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, {}

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

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

` See graph

` ` See pair graph

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

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

See Matrix
 

[y1, y1, y2, y2]

  p' = - 1 + s 2   p' = - s + s 3

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

See Matrix
 

[0, 0, y1, y1]

  p = s - s 2


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] ] $

τ= 8 , r'= 1/2

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

Ranges

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

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

β({1, 2}) = 1/3
β({3, 4}) = 2/3

Partitions

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

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

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

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

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

` ` See level-2 partition graph.

`

Sandwich
Coloring {}
Rank2
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

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

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

See Matrix
 

[y1, y2, 0, y3]

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

See Matrix
 

[0, 0, y2, y1]


 
3 . Coloring, {3}

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

` See graph

` ` See pair graph

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

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

See Matrix
 

[0, y3, y1, y2]

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

See Matrix
 

[y2 - y1, 0, y2, y1]

  p = s 2 - s 3


 
4 . Coloring, {4}

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

` See graph

` ` See pair graph

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

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

See Matrix
 

[y3, 0, y1, y2]

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

See Matrix
 

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

  p = s 2 - s 3


 
5 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

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

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

See Matrix
 

[0, y1, 0, y2]

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

See Matrix
 

[y1, 0, y2, 0]


 
6 . Coloring, {2, 4}

Ωp(Δ)=0:     p = s 2   p' = s 2

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

` See graph

` ` See pair graph

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

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

See Matrix
 

[y1, 0, y1, y1]

  p = - s + s 2   p = - s + s 3

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

See Matrix
 

[0, y1, y1, y1]

  p = - s + s 2   p = - s + s 3


` See 3-level graph

`

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] ] $

τ= 6 , r'= 2/3

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

Ranges

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

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

β({1, 3, 4}) = 1/2
β({2, 3, 4}) = 1/2

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}
Rank3
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}

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

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

` See graph

` ` See pair graph

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

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

See Matrix
 

[0, 0, y1, y1]

  p = - s + s 2

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

See Matrix
 

[y1, y1, y2, y2]

  p' = - s + s 3   p' = - 1 + s 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] ] $

τ= 8 , r'= 1/2

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

Ranges

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

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

β({1, 2}) = 1/3
β({3, 4}) = 2/3

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}
Rank2
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

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

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

See Matrix
 

[0, 0, y1, y2]

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

See Matrix
 

[y3, y2, y1, 0]


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

 
SANDWICH
Total 1
No .ColoringRank
1 {} 2

 
RT GROUPS
Total 2
No .ColoringRankSolv
1 {3, 4} 2 Solvable
2 {2, 4} 3 Solvable

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
5 0 5 , 3 5 , 3 3 8 8