]

p = - s ³ + s ⁵ p' = - s ³ + s ⁵

Omega Rank for B : cycles: {{1, 2, 9}} order: 6

See Matrix

[y₄, y₅, y₁, y₂, 0, 0, y₃, 0, y₆]

246 . Coloring, {3, 5, 6, 7, 8, 9}

R: [4, 4, 5, 7, 3, 8, 5, 6, 2]
B: [2, 9, 4, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 6

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

See Matrix

[0, y₅, y₄, y₃, y₂, y₁, -y₅ - y₄ - 15 y₁ + 4 y₃ + 4 y₂, y₃ + y₂ - 4 y₁, 0]

p = - s ⁴ + s ⁶ p' = s ⁴ - s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₃, y₄, 0, y₅, 0, 0, 3 y₅, y₁, y₂]

p = - s ³ + s ⁶

247 . Coloring, {4, 5, 6, 7, 8, 9}

R: [4, 4, 4, 8, 3, 8, 5, 6, 2]
B: [2, 9, 5, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[0, y₂, y₁, y₄, 3 y₂, y₃, 0, y₅, 0]

p = s ⁴ - s ⁶

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₁, y₃, 0, 0, y₂, 0, y₄, 0, y₅]

248 . Coloring, {2, 3, 4, 5, 6, 7, 8}

Ωp(Δ)=0: p = - s ³ + s ⁴ + 8s ⁷

R: [4, 9, 5, 8, 3, 8, 5, 6, 1]
B: [2, 4, 4, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

See Matrix

[y₂, 0, y₁, y₅, 2 y₂ - 3 y₁ + 2 y₃, y₄, 0, y₃, 3 y₂ - 4 y₁ - y₅ - y₄ + 3 y₃]

p' = - s ⁴ + s ⁶ p = - s ⁴ + s ⁶

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

See Matrix

[y₂, y₃, 0, y₁, 0, 0, y₄, 0, 0]

249 . Coloring, {2, 3, 4, 5, 6, 7, 9}

R: [4, 9, 5, 8, 3, 8, 5, 1, 2]
B: [2, 4, 4, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 5

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {1, 4, 8}} order: 6

See Matrix

[y₂, y₁, 2 y₁, -y₂ + 3 y₁ - y₄ + 3 y₃, 2 y₃, 0, 0, y₄, y₃]

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

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

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

250 . Coloring, {2, 3, 4, 5, 6, 8, 9}

R: [4, 9, 5, 8, 3, 8, 1, 6, 2]
B: [2, 4, 4, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 8 4 vs 5

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {6, 8}} order: 4

See Matrix

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

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

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

See Matrix

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

p = - s ⁴ + s ⁵

251 . Coloring, {2, 3, 4, 5, 7, 8, 9}

R: [4, 9, 5, 8, 3, 7, 5, 6, 2]
B: [2, 4, 4, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 6 vs 8 4 vs 5

Omega Rank for R : cycles: {{2, 9}, {3, 5}} order: 6

See Matrix

[0, y₆, y₃, y₄, y₅, -15 y₆ - y₄ - y₅ + 4 y₃ + 4 y₂ + 4 y₁, y₂, y₁, -4 y₆ + y₃ + y₂ + y₁]

p' = - s ⁵ + s ⁷ p = - s ⁵ + s ⁷

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

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

252 . Coloring, {2, 3, 4, 6, 7, 8, 9}

R: [4, 9, 5, 8, 7, 8, 5, 6, 2]
B: [2, 4, 4, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 3 vs 7 4 vs 5

Omega Rank for R : cycles: {{2, 9}, {6, 8}, {5, 7}} order: 2

See Matrix

[0, y₁ + y₂ - 2 y₃, 0, y₁, 2 y₃, y₂, 2 y₁ + 2 y₂ - 4 y₃, 2 y₁ + 2 y₂ - 3 y₃, y₃]

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

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

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

253 . Coloring, {2, 3, 5, 6, 7, 8, 9}

R: [4, 9, 5, 7, 3, 8, 5, 6, 2]
B: [2, 4, 4, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 9 vs 9 9 vs 9 4 vs 8 4 vs 5

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {6, 8}} order: 4

See Matrix

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

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

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

See Matrix

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

p = s ² - s ³ + s ⁴ - s ⁵

254 . Coloring, {2, 4, 5, 6, 7, 8, 9}

R: [4, 9, 4, 8, 3, 8, 5, 6, 2]
B: [2, 4, 5, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {6, 8}} order: 4

See Matrix

[0, y₁, y₄, y₅, y₃, -15 y₁ - y₅ - y₃ + 4 y₄ + 4 y₂, 0, y₂, -4 y₁ + y₄ + y₂]

p' = - s ⁴ + s ⁶ p = s ⁴ - s ⁶

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

See Matrix

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

p = - s ² + s ³ - s ⁴ + s ⁵

255 . Coloring, {3, 4, 5, 6, 7, 8, 9}

R: [4, 4, 5, 8, 3, 8, 5, 6, 2]
B: [2, 9, 4, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 5

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

See Matrix

[0, 3 y₂ - 4 y₃ + 3 y₁ - y₄, 2 y₂ - 3 y₃ + 2 y₁, y₂, y₃, y₁, 0, y₄, 0]

p = s ³ - s ⁵ p' = s ³ - s ⁵

Omega Rank for B : cycles: {{1, 2, 9}} order: 3

See Matrix

[y₁, y₂, 0, y₃, 0, 0, y₄, 0, y₅]

256 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9}

R: [4, 9, 5, 8, 3, 8, 5, 6, 2]
B: [2, 4, 4, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

Omega Rank for R : cycles: {{2, 9}, {3, 5}, {6, 8}} order: 2

See Matrix

[0, y₁ + y₂ - 2 y₃, 2 y₁ + 2 y₂ - 4 y₃, y₁, 2 y₃, y₂, 0, 2 y₁ + 2 y₂ - 3 y₃, y₃]

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

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

See Matrix

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

p = - s + s ² - s ³ + s ⁴

SUMMARY

Graph Type
CC

ν(A)
2

ν(Δ)
2

π
[3, 2, 1, 3, 2, 1, 3, 2, 1]

Dbly Stoch
false

SANDWICH
Total 4

No . Coloring Rank

1 {} 3

2 {2, 3, 5, 6, 8, 9} 6

3 {2, 5, 8} 3

4 {3, 6, 9} 3

RT GROUPS
Total 1

No . Coloring Rank Solv

1 {2, 4, 7, 9} 2 Not Solvable

CC Colorings
Total 1

No . Coloring Sandwich,Rank

1 {} true, 3

Δ-RANK'D SC'D !RK'D τ-RANK'D R/B RANK'D NOT SYNC'D Total Runs 2^n-1
213 0 243 , 247 28 , 44 5 256 256

Sandwich
Coloring	{2, 5, 8}
Rank	3
R,B	[4, 9, 4, 7, 3, 7, 1, 6, 1], [2, 4, 5, 8, 7, 8, 5, 1, 2]
π₂	[0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0]
u₂	[16, 10, 20, 16, 20, 20, 8, 10, 17, 12, 20, 12, 12, 20, 11, 20, 15, 20, 10, 8, 20, 14, 0, 20, 14, 20, 14, 10, 20, 11, 20, 14, 20, 18, 10, 18] (dim 1)
wpp	[3, 3, 3, 3, 3, 3, 3, 3, 3]
π₃	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 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, 2, 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]
u₃	[3, 8, 12, 8, 8, 4, 5, 10, 1, 10, 0, 6, 0, 10, 0, 20, 2, 10, 10, 6, 4, 5, 20, 2, 10, 6, 0, 0, 9, 12, 9, 5, 5, 8, 6, 0, 4, 6, 3, 6, 2, 20, 2, 4, 6, 3, 10, 3, 9, 9, 0, 10, 2, 20, 9, 5, 3, 6, 10, 2, 20, 0, 0, 6, 0, 4, 8, 5, 0, 0, 0, 12, 10, 12, 4, 8, 5, 8, 1, 9, 12, 10, 12, 6]

Sandwich
Coloring	{3, 6, 9}
Rank	3
R,B	[4, 4, 5, 7, 7, 8, 1, 1, 2], [2, 9, 4, 8, 3, 7, 5, 6, 1]
π₂	[0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0]
u₂	[186, 280, 535, 414, 480, 535, 470, 310, 293, 434, 535, 405, 450, 535, 372, 520, 525, 535, 270, 252, 535, 126, 90, 535, 510, 460, 180, 530, 535, 365, 500, 485, 535, 90, 300, 333] (dim 1)
wpp	[3, 3, 3, 3, 3, 3, 3, 3, 3]
π₃	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 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, 2, 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]
u₃	[323, 255, 195, 235, 303, 363, 0, 795, 447, 675, 45, 70, 165, 15, 105, 1605, 1335, 705, 110, 1227, 1047, 685, 1335, 1095, 765, 75, 225, 245, 531, 849, 489, 25, 30, 390, 75, 125, 1047, 1227, 646, 150, 1335, 1605, 606, 855, 1065, 726, 15, 470, 510, 343, 225, 765, 686, 1335, 510, 785, 726, 1065, 705, 606, 1605, 0, 105, 150, 0, 363, 303, 35, 165, 145, 45, 195, 675, 699, 430, 390, 30, 255, 375, 489, 215, 795, 849, 55]

Right Group
Coloring	{2, 4, 7, 9}
Rank	2
R,B	[4, 9, 4, 8, 7, 7, 5, 1, 2], [2, 4, 5, 7, 3, 8, 1, 6, 1]
π₂	[142, 0, 0, 104, 87, 191, 0, 46, 52, 94, 0, 0, 0, 92, 0, 0, 0, 57, 81, 0, 0, 162, 46, 171, 0, 97, 0, 0, 114, 0, 0, 0, 0, 127, 0, 47]
u₂	[1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1] (dim 1)
wpp	[4, 5, 4, 4, 5, 5, 5, 4, 5]

Sandwich
Coloring	{2, 3, 5, 6, 8, 9}
Rank	6
R,B	[4, 9, 5, 7, 3, 8, 1, 6, 2], [2, 4, 4, 8, 7, 7, 5, 1, 1]
π₂	[8, 4, 15, 8, 4, 15, 8, 4, 0, 8, 10, 0, 8, 10, 0, 4, 0, 5, 4, 0, 5, 8, 4, 15, 8, 4, 0, 8, 10, 0, 4, 0, 5, 8, 4, 0]
u₂	[3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 1, 3, 3, 2, 3, 3, 3, 3, 3, 1] (dim 2)
wpp	[1, 2, 2, 1, 2, 2, 1, 2, 2]
π₆	[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u₆	[0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0]

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

No .	Coloring	Rank
SANDWICH		Total 4
1	{}	3
2	{2, 3, 5, 6, 8, 9}	6
3	{2, 5, 8}	3
4	{3, 6, 9}	3