New Graph

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

 

 

π = [1, 1, 1, 1, 1, 1]

POSSIBLE RANKS

1 x 6
2 x 3

BASE DETERMINANT 3087149/33554432, .9200420976e-1

NullSpace of Δ

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

Nullspace of A

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

 
1 . Coloring, {}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 5 vs 5 5 vs 5

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

See Matrix
 

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

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

See Matrix
 

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


 
2 . Coloring, {2}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 2 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 2 + s 4   p = - s 2 + s 3

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

See Matrix
 

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

  p = - s 3 + s 4


 
3 . Coloring, {3}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 5 vs 5 4 vs 5

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

See Matrix
 

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

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

See Matrix
 

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

  p = - s 4 + s 5


 
4 . Coloring, {4}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 4 vs 5 5 vs 5

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

See Matrix
 

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

  p = - s 4 + s 5

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

See Matrix
 

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


 
5 . Coloring, {5}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 5 vs 5 4 vs 5

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

See Matrix
 

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

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

See Matrix
 

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

  p = - s 4 + s 5


 
6 . Coloring, {6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 4 vs 5 5 vs 5

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

See Matrix
 

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

  p = - s 4 + s 5

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

See Matrix
 

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


 
7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 vs 5 2 vs 5

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

See Matrix
 

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

  p = - s 3 + s 4   p = - s 3 + s 5

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

See Matrix
 

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

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


 
8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 3 + s 4

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

See Matrix
 

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

  p = - s 3 + s 4


 
9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 vs 5 2 vs 5

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

See Matrix
 

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

  p = - s 3 + s 4   p = - s 3 + s 5

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

See Matrix
 

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

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


 
10 . Coloring, {2, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 vs 4 3 vs 4

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

See Matrix
 

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

  p = s 3 - s 4

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

See Matrix
 

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

  p = - s 3 + s 4


 
11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 5 vs 5 5 vs 5

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

See Matrix
 

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

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

See Matrix
 

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


 
12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 vs 5 2 vs 5

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

See Matrix
 

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

  p = - s 3 + s 4   p = - s 3 + s 5

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

See Matrix
 

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

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


 
13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 3 + s 4

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

See Matrix
 

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

  p = - s 3 + s 4


 
14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 3 + s 4

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

See Matrix
 

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

  p = - s 3 + s 4


 
15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 2 vs 5 3 vs 5

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

See Matrix
 

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

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

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

See Matrix
 

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

  p = - s 3 + s 4   p = - s 3 + s 5


 
16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 5 vs 5 5 vs 5

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

See Matrix
 

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

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

See Matrix
 

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


 
17 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 5 vs 5 4 vs 5

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

See Matrix
 

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

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

See Matrix
 

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

  p = - s 4 + s 5


 
18 . Coloring, {2, 3, 5}

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

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
0 vs 5 1 vs 6 1 vs 6 1 vs 6 1 vs 6

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

See Matrix
 

[y1, y1, y1, y1, y1, y1]

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

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

See Matrix
 

[y1, y1, y1, y1, y1, y1]

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


` See 6-level graph

`

M               N

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

τ= 6 , r'= 5/6

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

Ranges

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

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

β({1, 2, 3, 4, 5, 6}) = 1/1

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

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

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

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

` ` See level-6 partition graph.

`

Right Group
Coloring {2, 3, 5}
Rank6
R,B [3, 4, 5, 6, 1, 2], [2, 1, 4, 3, 6, 5]
π2 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
u2 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 3)
wpp [1, 1, 1, 1, 1, 1]
π6 [1]
u6 [1]

 

 
19 . Coloring, {2, 3, 6}

Ωp(Δ)=0:     p' = s - 8s 3 - 8s 4   p' = s 2 - 2s 3 - 4s 4   p = s - 24s 4 - 32s 5

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 5 3 vs 6 3 vs 6 2 vs 4 1 vs 4

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

See Matrix
 

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

  p = - s 2 + s 3   p = - s 2 + s 4

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

See Matrix
 

[2 y1, 2 y1, y1, y1, 0, 0]

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


M               N

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

τ= 18 , r'= 1/2

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

Ranges

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

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 3, 6}
Rank2
R,B [3, 4, 5, 6, 6, 5], [2, 1, 4, 3, 1, 2]
π2 [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
u2 [3, 2, 1, 2, 1, 1, 2, 1, 2, 3, 2, 1, 1, 2, 3] (dim 1)
wpp [3, 3, 3, 3, 3, 3]

 

 
20 . Coloring, {2, 4, 5}

Ωp(Δ)=0:     p = s - 24s 4 - 32s 5   p' = s - 8s 3 - 8s 4   p' = s 2 - 2s 3 - 4s 4

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 5 3 vs 6 3 vs 6 2 vs 4 1 vs 4

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

See Matrix
 

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

  p = - s 2 + s 4   p = - s 2 + s 3

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

See Matrix
 

[y1, y1, 0, 0, 2 y1, 2 y1]

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


M               N

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

τ= 18 , r'= 1/2

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

Ranges

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

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 4, 5}
Rank2
R,B [3, 4, 4, 3, 1, 2], [2, 1, 5, 6, 6, 5]
π2 [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
u2 [3, 2, 1, 2, 1, 1, 2, 1, 2, 3, 2, 1, 1, 2, 3] (dim 1)
wpp [3, 3, 3, 3, 3, 3]

 

 
21 . Coloring, {2, 4, 6}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 4s 4   p' = s - 8s 3 + 8s 4   p = s + 24s 4 - 32s 5

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 5 3 vs 6 3 vs 6 1 vs 4 2 vs 4

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

See Matrix
 

[0, 0, 2 y1, 2 y1, y1, y1]

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

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

See Matrix
 

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

  p = - s 2 + s 3   p = - s 2 + s 4


M               N

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

τ= 18 , r'= 1/2

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

Ranges

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

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

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

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

`

Sandwich
Coloring {2, 4, 6}
Rank2
R,B [3, 4, 4, 3, 6, 5], [2, 1, 5, 6, 1, 2]
π2 [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
u2 [3, 2, 1, 2, 1, 1, 2, 1, 2, 3, 2, 1, 1, 2, 3] (dim 1)
wpp [3, 3, 3, 3, 3, 3]

 

 
22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 5 vs 5 4 vs 5

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

See Matrix
 

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

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

See Matrix
 

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

  p = - s 4 + s 5


 
23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 2 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 2 + s 4   p = - s 2 + s 3

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

See Matrix
 

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

  p = - s 3 + s 4


 
24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 vs 4 2 vs 4

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

See Matrix
 

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

  p = - s 3 + s 4

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

See Matrix
 

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

  p = - s 2 + s 3   p = - s 2 + s 4


 
25 . Coloring, {3, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 2 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 2 + s 3   p = - s 2 + s 4

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

See Matrix
 

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

  p = - s 3 + s 4


 
26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 vs 4 2 vs 4

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

See Matrix
 

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

  p = s 3 - s 4

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

See Matrix
 

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

  p = - s 2 + s 3   p = - s 2 + s 4


 
27 . Coloring, {2, 3, 4, 5}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 vs 5 2 vs 5

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

See Matrix
 

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

  p = - s 3 + s 4   p = - s 3 + s 5

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

See Matrix
 

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

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


 
28 . Coloring, {2, 3, 4, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 vs 4 3 vs 4

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

See Matrix
 

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

  p = s 3 - s 4

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

See Matrix
 

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

  p = - s 3 + s 4


 
29 . Coloring, {2, 3, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 3 vs 5 2 vs 5

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

See Matrix
 

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

  p = - s 3 + s 4   p = - s 3 + s 5

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

See Matrix
 

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

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


 
30 . Coloring, {2, 4, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 5 vs 5 5 vs 5 3 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 3 + s 4

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

See Matrix
 

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

  p = - s 3 + s 4


 
31 . Coloring, {3, 4, 5, 6}

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

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 5 1 vs 5 1 vs 5 1 vs 3 1 vs 3

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

See Matrix
 

[y1, 0, y1, 0, y1, 0]

  p = - s + s 2   p = - s + s 3

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

See Matrix
 

[0, y1, 0, y1, 0, y1]

  p = - s + s 3   p = - s + s 2


` See 3-level graph

`

M               N

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

τ= 12 , r'= 2/3

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

Ranges

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

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

β({1, 3, 5}) = 1/2
β({2, 4, 6}) = 1/2

Partitions

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

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

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

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

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

` ` See level-3 partition graph.

`

Sandwich
Coloring {3, 4, 5, 6}
Rank3
R,B [3, 1, 5, 3, 1, 5], [2, 4, 4, 6, 6, 2]
π2 [0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0]
u2 [2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2] (dim 1)
wpp [2, 2, 2, 2, 2, 2]
π3 [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
u3 [1, 1, 1, 1, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 1]

 

 
32 . Coloring, {2, 3, 4, 5, 6}

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

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 2 vs 4 3 vs 4

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

See Matrix
 

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

  p = - s 2 + s 3   p = - s 2 + s 4

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

See Matrix
 

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

  p = - s 3 + s 4


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

 
SANDWICH
Total 4
No .ColoringRank
1 {2, 4, 5} 2
2 {2, 4, 6} 2
3 {2, 3, 6} 2
4 {3, 4, 5, 6} 3

 
RT GROUPS
Total 1
No .ColoringRankSolv
1 {2, 3, 5} 6 ["group", Not Solvable]

 
CC Colorings
Total 1
No .ColoringSandwich,Rank
1 {3, 4, 5, 6} true, 3

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
27 0 27 , 27 7 , 5 5 32 32