New Graph

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

π = [1, 1, 2, 2]

POSSIBLE RANKS

1 x 6
2 x 3

BASE DETERMINANT 3/16, .1875000000

NullSpace of Δ

{1, 2, 3, 4}

Nullspace of A

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

1 . Coloring, {}

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

` See graph

` ` See pair graph

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

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

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

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

[1, 1, 2, 2]

+ - Δ

See Matrices

[y₁, y₁, -y₁, -y₁]

p' = s + 2s ² p = s - 4s ³

S+ S- NM

See Matrices

CmmCk true, true, true

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

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

See Matrix

[y₁, y₁, y₂, y₂]

p' = - 1 + s ² p' = - s + s ³

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

See Matrix

[0, 0, y₁, y₁]

p = s - s ²

« NOT SYNC'D »

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

Range of {ΩΔⁱ}: [-μ₁, -μ₁, μ₁, μ₁]

rank of M is 4 , rank of N is 3

M N

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

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

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

τ= 8 , r'= 1/2

Ranges

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

Sandwich

Coloring {}

Rank 2

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

π₂ [1, 0, 0, 0, 0, 2]

u₂ [3, 2, 1, 1, 2, 3] (dim 1)

wpp [2, 2, 2, 2]

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

2 . Coloring, {2}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-21, -39, -28, -52] . FixedPtCheck, [21, 39, 28, 52]

det(A + τ Δ) = 0

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

See Matrix for A+τΔ

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

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

[y₁, y₂, 0, y₃]

See Matrices

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

[0, 0, y₁, y₂]

See Matrices

» SYNC'D 1/4 , 0.2500000000

3 . Coloring, {3}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-13, -45, -52, -60] . FixedPtCheck, [13, 45, 52, 60]

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

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

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

[0, y₁, y₂, y₃]

See Matrices

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

See Matrix

[y₁, 0, y₁ + y₂, y₂]

p = - s ² + s ³

» SYNC'D 1/4 , 0.2500000000

4 . Coloring, {4}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-45, -13, -60, -52] . FixedPtCheck, [45, 13, 60, 52]

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

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

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

[y₃, 0, y₂, y₁]

See Matrices

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

See Matrix

[0, y₁, y₂, y₁ + y₂]

p = - s ² + s ³

» SYNC'D 1/4 , 0.2500000000

5 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-7, -45, -28, -60] . FixedPtCheck, [7, 45, 28, 60]

det(A + τ Δ) = 0

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

See Matrix for A+τΔ

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

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

[0, y₁, 0, y₂]

See Matrices

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

[y₂, 0, y₁, 0]

See Matrices

» SYNC'D 1/4 , 0.2500000000

6 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

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

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

det(A + τ Δ) = 0

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

[1, 1, 2, 2]

+ - Δ

See Matrices

[-y₁, y₁, 0, 0]

p = s ²

S+ S- NM

See Matrices

CmmCk true, true, true

p' = s ²

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

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

See Matrix

[y₁, 0, y₁, y₁]

p' = s - s ² p = s - s ³

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

See Matrix

[0, y₁, y₁, y₁]

p = - s + s ³ p = - s + s ²

« NOT SYNC'D »

Nullspace of {Ω&Deltaⁱ} :
[0, x₁, x₂]
For A+2Δ : [y₁, -3 y₁ - 4 y₂ - 4 y₃, y₂, y₃]
For A-2Δ : [-3 y₁ - 4 y₂ - 4 y₃, y₁, y₂, y₃]

Range of {ΩΔⁱ}: [-μ₁, μ₁, 0, 0]

rank of M is 3 , rank of N is 3

M N

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

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

ker M, [λ₁, -λ₁, 0, 0]
Range M, [x₁, x₁, x₂, x₃]

τ= 6 , r'= 2/3

Ranges

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

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

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

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

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

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

` ` See level-3 partition graph.

Right Group

Coloring {2, 4}

Rank 3

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

π₂ [0, 1, 1, 1, 1, 2]

u₂ [0, 1, 1, 1, 1, 1] (dim 1)

wpp [2, 2, 1, 1]

π₃ [0, 0, 1, 1]

u₃ [0, 0, 1, 1]

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

7 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

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

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

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

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

[1, 1, 2, 2]

+ - Δ

See Matrices

[y₁, y₁, -y₁, -y₁]

p' = s - 2s ² p = s - 4s ³

S+ S- NM

See Matrices

CmmCk true, true, true

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

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

See Matrix

[0, 0, y₁, y₁]

p = s - s ²

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

See Matrix

[y₁, y₁, y₂, y₂]

p' = - 1 + s ² p' = - s + s ³

« NOT SYNC'D »

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

Range of {ΩΔⁱ}: [-μ₁, -μ₁, μ₁, μ₁]

rank of M is 4 , rank of N is 2

M N

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

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

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

τ= 8 , r'= 1/2

Ranges

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

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

Partitions
α([{1, 3}, {2, 4}]) = 1/1

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

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

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

` ` See level-2 partition graph.

Right Group

Coloring {3, 4}

Rank 2

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

π₂ [1, 0, 0, 0, 0, 2]

u₂ [1, 0, 1, 1, 0, 1] (dim 1)

wpp [2, 2, 2, 2]

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

8 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

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

For τ=1/2, [-13, -15, -52, -60] . FixedPtCheck, [13, 15, 52, 60]

det(A + τ Δ) = 0

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

See Matrix for A+τΔ

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

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

[0, 0, y₁, y₂]

See Matrices

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

[y₃, y₂, y₁, 0]

See Matrices

» SYNC'D 1/4 , 0.2500000000

SUMMARY

Graph Type
CC

ν(A)
1

ν(Δ)
1

π
[1, 1, 2, 2]

Dbly Stoch
false

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

SANDWICH
Total 1

No . Coloring Rank

1 {} 2

SANDWICH		Total 1
No .	Coloring	Rank
1	{}	2

RT GROUPS
Total 2

No . Coloring Rank Solv

1 {3, 4} 2 Solvable

2 {2, 4} 3 Solvable

RT GROUPS		Total 2
No .	Coloring	Rank	Solv
1	{3, 4}	2	Solvable
2	{2, 4}	3	Solvable

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