New Graph
[3, 1, 1, 1, 2, 3], [5, 4, 4, 6, 4, 4]
π =
[4, 1, 3, 4, 2, 2]
POSSIBLE RANKS
1 x 16
2 x 8
4 x 4
BASE DETERMINANT
2831/16384, .1727905273
NullSpace of Δ
{1, 2, 3, 4, 5, 6}
Nullspace of A
[{1, 2, 3},{4, 5, 6}]
1
.
Coloring, {}
Ωp(Δ)=0:
p =
s 4
p' =
s 4
R:
[3, 1, 1, 1, 2, 3]
B:
[5, 4, 4, 6, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 5 |
3 vs 5 |
3 vs 5 |
2 vs 3 |
2 vs 3 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[8, 2, 6, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
] $
[y1 + y2, y1, y2, 0, 0, 0]
p =
s 2 - s 3
Omega Rank for B :
cycles:
{{4, 6}}
order:
2
See Matrix
$ [
[0, 0, 0, 8, 4, 4]
,
[0, 0, 0, 8, 0, 8]
,
[0, 0, 0, 8, 0, 8]
] $
[0, 0, 0, y1, y1 - y2, y2]
p =
s 2 - s 3
M
N
$ [
[0, 1, 3, 0, 0, 0]
,
[1, 0, 0, 0, 0, 0]
,
[3, 0, 0, 0, 0, 0]
,
[0, 0, 0, 0, 2, 2]
,
[0, 0, 0, 2, 0, 0]
,
[0, 0, 0, 2, 0, 0]
] $
$ [
[0, 2, 2, 1, 1, 1]
,
[2, 0, 0, 1, 1, 1]
,
[2, 0, 0, 1, 1, 1]
,
[1, 1, 1, 0, 2, 2]
,
[1, 1, 1, 2, 0, 0]
,
[1, 1, 1, 2, 0, 0]
] $
R:
[3, 1, 1, 1, 2, 3]
B:
[5, 4, 4, 6, 4, 4]
Ranges
Action of R on ranges, [[2], [2], [1], [2]]
Action of B on ranges, [[3], [3], [4], [4]]
Cycles:
R , {{1, 3}}, B , {{4, 6}}
β({1, 2})
=
1/8
β({1, 3})
=
3/8
β({4, 5})
=
1/4
β({4, 6})
=
1/4
Partitions
Action of R on partitions, [[1], [1]]
Action of B on partitions, [[2], [2]]
α([{2, 3, 4}, {1, 5, 6}]) = 1/2
α([{1, 4}, {2, 3, 5, 6}]) = 1/2
b1 = {2, 3, 4}
` , ` b2 = {1, 5, 6}
` , ` b3 = {1, 4}
` , ` b4 = {2, 3, 5, 6}
Action of R and B on the blocks of the partitions:
=
[2, 1, 1, 2]
[4, 3, 4, 3]
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 |
[3, 1, 1, 1, 2, 3], [5, 4, 4, 6, 4, 4]
|
|---|
| π2 |
[1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0]
|
|---|
| u2 |
[2, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 0]
(dim 1) |
|---|
| wpp |
[5, 7, 7, 5, 7, 7]
|
|---|
2
.
Coloring, {2}
R:
[3, 4, 1, 1, 2, 3]
B:
[5, 1, 4, 6, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
3 vs 4 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
4
See Matrix
$ [
[7, 2, 6, 1, 0, 0]
,
[7, 0, 7, 2, 0, 0]
,
[9, 0, 7, 0, 0, 0]
,
[7, 0, 9, 0, 0, 0]
] $
[y2, y1, y3, y4, 0, 0]
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[1, 0, 0, 7, 4, 4]
,
[0, 0, 0, 8, 1, 7]
,
[0, 0, 0, 8, 0, 8]
,
[0, 0, 0, 8, 0, 8]
] $
[-y3 + y1 + y2, 0, 0, y3, y1, y2]
p =
- s 3 + s 4
3
.
Coloring, {3}
R:
[3, 1, 4, 1, 2, 3]
B:
[5, 4, 1, 6, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
3 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[5, 2, 6, 3, 0, 0]
,
[5, 0, 5, 6, 0, 0]
,
[6, 0, 5, 5, 0, 0]
,
[5, 0, 6, 5, 0, 0]
] $
[y2, y1, y3, y4, 0, 0]
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[3, 0, 0, 5, 4, 4]
,
[0, 0, 0, 8, 3, 5]
,
[0, 0, 0, 8, 0, 8]
,
[0, 0, 0, 8, 0, 8]
] $
[y1, 0, 0, -y1 + y3 + y2, y3, y2]
p =
- s 3 + s 4
4
.
Coloring, {4}
R:
[3, 1, 1, 6, 2, 3]
B:
[5, 4, 4, 1, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
4 vs 5 |
2 vs 4 |
3 vs 3 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[4, 2, 6, 0, 0, 4]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
] $
[-y1 + y2, y1, y2, 0, 0, 2 y1]
p' =
s 2 - s 3
p =
s 2 - s 4
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[4, 0, 0, 8, 4, 0]
,
[8, 0, 0, 4, 4, 0]
,
[4, 0, 0, 4, 8, 0]
] $
[y1, 0, 0, y2, y3, 0]
5
.
Coloring, {5}
R:
[3, 1, 1, 1, 4, 3]
B:
[5, 4, 4, 6, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
2 vs 3 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[8, 0, 6, 2, 0, 0]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
] $
[y1 + y2, 0, y1, y2, 0, 0]
p =
s 2 - s 3
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[0, 2, 0, 6, 4, 4]
,
[0, 4, 0, 6, 0, 6]
,
[0, 0, 0, 10, 0, 6]
,
[0, 0, 0, 6, 0, 10]
] $
[0, y2, 0, y1, y4, y3]
6
.
Coloring, {6}
R:
[3, 1, 1, 1, 2, 4]
B:
[5, 4, 4, 6, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
2 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[8, 2, 4, 2, 0, 0]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
] $
[2 y1 + y2, y1, y2, y1, 0, 0]
p =
s 2 - s 4
p' =
s 2 - s 3
Omega Rank for B :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[0, 0, 2, 6, 4, 4]
,
[0, 0, 4, 6, 0, 6]
,
[0, 0, 6, 4, 0, 6]
,
[0, 0, 6, 6, 0, 4]
] $
[0, 0, y4, y3, y2, y1]
7
.
Coloring, {2, 3}
R:
[3, 4, 4, 1, 2, 3]
B:
[5, 1, 1, 6, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
4 vs 4 |
3 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[4, 2, 6, 4, 0, 0]
,
[4, 0, 4, 8, 0, 0]
,
[8, 0, 4, 4, 0, 0]
,
[4, 0, 8, 4, 0, 0]
] $
[y3, y4, y2, y1, 0, 0]
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[4, 0, 0, 4, 4, 4]
,
[0, 0, 0, 8, 4, 4]
,
[0, 0, 0, 8, 0, 8]
,
[0, 0, 0, 8, 0, 8]
] $
[-y1 + y2 + y3, 0, 0, y1, y2, y3]
p =
- s 3 + s 4
8
.
Coloring, {2, 4}
R:
[3, 4, 1, 6, 2, 3]
B:
[5, 1, 4, 1, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
3 vs 3 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
4
See Matrix
$ [
[3, 2, 6, 1, 0, 4]
,
[6, 0, 7, 2, 0, 1]
,
[7, 0, 7, 0, 0, 2]
,
[7, 0, 9, 0, 0, 0]
,
[9, 0, 7, 0, 0, 0]
] $
[y1, y2, y3, y4, 0, y5]
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[5, 0, 0, 7, 4, 0]
,
[7, 0, 0, 4, 5, 0]
,
[4, 0, 0, 5, 7, 0]
] $
[y2, 0, 0, y1, y3, 0]
9
.
Coloring, {2, 5}
R:
[3, 4, 1, 1, 4, 3]
B:
[5, 1, 4, 6, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
3 vs 3 |
4 vs 5 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[7, 0, 6, 3, 0, 0]
,
[9, 0, 7, 0, 0, 0]
,
[7, 0, 9, 0, 0, 0]
] $
[y1, 0, y2, y3, 0, 0]
Omega Rank for B :
cycles:
{{1, 2, 5}, {4, 6}}
order:
6
See Matrix
$ [
[1, 2, 0, 5, 4, 4]
,
[2, 4, 0, 4, 1, 5]
,
[4, 1, 0, 5, 2, 4]
,
[1, 2, 0, 4, 4, 5]
,
[2, 4, 0, 5, 1, 4]
] $
[7 y1, 7 y4, 0, 7 y3, 7 y2, 9 y1 + 9 y4 - 7 y3 + 9 y2]
p =
- s - s 2 + s 4 + s 5
10
.
Coloring, {2, 6}
R:
[3, 4, 1, 1, 2, 4]
B:
[5, 1, 4, 6, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
4
See Matrix
$ [
[7, 2, 4, 3, 0, 0]
,
[7, 0, 7, 2, 0, 0]
,
[9, 0, 7, 0, 0, 0]
,
[7, 0, 9, 0, 0, 0]
] $
[y4, y2, y3, y1, 0, 0]
Omega Rank for B :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[1, 0, 2, 5, 4, 4]
,
[0, 0, 4, 6, 1, 5]
,
[0, 0, 5, 5, 0, 6]
,
[0, 0, 6, 5, 0, 5]
,
[0, 0, 5, 6, 0, 5]
] $
[y1, 0, y2, y3, y4, y5]
11
.
Coloring, {3, 4}
R:
[3, 1, 4, 6, 2, 3]
B:
[5, 4, 1, 1, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
3 vs 3 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[1, 2, 6, 3, 0, 4]
,
[2, 0, 5, 6, 0, 3]
,
[0, 0, 5, 5, 0, 6]
,
[0, 0, 6, 5, 0, 5]
,
[0, 0, 5, 6, 0, 5]
] $
[y1, y4, y3, y2, 0, y5]
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[7, 0, 0, 5, 4, 0]
,
[5, 0, 0, 4, 7, 0]
,
[4, 0, 0, 7, 5, 0]
] $
[y1, 0, 0, y2, y3, 0]
12
.
Coloring, {3, 5}
R:
[3, 1, 4, 1, 4, 3]
B:
[5, 4, 1, 6, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
3 vs 3 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[5, 0, 6, 5, 0, 0]
,
[5, 0, 5, 6, 0, 0]
,
[6, 0, 5, 5, 0, 0]
] $
[y3, 0, y1, y2, 0, 0]
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[3, 2, 0, 3, 4, 4]
,
[0, 4, 0, 6, 3, 3]
,
[0, 3, 0, 7, 0, 6]
,
[0, 0, 0, 9, 0, 7]
,
[0, 0, 0, 7, 0, 9]
] $
[y5, y4, 0, y3, y1, y2]
13
.
Coloring, {3, 6}
R:
[3, 1, 4, 1, 2, 4]
B:
[5, 4, 1, 6, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[5, 2, 4, 5, 0, 0]
,
[7, 0, 5, 4, 0, 0]
,
[4, 0, 7, 5, 0, 0]
,
[5, 0, 4, 7, 0, 0]
] $
[y1, y2, y3, y4, 0, 0]
Omega Rank for B :
cycles:
{{1, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[3, 0, 2, 3, 4, 4]
,
[2, 0, 4, 4, 3, 3]
,
[4, 0, 3, 3, 2, 4]
,
[3, 0, 4, 2, 4, 3]
,
[4, 0, 3, 4, 3, 2]
] $
[y5, 0, y4, y3, y1, y2]
14
.
Coloring, {4, 5}
R:
[3, 1, 1, 6, 4, 3]
B:
[5, 4, 4, 1, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
3 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
4
See Matrix
$ [
[4, 0, 6, 2, 0, 4]
,
[6, 0, 8, 0, 0, 2]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
] $
[y2 + y3 - y1, 0, y2, y3, 0, y1]
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[4, 2, 0, 6, 4, 0]
,
[6, 4, 0, 2, 4, 0]
,
[2, 4, 0, 4, 6, 0]
,
[4, 6, 0, 4, 2, 0]
] $
[y3, y4, 0, y1, y2, 0]
15
.
Coloring, {4, 6}
R:
[3, 1, 1, 6, 2, 4]
B:
[5, 4, 4, 1, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
3 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3}, {4, 6}}
order:
2
See Matrix
$ [
[4, 2, 4, 2, 0, 4]
,
[6, 0, 4, 4, 0, 2]
,
[4, 0, 6, 2, 0, 4]
,
[6, 0, 4, 4, 0, 2]
,
[4, 0, 6, 2, 0, 4]
] $
[y3, 4 y3 - y2 - 5 y1, y2, y1, 0, 3 y3 - 4 y1]
p' =
s 2 - s 4
p =
- s 2 + s 4
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[4, 0, 2, 6, 4, 0]
,
[6, 0, 0, 6, 4, 0]
,
[6, 0, 0, 4, 6, 0]
,
[4, 0, 0, 6, 6, 0]
] $
[y1, 0, y2, y3, y4, 0]
16
.
Coloring, {5, 6}
R:
[3, 1, 1, 1, 4, 4]
B:
[5, 4, 4, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
2 vs 3 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[8, 0, 4, 4, 0, 0]
,
[8, 0, 8, 0, 0, 0]
,
[8, 0, 8, 0, 0, 0]
] $
[y1 + y2, 0, y1, y2, 0, 0]
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[0, 2, 2, 4, 4, 4]
,
[0, 4, 4, 4, 0, 4]
,
[0, 0, 4, 8, 0, 4]
,
[0, 0, 4, 4, 0, 8]
,
[0, 0, 8, 4, 0, 4]
] $
[0, y1, y2, y3, y4, y5]
17
.
Coloring, {2, 3, 4}
R:
[3, 4, 4, 6, 2, 3]
B:
[5, 1, 1, 1, 4, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
4 vs 4 |
3 vs 3 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[0, 2, 6, 4, 0, 4]
,
[0, 0, 4, 8, 0, 4]
,
[0, 0, 4, 4, 0, 8]
,
[0, 0, 8, 4, 0, 4]
] $
[0, y3, y4, y2, 0, y1]
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[8, 0, 0, 4, 4, 0]
,
[4, 0, 0, 4, 8, 0]
,
[4, 0, 0, 8, 4, 0]
] $
[y1, 0, 0, y2, y3, 0]
18
.
Coloring, {2, 3, 5}
R:
[3, 4, 4, 1, 4, 3]
B:
[5, 1, 1, 6, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
3 vs 3 |
4 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[4, 0, 6, 6, 0, 0]
,
[6, 0, 4, 6, 0, 0]
,
[6, 0, 6, 4, 0, 0]
] $
[y1, 0, y2, y3, 0, 0]
Omega Rank for B :
cycles:
{{1, 2, 5}, {4, 6}}
order:
6
See Matrix
$ [
[4, 2, 0, 2, 4, 4]
,
[2, 4, 0, 4, 4, 2]
,
[4, 4, 0, 2, 2, 4]
,
[4, 2, 0, 4, 4, 2]
,
[2, 4, 0, 2, 4, 4]
] $
[-3 y2 + 5 y1 - 3 y3 + 5 y4, 3 y2, 0, 3 y1, 3 y3, 3 y4]
p =
- s - s 2 + s 4 + s 5
19
.
Coloring, {2, 3, 6}
R:
[3, 4, 4, 1, 2, 4]
B:
[5, 1, 1, 6, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[4, 2, 4, 6, 0, 0]
,
[6, 0, 4, 6, 0, 0]
,
[6, 0, 6, 4, 0, 0]
,
[4, 0, 6, 6, 0, 0]
] $
[y1, y2, y3, y4, 0, 0]
Omega Rank for B :
cycles:
{{1, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[4, 0, 2, 2, 4, 4]
,
[2, 0, 4, 4, 4, 2]
,
[4, 0, 2, 4, 2, 4]
,
[2, 0, 4, 2, 4, 4]
,
[4, 0, 4, 4, 2, 2]
] $
[y2, 0, y1, y3, y4, y5]
20
.
Coloring, {2, 4, 5}
R:
[3, 4, 1, 6, 4, 3]
B:
[5, 1, 4, 1, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
4
See Matrix
$ [
[3, 0, 6, 3, 0, 4]
,
[6, 0, 7, 0, 0, 3]
,
[7, 0, 9, 0, 0, 0]
,
[9, 0, 7, 0, 0, 0]
] $
[y1, 0, y2, y3, 0, y4]
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[5, 2, 0, 5, 4, 0]
,
[7, 4, 0, 0, 5, 0]
,
[4, 5, 0, 0, 7, 0]
,
[5, 7, 0, 0, 4, 0]
] $
[y4, y3, 0, y2, y1, 0]
21
.
Coloring, {2, 4, 6}
R:
[3, 4, 1, 6, 2, 4]
B:
[5, 1, 4, 1, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
3 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3}, {4, 6}}
order:
2
See Matrix
$ [
[3, 2, 4, 3, 0, 4]
,
[4, 0, 3, 6, 0, 3]
,
[3, 0, 4, 3, 0, 6]
,
[4, 0, 3, 6, 0, 3]
,
[3, 0, 4, 3, 0, 6]
] $
[-7 y1 + 15 y2 - 7 y3, 6 y1, 6 y2, -15 y1 + 27 y2 - 15 y3, 0,
6 y3]
p =
- s 2 + s 4
p' =
- s 2 + s 4
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[5, 0, 2, 5, 4, 0]
,
[5, 0, 0, 6, 5, 0]
,
[6, 0, 0, 5, 5, 0]
,
[5, 0, 0, 5, 6, 0]
] $
[y2, 0, y1, y3, y4, 0]
22
.
Coloring, {2, 5, 6}
R:
[3, 4, 1, 1, 4, 4]
B:
[5, 1, 4, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
3 vs 3 |
3 vs 6 |
Omega Rank for R :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[7, 0, 4, 5, 0, 0]
,
[9, 0, 7, 0, 0, 0]
,
[7, 0, 9, 0, 0, 0]
] $
[y3, 0, y1, y2, 0, 0]
Omega Rank for B :
cycles:
{{1, 2, 5}, {3, 4, 6}}
order:
3
See Matrix
$ [
[1, 2, 2, 3, 4, 4]
,
[2, 4, 4, 2, 1, 3]
,
[4, 1, 3, 4, 2, 2]
,
[1, 2, 2, 3, 4, 4]
,
[2, 4, 4, 2, 1, 3]
,
[4, 1, 3, 4, 2, 2]
] $
[28 y1 + 25 y2 - 35 y3, -20 y2 + 28 y3, 16 y1,
20 y1 + 27 y2 - 25 y3, 16 y2, 16 y3]
p' =
1 - s 3
p' =
s - s 4
p' =
s 2 - s 5
23
.
Coloring, {3, 4, 5}
R:
[3, 1, 4, 6, 4, 3]
B:
[5, 4, 1, 1, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[1, 0, 6, 5, 0, 4]
,
[0, 0, 5, 6, 0, 5]
,
[0, 0, 5, 5, 0, 6]
,
[0, 0, 6, 5, 0, 5]
] $
[y1, 0, y2, y3, 0, y4]
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[7, 2, 0, 3, 4, 0]
,
[3, 4, 0, 2, 7, 0]
,
[2, 7, 0, 4, 3, 0]
,
[4, 3, 0, 7, 2, 0]
] $
[y1, y2, 0, y3, y4, 0]
24
.
Coloring, {3, 4, 6}
R:
[3, 1, 4, 6, 2, 4]
B:
[5, 4, 1, 1, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[1, 2, 4, 5, 0, 4]
,
[2, 0, 1, 8, 0, 5]
,
[0, 0, 2, 6, 0, 8]
,
[0, 0, 0, 10, 0, 6]
,
[0, 0, 0, 6, 0, 10]
] $
[y4, y1, y2, y3, 0, y5]
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[7, 0, 2, 3, 4, 0]
,
[5, 0, 0, 4, 7, 0]
,
[4, 0, 0, 7, 5, 0]
,
[7, 0, 0, 5, 4, 0]
] $
[y3, 0, y1, y2, y4, 0]
25
.
Coloring, {3, 5, 6}
R:
[3, 1, 4, 1, 4, 4]
B:
[5, 4, 1, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
3 vs 3 |
6 vs 6 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[5, 0, 4, 7, 0, 0]
,
[7, 0, 5, 4, 0, 0]
,
[4, 0, 7, 5, 0, 0]
] $
[y3, 0, y1, y2, 0, 0]
Omega Rank for B :
cycles:
{{1, 2, 3, 4, 5, 6}}
order:
6
See Matrix
$ [
[3, 2, 2, 1, 4, 4]
,
[2, 4, 4, 2, 3, 1]
,
[4, 3, 1, 4, 2, 2]
,
[1, 2, 2, 3, 4, 4]
,
[2, 4, 4, 2, 1, 3]
,
[4, 1, 3, 4, 2, 2]
] $
[y1, y2, y3, y4, y5, y6]
26
.
Coloring, {4, 5, 6}
Ωp(Δ)=0:
p =
s 3
p' =
s 3
p' =
s 4
R:
[3, 1, 1, 6, 4, 4]
B:
[5, 4, 4, 1, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 2 vs 5 |
2 vs 5 |
2 vs 5 |
1 vs 4 |
2 vs 5 |
Omega Rank for R :
cycles:
{{1, 3}, {4, 6}}
order:
2
See Matrix
$ [
[4, 0, 4, 4, 0, 4]
,
[4, 0, 4, 4, 0, 4]
,
[4, 0, 4, 4, 0, 4]
,
[4, 0, 4, 4, 0, 4]
] $
[y1, 0, y1, y1, 0, y1]
p' =
- s + s 3
p =
s - s 2
p' =
- s + s 2
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[4, 2, 2, 4, 4, 0]
,
[4, 4, 0, 4, 4, 0]
,
[4, 4, 0, 4, 4, 0]
,
[4, 4, 0, 4, 4, 0]
,
[4, 4, 0, 4, 4, 0]
] $
[y2, y2 - y1, y1, y2, y2, 0]
p =
- s 2 + s 3
p =
- s 2 + s 4
p =
- s 2 + s 5
` See 4-level graph `
M
N
$ [
[0, 8, 24, 36, 18, 18]
,
[8, 0, 0, 9, 9, 0]
,
[24, 0, 0, 27, 9, 18]
,
[36, 9, 27, 0, 16, 16]
,
[18, 9, 9, 16, 0, 0]
,
[18, 0, 18, 16, 0, 0]
] $
$ [
[0, 1, 1, 1, 1, 1]
,
[1, 0, 0, 1, 1, 1]
,
[1, 0, 0, 1, 1, 1]
,
[1, 1, 1, 0, 1, 1]
,
[1, 1, 1, 1, 0, 0]
,
[1, 1, 1, 1, 0, 0]
] $
R:
[3, 1, 1, 6, 4, 4]
B:
[5, 4, 4, 1, 2, 3]
Ranges
Action of R on ranges, [[3], [3], [3]]
Action of B on ranges, [[1], [1], [2]]
Cycles:
R , {{1, 3}, {4, 6}}, B , {{1, 2, 4, 5}}
β({1, 2, 4, 5})
=
1/4
β({1, 3, 4, 5})
=
1/4
β({1, 3, 4, 6})
=
1/2
Partitions
α([{1}, {5, 6}, {4}, {2, 3}]) = 1/1
b1 = {1}
` , ` b2 = {5, 6}
` , ` b3 = {4}
` , ` b4 = {2, 3}
Action of R and B on the blocks of the partitions:
=
[4, 3, 2, 1]
[3, 1, 4, 2]
with invariant measure
[1, 1, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-4 partition graph. `
| Right Group |
|---|
| Coloring |
{4, 5, 6}
|
|---|
| Rank | 4 |
|---|
| R,B |
[3, 1, 1, 6, 4, 4], [5, 4, 4, 1, 2, 3]
|
|---|
| π2 |
[8, 24, 36, 18, 18, 0, 9, 9, 0, 27, 9, 18, 16, 16, 0]
|
|---|
| u2 |
[1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0]
(dim 2) |
|---|
| wpp |
[1, 2, 2, 1, 2, 2]
|
|---|
| π4 |
[0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0]
|
|---|
| u4 |
[0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0]
|
|---|
27
.
Coloring, {2, 3, 4, 5}
R:
[3, 4, 4, 6, 4, 3]
B:
[5, 1, 1, 1, 2, 4]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
3 vs 3 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[0, 0, 6, 6, 0, 4]
,
[0, 0, 4, 6, 0, 6]
,
[0, 0, 6, 4, 0, 6]
] $
[0, 0, y1, y2, 0, y3]
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[8, 2, 0, 2, 4, 0]
,
[4, 4, 0, 0, 8, 0]
,
[4, 8, 0, 0, 4, 0]
,
[8, 4, 0, 0, 4, 0]
] $
[y4, y3, 0, y2, y1, 0]
28
.
Coloring, {2, 3, 4, 6}
R:
[3, 4, 4, 6, 2, 4]
B:
[5, 1, 1, 1, 4, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
3 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{4, 6}}
order:
2
See Matrix
$ [
[0, 2, 4, 6, 0, 4]
,
[0, 0, 0, 10, 0, 6]
,
[0, 0, 0, 6, 0, 10]
,
[0, 0, 0, 10, 0, 6]
] $
[0, y1, 2 y1, y3, 0, y2]
p =
s 2 - s 4
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[8, 0, 2, 2, 4, 0]
,
[4, 0, 0, 4, 8, 0]
,
[4, 0, 0, 8, 4, 0]
,
[8, 0, 0, 4, 4, 0]
] $
[y1, 0, y4, y2, y3, 0]
29
.
Coloring, {2, 3, 5, 6}
R:
[3, 4, 4, 1, 4, 4]
B:
[5, 1, 1, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
5 vs 5 |
5 vs 5 |
3 vs 3 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[4, 0, 4, 8, 0, 0]
,
[8, 0, 4, 4, 0, 0]
,
[4, 0, 8, 4, 0, 0]
] $
[y1, 0, y3, y2, 0, 0]
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[4, 2, 2, 0, 4, 4]
,
[4, 4, 4, 0, 4, 0]
,
[8, 4, 0, 0, 4, 0]
,
[4, 4, 0, 0, 8, 0]
,
[4, 8, 0, 0, 4, 0]
] $
[y5, y3, y4, 0, y1, y2]
30
.
Coloring, {2, 4, 5, 6}
R:
[3, 4, 1, 6, 4, 4]
B:
[5, 1, 4, 1, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
2 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3}, {4, 6}}
order:
2
See Matrix
$ [
[3, 0, 4, 5, 0, 4]
,
[4, 0, 3, 4, 0, 5]
,
[3, 0, 4, 5, 0, 4]
,
[4, 0, 3, 4, 0, 5]
] $
[-8 y2 + 7 y1, 0, y2, y1, 0, -9 y2 + 8 y1]
p =
- s + s 3
p' =
- s + s 3
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[5, 2, 2, 3, 4, 0]
,
[5, 4, 0, 2, 5, 0]
,
[6, 5, 0, 0, 5, 0]
,
[5, 5, 0, 0, 6, 0]
,
[5, 6, 0, 0, 5, 0]
] $
[y4, y5, y2, y1, y3, 0]
31
.
Coloring, {3, 4, 5, 6}
R:
[3, 1, 4, 6, 4, 4]
B:
[5, 4, 1, 1, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
6 vs 6 |
6 vs 6 |
3 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[1, 0, 4, 7, 0, 4]
,
[0, 0, 1, 8, 0, 7]
,
[0, 0, 0, 8, 0, 8]
,
[0, 0, 0, 8, 0, 8]
] $
[y3, 0, y3 + y1 - y2, y1, 0, y2]
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[7, 2, 2, 1, 4, 0]
,
[3, 4, 0, 2, 7, 0]
,
[2, 7, 0, 4, 3, 0]
,
[4, 3, 0, 7, 2, 0]
,
[7, 2, 0, 3, 4, 0]
] $
[y2, y1, y4, y5, y3, 0]
32
.
Coloring, {2, 3, 4, 5, 6}
R:
[3, 4, 4, 6, 4, 4]
B:
[5, 1, 1, 1, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 5 vs 5 |
4 vs 5 |
5 vs 5 |
2 vs 3 |
4 vs 4 |
Omega Rank for R :
cycles:
{{4, 6}}
order:
2
See Matrix
$ [
[0, 0, 4, 8, 0, 4]
,
[0, 0, 0, 8, 0, 8]
,
[0, 0, 0, 8, 0, 8]
] $
[0, 0, y1 - y2, y1, 0, y2]
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[8, 2, 2, 0, 4, 0]
,
[4, 4, 0, 0, 8, 0]
,
[4, 8, 0, 0, 4, 0]
,
[8, 4, 0, 0, 4, 0]
] $
[y3, y2, y1, 0, y4, 0]
| SUMMARY |
|---|
| Graph Type |
| CC |
|---|
| ν(A) |
|
1
|
|---|
| ν(Δ) |
|
1
|
|---|
| π |
|
[4, 1, 3, 4, 2, 2]
|
|---|
| Dbly Stoch |
| false |
|---|
| SANDWICH |
| Total
1
|
|---|
| No . | Coloring | Rank |
|---|
| 1 |
{}
|
2
|
|---|
| RT GROUPS |
| Total
1
|
|---|
| No . | Coloring | Rank | Solv |
|---|
| 1 |
{4, 5, 6}
|
4
|
Solvable
|
|---|
| 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 | 2n-1 |
|---|
| 30 |
0 |
29 , 29 |
19 , 24 |
2 |
32 |
32 |