New Graph

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

π = [1, 1, 1, 1]

POSSIBLE RANKS

1 x 4
2 x 2

BASE DETERMINANT 117/512, .2285156250

NullSpace of Δ

{1, 2, 3, 4}

Nullspace of A

[{1, 4},{2, 3}]

1 . Coloring, {}

Ωp(Δ)=0: p' = s ² p = s ²

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[0, y₁, y₁, 0]

p = - s + s ²

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

See Matrix

[y₁, 0, 0, y₁]

p = - s + s ²

M N

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

τ= 8 , r'= 1/2

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

Ranges

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

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

β({1, 4}) = 1/2
β({2, 3}) = 1/2

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

Sandwich

Coloring {}

Rank 2

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

π₂ [0, 0, 1, 1, 0, 0]

u₂ [1, 1, 2, 2, 1, 1] (dim 1)

wpp [2, 2, 2, 2]

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

2 . Coloring, {2}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[0, y₂, y₃, y₁]

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

See Matrix

[y₂, 0, y₂ - y₁, y₁]

p = - s ² + s ³

3 . Coloring, {3}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₂, y₃, y₁, 0]

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

See Matrix

[-y₁ + y₂, y₁, 0, y₂]

p = s ² - s ³

4 . Coloring, {4}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁ - y₂, y₁, y₂, 0]

p = s ² - s ³

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

See Matrix

[y₁, 0, y₃, y₂]

5 . Coloring, {2, 3}

Ωp(Δ)=0: p' = s ² p' = s p = s

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, y₁, y₁, y₁]

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

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

See Matrix

[y₁, y₁, y₁, y₁]

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

` See 4-level graph

M N

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

τ= 4 , r'= 3/4

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

Ranges

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

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

β({1, 2, 3, 4}) = 1/1

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

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

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

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

` ` See level-4 partition graph.

Right Group

Coloring {2, 3}

Rank 4

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

π₂ [1, 1, 1, 1, 1, 1]

u₂ [1, 1, 1, 1, 1, 1] (dim 2)

wpp [1, 1, 1, 1]

π₄ [1]

u₄ [1]

Right Group
Coloring	{2, 3}
Rank	4
R,B	[2, 4, 1, 3], [4, 3, 2, 1]
π₂	[1, 1, 1, 1, 1, 1]
u₂	[1, 1, 1, 1, 1, 1] (dim 2)
wpp	[1, 1, 1, 1]
π₄	[1]
u₄	[1]

6 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 3 3 vs 3 3 vs 3 3 vs 3 3 vs 3

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

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

See Matrix

[y₂, y₃, 0, y₁]

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

See Matrix

[y₁, 0, y₂, y₃]

7 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

3 vs 3 3 vs 3 3 vs 3 3 vs 3 3 vs 3

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

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

See Matrix

[y₁, y₃, y₂, 0]

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

See Matrix

[0, y₃, y₂, y₁]

8 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

`
Δ-Rank A+(1/2)Δ A-(1/2)Δ R B

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

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

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

See Matrix

[y₁, y₃, 0, y₂]

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

See Matrix

[0, y₁, y₂, -y₁ + y₂]

p = s ² - s ³

SUMMARY

Graph Type
CC

ν(A)
1

ν(Δ)
1

π
[1, 1, 1, 1]

Dbly Stoch
true

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

SANDWICH
Total 1

No . Coloring Rank

1 {} 2

SANDWICH		Total 1
No .	Coloring	Rank
1	{}	2

RT GROUPS
Total 1

No . Coloring Rank Solv

1 {2, 3} 4 ["group", Not Solvable]

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

CC Colorings
Total 1

No . Coloring Sandwich,Rank

1 {} true, 2

CC Colorings		Total 1
No .	Coloring	Sandwich,Rank
1	{}	true, 2

Δ-RANK'D SC'D !RK'D τ-RANK'D R/B RANK'D NOT SYNC'D Total Runs 2^n-1
6 0 6 , 6 5 , 3 2 8 8