New Graph
[4, 5, 1, 3, 1, 4], [2, 4, 5, 6, 3, 1]
π =
[6, 3, 5, 6, 4, 3]
POSSIBLE RANKS
1 x 27
3 x 9
BASE DETERMINANT
161757/1048576, .1542634964
NullSpace of Δ
{1, 2, 3, 4, 5, 6}
Nullspace of A
` det(A) = ` 1/32
1
.
Coloring, {}
Ωp(Δ)=0:
p' =
s - 12s 3 + 16s 4
p =
s - 32s 4 + 48s 5
p' =
s 2 - 4s 3 + 4s 4
R:
[4, 5, 1, 3, 1, 4]
B:
[2, 4, 5, 6, 3, 1]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 2 vs 5 |
3 vs 6 |
3 vs 6 |
2 vs 4 |
2 vs 6 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[9, 0, 6, 9, 3, 0]
,
[9, 0, 9, 9, 0, 0]
,
[9, 0, 9, 9, 0, 0]
,
[9, 0, 9, 9, 0, 0]
] $
[y2, 0, y1, y2, y2 - y1, 0]
p =
- s 2 + s 4
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{3, 5}, {1, 2, 4, 6}}
order:
4
See Matrix
$ [
[3, 6, 4, 3, 5, 6]
,
[6, 3, 5, 6, 4, 3]
,
[3, 6, 4, 3, 5, 6]
,
[6, 3, 5, 6, 4, 3]
,
[3, 6, 4, 3, 5, 6]
,
[6, 3, 5, 6, 4, 3]
] $
[-2 y2 + 3 y1, y2, -y2 + 2 y1, -2 y2 + 3 y1, y1, y2]
p' =
- s + s 3
p' =
- 1 + s 4
p' =
- s + s 5
p' =
- 1 + s 2
` See 3-level graph `
M
N
$ [
[0, 0, 4, 6, 2, 0]
,
[0, 0, 1, 0, 2, 3]
,
[4, 1, 0, 4, 0, 1]
,
[6, 0, 4, 0, 2, 0]
,
[2, 2, 0, 2, 0, 2]
,
[0, 3, 1, 0, 2, 0]
] $
$ [
[0, 2, 3, 3, 3, 1]
,
[2, 0, 3, 1, 3, 3]
,
[3, 3, 0, 3, 0, 3]
,
[3, 1, 3, 0, 3, 2]
,
[3, 3, 0, 3, 0, 3]
,
[1, 3, 3, 2, 3, 0]
] $
R:
[4, 5, 1, 3, 1, 4]
B:
[2, 4, 5, 6, 3, 1]
Ranges
Action of R on ranges, [[1], [1], [2], [2]]
Action of B on ranges, [[4], [3], [2], [1]]
Cycles:
R , {{1, 3, 4}}, B , {{3, 5}, {1, 2, 4, 6}}
β({1, 3, 4})
=
4/9
β({1, 4, 5})
=
2/9
β({2, 3, 6})
=
1/9
β({2, 5, 6})
=
2/9
Partitions
Action of R on partitions, [[1], [1]]
Action of B on partitions, [[2], [1]]
α([{2, 4}, {3, 5}, {1, 6}]) = 2/3
α([{1, 2}, {3, 5}, {4, 6}]) = 1/3
b1 = {2, 4}
` , ` b2 = {1, 2}
` , ` b3 = {3, 5}
` , ` b4 = {4, 6}
` , ` b5 = {1, 6}
Action of R and B on the blocks of the partitions:
=
[5, 3, 1, 5, 3]
[2, 5, 3, 1, 4]
with invariant measure
[2, 1, 3, 1, 2]
N by blocks,
check:
true
.
` See partition graph. `
` See level-3 partition graph. `
| Sandwich |
|---|
| Coloring |
{}
|
|---|
| Rank | 3 |
|---|
| R,B |
[4, 5, 1, 3, 1, 4], [2, 4, 5, 6, 3, 1]
|
|---|
| π2 |
[0, 4, 6, 2, 0, 1, 0, 2, 3, 4, 0, 1, 2, 0, 2]
|
|---|
| u2 |
[2, 3, 3, 3, 1, 3, 1, 3, 3, 3, 0, 3, 3, 2, 3]
(dim 1) |
|---|
| wpp |
[2, 2, 2, 2, 2, 2]
|
|---|
| π3 |
[0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0]
|
|---|
| u3 |
[2, 0, 2, 0, 3, 0, 1, 3, 0, 1, 1, 0, 3, 1, 0, 3, 0, 2, 0, 2]
|
|---|
2
.
Coloring, {2}
R:
[4, 4, 1, 3, 1, 4]
B:
[2, 5, 5, 6, 3, 1]
` 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
$ [
[9, 0, 6, 12, 0, 0]
,
[6, 0, 12, 9, 0, 0]
,
[12, 0, 9, 6, 0, 0]
] $
[y1, 0, y2, y3, 0, 0]
Omega Rank for B :
cycles:
{{3, 5}}
order:
4
See Matrix
$ [
[3, 6, 4, 0, 8, 6]
,
[6, 3, 8, 0, 10, 0]
,
[0, 6, 10, 0, 11, 0]
,
[0, 0, 11, 0, 16, 0]
,
[0, 0, 16, 0, 11, 0]
] $
[y2, y1, y5, 0, y4, y3]
3
.
Coloring, {3}
R:
[4, 5, 5, 3, 1, 4]
B:
[2, 4, 1, 6, 3, 1]
` 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, 5}}
order:
4
See Matrix
$ [
[4, 0, 6, 9, 8, 0]
,
[8, 0, 9, 4, 6, 0]
,
[6, 0, 4, 8, 9, 0]
,
[9, 0, 8, 6, 4, 0]
] $
[y3, 0, y1, y4, y2, 0]
Omega Rank for B :
cycles:
{{1, 2, 4, 6}}
order:
4
See Matrix
$ [
[8, 6, 4, 3, 0, 6]
,
[10, 8, 0, 6, 0, 3]
,
[3, 10, 0, 8, 0, 6]
,
[6, 3, 0, 10, 0, 8]
,
[8, 6, 0, 3, 0, 10]
] $
[y5, y4, y3, y2, 0, y1]
4
.
Coloring, {4}
R:
[4, 5, 1, 6, 1, 4]
B:
[2, 4, 5, 3, 3, 1]
` 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:
{{4, 6}}
order:
4
See Matrix
$ [
[9, 0, 0, 9, 3, 6]
,
[3, 0, 0, 15, 0, 9]
,
[0, 0, 0, 12, 0, 15]
,
[0, 0, 0, 15, 0, 12]
] $
[y3, 0, 0, y1, y2, y4]
Omega Rank for B :
cycles:
{{3, 5}}
order:
4
See Matrix
$ [
[3, 6, 10, 3, 5, 0]
,
[0, 3, 8, 6, 10, 0]
,
[0, 0, 16, 3, 8, 0]
,
[0, 0, 11, 0, 16, 0]
,
[0, 0, 16, 0, 11, 0]
] $
[y1, y2, y3, y4, y5, 0]
5
.
Coloring, {5}
R:
[4, 5, 1, 3, 3, 4]
B:
[2, 4, 5, 6, 1, 1]
` 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, 0, 10, 9, 3, 0]
,
[10, 0, 12, 5, 0, 0]
,
[12, 0, 5, 10, 0, 0]
,
[5, 0, 10, 12, 0, 0]
] $
[y3, 0, y1, y2, y4, 0]
Omega Rank for B :
cycles:
{{1, 2, 4, 6}}
order:
4
See Matrix
$ [
[7, 6, 0, 3, 5, 6]
,
[11, 7, 0, 6, 0, 3]
,
[3, 11, 0, 7, 0, 6]
,
[6, 3, 0, 11, 0, 7]
,
[7, 6, 0, 3, 0, 11]
] $
[y1, y2, 0, y3, y4, y5]
6
.
Coloring, {6}
R:
[4, 5, 1, 3, 1, 1]
B:
[2, 4, 5, 6, 3, 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 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[12, 0, 6, 6, 3, 0]
,
[9, 0, 6, 12, 0, 0]
,
[6, 0, 12, 9, 0, 0]
,
[12, 0, 9, 6, 0, 0]
] $
[y1, 0, y3, y4, y2, 0]
Omega Rank for B :
cycles:
{{3, 5}, {4, 6}}
order:
2
See Matrix
$ [
[0, 6, 4, 6, 5, 6]
,
[0, 0, 5, 12, 4, 6]
,
[0, 0, 4, 6, 5, 12]
,
[0, 0, 5, 12, 4, 6]
,
[0, 0, 4, 6, 5, 12]
] $
[0, -2 y2 + 4 y3 - y1, y2, 4 y2 - 2 y3, y3, y1]
p =
- s 2 + s 4
p' =
- s 2 + s 4
7
.
Coloring, {2, 3}
R:
[4, 4, 5, 3, 1, 4]
B:
[2, 5, 1, 6, 3, 1]
` 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, 5}}
order:
4
See Matrix
$ [
[4, 0, 6, 12, 5, 0]
,
[5, 0, 12, 4, 6, 0]
,
[6, 0, 4, 5, 12, 0]
,
[12, 0, 5, 6, 4, 0]
] $
[y3, 0, y4, y2, y1, 0]
Omega Rank for B :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[8, 6, 4, 0, 3, 6]
,
[10, 8, 3, 0, 6, 0]
,
[3, 10, 6, 0, 8, 0]
,
[6, 3, 8, 0, 10, 0]
,
[8, 6, 10, 0, 3, 0]
] $
[y1, y5, y3, 0, y4, y2]
8
.
Coloring, {2, 4}
R:
[4, 4, 1, 6, 1, 4]
B:
[2, 5, 5, 3, 3, 1]
` 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 4 |
Omega Rank for R :
cycles:
{{4, 6}}
order:
2
See Matrix
$ [
[9, 0, 0, 12, 0, 6]
,
[0, 0, 0, 15, 0, 12]
,
[0, 0, 0, 12, 0, 15]
] $
[y3, 0, 0, y2, 0, y1]
Omega Rank for B :
cycles:
{{3, 5}}
order:
4
See Matrix
$ [
[3, 6, 10, 0, 8, 0]
,
[0, 3, 8, 0, 16, 0]
,
[0, 0, 16, 0, 11, 0]
,
[0, 0, 11, 0, 16, 0]
] $
[y1, y3, y4, 0, y2, 0]
9
.
Coloring, {2, 5}
R:
[4, 4, 1, 3, 3, 4]
B:
[2, 5, 5, 6, 1, 1]
` 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 4 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[5, 0, 10, 12, 0, 0]
,
[10, 0, 12, 5, 0, 0]
,
[12, 0, 5, 10, 0, 0]
] $
[y1, 0, y2, y3, 0, 0]
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[7, 6, 0, 0, 8, 6]
,
[14, 7, 0, 0, 6, 0]
,
[6, 14, 0, 0, 7, 0]
,
[7, 6, 0, 0, 14, 0]
] $
[y3, y4, 0, 0, y2, y1]
10
.
Coloring, {2, 6}
R:
[4, 4, 1, 3, 1, 1]
B:
[2, 5, 5, 6, 3, 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 |
3 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[12, 0, 6, 9, 0, 0]
,
[6, 0, 9, 12, 0, 0]
,
[9, 0, 12, 6, 0, 0]
] $
[y1, 0, y3, y2, 0, 0]
Omega Rank for B :
cycles:
{{3, 5}, {4, 6}}
order:
2
See Matrix
$ [
[0, 6, 4, 3, 8, 6]
,
[0, 0, 8, 6, 10, 3]
,
[0, 0, 10, 3, 8, 6]
,
[0, 0, 8, 6, 10, 3]
,
[0, 0, 10, 3, 8, 6]
] $
[0, -2 y1 - 4 y2 + 4 y3, 2 y1, 2 y2, 2 y3, -4 y2 + 3 y3]
p =
- s 2 + s 4
p' =
- s 2 + s 4
11
.
Coloring, {3, 4}
R:
[4, 5, 5, 6, 1, 4]
B:
[2, 4, 1, 3, 3, 1]
` 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:
{{4, 6}}
order:
4
See Matrix
$ [
[4, 0, 0, 9, 8, 6]
,
[8, 0, 0, 10, 0, 9]
,
[0, 0, 0, 17, 0, 10]
,
[0, 0, 0, 10, 0, 17]
] $
[y1, 0, 0, y3, y4, y2]
Omega Rank for B :
cycles:
{{1, 2, 3, 4}}
order:
4
See Matrix
$ [
[8, 6, 10, 3, 0, 0]
,
[10, 8, 3, 6, 0, 0]
,
[3, 10, 6, 8, 0, 0]
,
[6, 3, 8, 10, 0, 0]
] $
[y1, y2, y3, y4, 0, 0]
12
.
Coloring, {3, 5}
R:
[4, 5, 5, 3, 3, 4]
B:
[2, 4, 1, 6, 1, 1]
` 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 4 |
Omega Rank for R :
cycles:
{{3, 5}}
order:
2
See Matrix
$ [
[0, 0, 10, 9, 8, 0]
,
[0, 0, 17, 0, 10, 0]
,
[0, 0, 10, 0, 17, 0]
] $
[0, 0, y2, y1, y3, 0]
Omega Rank for B :
cycles:
{{1, 2, 4, 6}}
order:
4
See Matrix
$ [
[12, 6, 0, 3, 0, 6]
,
[6, 12, 0, 6, 0, 3]
,
[3, 6, 0, 12, 0, 6]
,
[6, 3, 0, 6, 0, 12]
] $
[y2, y3, 0, y1, 0, y4]
13
.
Coloring, {3, 6}
R:
[4, 5, 5, 3, 1, 1]
B:
[2, 4, 1, 6, 3, 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 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4, 5}}
order:
4
See Matrix
$ [
[7, 0, 6, 6, 8, 0]
,
[8, 0, 6, 7, 6, 0]
,
[6, 0, 7, 8, 6, 0]
,
[6, 0, 8, 6, 7, 0]
] $
[y4, 0, y3, y2, y1, 0]
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[5, 6, 4, 6, 0, 6]
,
[4, 5, 0, 12, 0, 6]
,
[0, 4, 0, 11, 0, 12]
,
[0, 0, 0, 16, 0, 11]
,
[0, 0, 0, 11, 0, 16]
] $
[y1, y2, y3, y4, 0, y5]
14
.
Coloring, {4, 5}
R:
[4, 5, 1, 6, 3, 4]
B:
[2, 4, 5, 3, 1, 1]
` 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 |
5 vs 5 |
Omega Rank for R :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[5, 0, 4, 9, 3, 6]
,
[4, 0, 3, 11, 0, 9]
,
[3, 0, 0, 13, 0, 11]
,
[0, 0, 0, 14, 0, 13]
,
[0, 0, 0, 13, 0, 14]
] $
[y3, 0, y2, y1, y4, y5]
Omega Rank for B :
cycles:
{{1, 2, 3, 4, 5}}
order:
5
See Matrix
$ [
[7, 6, 6, 3, 5, 0]
,
[5, 7, 3, 6, 6, 0]
,
[6, 5, 6, 7, 3, 0]
,
[3, 6, 7, 5, 6, 0]
,
[6, 3, 5, 6, 7, 0]
] $
[y1, y2, y3, y5, y4, 0]
15
.
Coloring, {4, 6}
R:
[4, 5, 1, 6, 1, 1]
B:
[2, 4, 5, 3, 3, 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, 4, 6}}
order:
3
See Matrix
$ [
[12, 0, 0, 6, 3, 6]
,
[9, 0, 0, 12, 0, 6]
,
[6, 0, 0, 9, 0, 12]
,
[12, 0, 0, 6, 0, 9]
] $
[y4, 0, 0, y2, y3, y1]
Omega Rank for B :
cycles:
{{3, 5}}
order:
4
See Matrix
$ [
[0, 6, 10, 6, 5, 0]
,
[0, 0, 11, 6, 10, 0]
,
[0, 0, 16, 0, 11, 0]
,
[0, 0, 11, 0, 16, 0]
] $
[0, y1, y2, y3, y4, 0]
16
.
Coloring, {5, 6}
R:
[4, 5, 1, 3, 3, 1]
B:
[2, 4, 5, 6, 1, 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 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4}}
order:
3
See Matrix
$ [
[8, 0, 10, 6, 3, 0]
,
[10, 0, 9, 8, 0, 0]
,
[9, 0, 8, 10, 0, 0]
,
[8, 0, 10, 9, 0, 0]
] $
[y1, 0, y2, y3, y4, 0]
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[4, 6, 0, 6, 5, 6]
,
[5, 4, 0, 12, 0, 6]
,
[0, 5, 0, 10, 0, 12]
,
[0, 0, 0, 17, 0, 10]
,
[0, 0, 0, 10, 0, 17]
] $
[y2, y1, 0, y5, y4, y3]
17
.
Coloring, {2, 3, 4}
R:
[4, 4, 5, 6, 1, 4]
B:
[2, 5, 1, 3, 3, 1]
` 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:
{{4, 6}}
order:
4
See Matrix
$ [
[4, 0, 0, 12, 5, 6]
,
[5, 0, 0, 10, 0, 12]
,
[0, 0, 0, 17, 0, 10]
,
[0, 0, 0, 10, 0, 17]
] $
[y4, 0, 0, y1, y2, y3]
Omega Rank for B :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[8, 6, 10, 0, 3, 0]
,
[10, 8, 3, 0, 6, 0]
,
[3, 10, 6, 0, 8, 0]
,
[6, 3, 8, 0, 10, 0]
] $
[y3, y4, y1, 0, y2, 0]
18
.
Coloring, {2, 3, 5}
R:
[4, 4, 5, 3, 3, 4]
B:
[2, 5, 1, 6, 1, 1]
` 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 4 |
Omega Rank for R :
cycles:
{{3, 5}}
order:
2
See Matrix
$ [
[0, 0, 10, 12, 5, 0]
,
[0, 0, 17, 0, 10, 0]
,
[0, 0, 10, 0, 17, 0]
] $
[0, 0, y1, y2, y3, 0]
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[12, 6, 0, 0, 3, 6]
,
[9, 12, 0, 0, 6, 0]
,
[6, 9, 0, 0, 12, 0]
,
[12, 6, 0, 0, 9, 0]
] $
[y4, y3, 0, 0, y2, y1]
19
.
Coloring, {2, 3, 6}
R:
[4, 4, 5, 3, 1, 1]
B:
[2, 5, 1, 6, 3, 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 6 |
Omega Rank for R :
cycles:
{{1, 3, 4, 5}}
order:
4
See Matrix
$ [
[7, 0, 6, 9, 5, 0]
,
[5, 0, 9, 7, 6, 0]
,
[6, 0, 7, 5, 9, 0]
,
[9, 0, 5, 6, 7, 0]
] $
[y1, 0, y2, y4, y3, 0]
Omega Rank for B :
cycles:
{{4, 6}, {1, 2, 3, 5}}
order:
4
See Matrix
$ [
[5, 6, 4, 3, 3, 6]
,
[4, 5, 3, 6, 6, 3]
,
[3, 4, 6, 3, 5, 6]
,
[6, 3, 5, 6, 4, 3]
,
[5, 6, 4, 3, 3, 6]
,
[4, 5, 3, 6, 6, 3]
] $
[4 y1 + 4 y2 - 2 y3 - 4 y4, 2 y1, 2 y2, 3 y1 + 3 y2 - 4 y4,
2 y3, 2 y4]
p' =
- s + s 5
p' =
- 1 + s 4
20
.
Coloring, {2, 4, 5}
R:
[4, 4, 1, 6, 3, 4]
B:
[2, 5, 5, 3, 1, 1]
` 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:
{{4, 6}}
order:
4
See Matrix
$ [
[5, 0, 4, 12, 0, 6]
,
[4, 0, 0, 11, 0, 12]
,
[0, 0, 0, 16, 0, 11]
,
[0, 0, 0, 11, 0, 16]
] $
[y1, 0, y3, y4, 0, y2]
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[7, 6, 6, 0, 8, 0]
,
[8, 7, 0, 0, 12, 0]
,
[12, 8, 0, 0, 7, 0]
,
[7, 12, 0, 0, 8, 0]
] $
[y1, y2, y3, 0, y4, 0]
21
.
Coloring, {2, 4, 6}
R:
[4, 4, 1, 6, 1, 1]
B:
[2, 5, 5, 3, 3, 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 |
3 vs 4 |
Omega Rank for R :
cycles:
{{1, 4, 6}}
order:
3
See Matrix
$ [
[12, 0, 0, 9, 0, 6]
,
[6, 0, 0, 12, 0, 9]
,
[9, 0, 0, 6, 0, 12]
] $
[y1, 0, 0, y3, 0, y2]
Omega Rank for B :
cycles:
{{3, 5}}
order:
2
See Matrix
$ [
[0, 6, 10, 3, 8, 0]
,
[0, 0, 11, 0, 16, 0]
,
[0, 0, 16, 0, 11, 0]
,
[0, 0, 11, 0, 16, 0]
] $
[0, 2 y2, y1, y2, y3, 0]
p =
s 2 - s 4
22
.
Coloring, {2, 5, 6}
R:
[4, 4, 1, 3, 3, 1]
B:
[2, 5, 5, 6, 1, 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, 4}}
order:
3
See Matrix
$ [
[8, 0, 10, 9, 0, 0]
,
[10, 0, 9, 8, 0, 0]
,
[9, 0, 8, 10, 0, 0]
] $
[y3, 0, y1, y2, 0, 0]
Omega Rank for B :
cycles:
{{4, 6}, {1, 2, 5}}
order:
6
See Matrix
$ [
[4, 6, 0, 3, 8, 6]
,
[8, 4, 0, 6, 6, 3]
,
[6, 8, 0, 3, 4, 6]
,
[4, 6, 0, 6, 8, 3]
,
[8, 4, 0, 3, 6, 6]
] $
[-y1 + 2 y2 - y3 + 2 y4, y1, 0, y2, y3, y4]
p =
- s - s 2 + s 4 + s 5
23
.
Coloring, {3, 4, 5}
R:
[4, 5, 5, 6, 3, 4]
B:
[2, 4, 1, 3, 1, 1]
` 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 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 5}, {4, 6}}
order:
2
See Matrix
$ [
[0, 0, 4, 9, 8, 6]
,
[0, 0, 8, 6, 4, 9]
,
[0, 0, 4, 9, 8, 6]
,
[0, 0, 8, 6, 4, 9]
] $
[0, 0, 4 y2, -15 y2 + 16 y1, -16 y2 + 16 y1, 4 y1]
p' =
s - s 3
p =
s - s 3
Omega Rank for B :
cycles:
{{1, 2, 3, 4}}
order:
4
See Matrix
$ [
[12, 6, 6, 3, 0, 0]
,
[6, 12, 3, 6, 0, 0]
,
[3, 6, 6, 12, 0, 0]
,
[6, 3, 12, 6, 0, 0]
] $
[y1, y2, y3, y4, 0, 0]
24
.
Coloring, {3, 4, 6}
R:
[4, 5, 5, 6, 1, 1]
B:
[2, 4, 1, 3, 3, 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, 4, 6}}
order:
3
See Matrix
$ [
[7, 0, 0, 6, 8, 6]
,
[14, 0, 0, 7, 0, 6]
,
[6, 0, 0, 14, 0, 7]
,
[7, 0, 0, 6, 0, 14]
] $
[y3, 0, 0, y2, y1, y4]
Omega Rank for B :
cycles:
{{1, 2, 3, 4}}
order:
4
See Matrix
$ [
[5, 6, 10, 6, 0, 0]
,
[10, 5, 6, 6, 0, 0]
,
[6, 10, 6, 5, 0, 0]
,
[6, 6, 5, 10, 0, 0]
] $
[y1, y2, y3, y4, 0, 0]
25
.
Coloring, {3, 5, 6}
R:
[4, 5, 5, 3, 3, 1]
B:
[2, 4, 1, 6, 1, 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, 5}}
order:
4
See Matrix
$ [
[3, 0, 10, 6, 8, 0]
,
[0, 0, 14, 3, 10, 0]
,
[0, 0, 13, 0, 14, 0]
,
[0, 0, 14, 0, 13, 0]
] $
[y3, 0, y2, y1, y4, 0]
Omega Rank for B :
cycles:
{{4, 6}}
order:
4
See Matrix
$ [
[9, 6, 0, 6, 0, 6]
,
[0, 9, 0, 12, 0, 6]
,
[0, 0, 0, 15, 0, 12]
,
[0, 0, 0, 12, 0, 15]
] $
[y1, y3, 0, y2, 0, y4]
26
.
Coloring, {4, 5, 6}
R:
[4, 5, 1, 6, 3, 1]
B:
[2, 4, 5, 3, 1, 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 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 4, 6}}
order:
3
See Matrix
$ [
[8, 0, 4, 6, 3, 6]
,
[10, 0, 3, 8, 0, 6]
,
[9, 0, 0, 10, 0, 8]
,
[8, 0, 0, 9, 0, 10]
,
[10, 0, 0, 8, 0, 9]
] $
[y1, 0, y3, y2, y4, y5]
Omega Rank for B :
cycles:
{{1, 2, 3, 4, 5}}
order:
5
See Matrix
$ [
[4, 6, 6, 6, 5, 0]
,
[5, 4, 6, 6, 6, 0]
,
[6, 5, 6, 4, 6, 0]
,
[6, 6, 4, 5, 6, 0]
,
[6, 6, 5, 6, 4, 0]
] $
[y1, y2, y3, y4, y5, 0]
27
.
Coloring, {2, 3, 4, 5}
R:
[4, 4, 5, 6, 3, 4]
B:
[2, 5, 1, 3, 1, 1]
` 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 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 5}, {4, 6}}
order:
2
See Matrix
$ [
[0, 0, 4, 12, 5, 6]
,
[0, 0, 5, 6, 4, 12]
,
[0, 0, 4, 12, 5, 6]
,
[0, 0, 5, 6, 4, 12]
] $
[0, 0, y1, -2 y1 + 4 y2, y2, 4 y1 - 2 y2]
p' =
- s + s 3
p =
- s + s 3
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[12, 6, 6, 0, 3, 0]
,
[9, 12, 0, 0, 6, 0]
,
[6, 9, 0, 0, 12, 0]
,
[12, 6, 0, 0, 9, 0]
] $
[y3, y4, y2, 0, y1, 0]
28
.
Coloring, {2, 3, 4, 6}
R:
[4, 4, 5, 6, 1, 1]
B:
[2, 5, 1, 3, 3, 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 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 4, 6}}
order:
3
See Matrix
$ [
[7, 0, 0, 9, 5, 6]
,
[11, 0, 0, 7, 0, 9]
,
[9, 0, 0, 11, 0, 7]
,
[7, 0, 0, 9, 0, 11]
] $
[y1, 0, 0, y2, y3, y4]
Omega Rank for B :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[5, 6, 10, 3, 3, 0]
,
[10, 5, 6, 0, 6, 0]
,
[6, 10, 6, 0, 5, 0]
,
[6, 6, 5, 0, 10, 0]
,
[5, 6, 10, 0, 6, 0]
] $
[y1, y2, y5, y3, y4, 0]
29
.
Coloring, {2, 3, 5, 6}
R:
[4, 4, 5, 3, 3, 1]
B:
[2, 5, 1, 6, 1, 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 5 |
Omega Rank for R :
cycles:
{{3, 5}}
order:
4
See Matrix
$ [
[3, 0, 10, 9, 5, 0]
,
[0, 0, 14, 3, 10, 0]
,
[0, 0, 13, 0, 14, 0]
,
[0, 0, 14, 0, 13, 0]
] $
[y4, 0, y1, y2, y3, 0]
Omega Rank for B :
cycles:
{{4, 6}, {1, 2, 5}}
order:
6
See Matrix
$ [
[9, 6, 0, 3, 3, 6]
,
[3, 9, 0, 6, 6, 3]
,
[6, 3, 0, 3, 9, 6]
,
[9, 6, 0, 6, 3, 3]
,
[3, 9, 0, 3, 6, 6]
] $
[y1, -y1 + 2 y4 - y3 + 2 y2, 0, y4, y3, y2]
p =
s + s 2 - s 4 - s 5
30
.
Coloring, {2, 4, 5, 6}
R:
[4, 4, 1, 6, 3, 1]
B:
[2, 5, 5, 3, 1, 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 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 4, 6}}
order:
3
See Matrix
$ [
[8, 0, 4, 9, 0, 6]
,
[10, 0, 0, 8, 0, 9]
,
[9, 0, 0, 10, 0, 8]
,
[8, 0, 0, 9, 0, 10]
] $
[y3, 0, y1, y2, 0, y4]
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[4, 6, 6, 3, 8, 0]
,
[8, 4, 3, 0, 12, 0]
,
[12, 8, 0, 0, 7, 0]
,
[7, 12, 0, 0, 8, 0]
,
[8, 7, 0, 0, 12, 0]
] $
[y2, y1, y5, y4, y3, 0]
31
.
Coloring, {3, 4, 5, 6}
R:
[4, 5, 5, 6, 3, 1]
B:
[2, 4, 1, 3, 1, 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 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 5}, {1, 4, 6}}
order:
6
See Matrix
$ [
[3, 0, 4, 6, 8, 6]
,
[6, 0, 8, 3, 4, 6]
,
[6, 0, 4, 6, 8, 3]
,
[3, 0, 8, 6, 4, 6]
,
[6, 0, 4, 3, 8, 6]
] $
[5 y1 - 4 y3 + 5 y4 - 4 y2, 0, 4 y1, 4 y3, 4 y4, 4 y2]
p =
- s - s 2 + s 4 + s 5
Omega Rank for B :
cycles:
{{1, 2, 3, 4}}
order:
4
See Matrix
$ [
[9, 6, 6, 6, 0, 0]
,
[6, 9, 6, 6, 0, 0]
,
[6, 6, 6, 9, 0, 0]
,
[6, 6, 9, 6, 0, 0]
] $
[y2, y3, y1, y4, 0, 0]
32
.
Coloring, {2, 3, 4, 5, 6}
R:
[4, 4, 5, 6, 3, 1]
B:
[2, 5, 1, 3, 1, 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 5 |
3 vs 5 |
Omega Rank for R :
cycles:
{{3, 5}, {1, 4, 6}}
order:
6
See Matrix
$ [
[3, 0, 4, 9, 5, 6]
,
[6, 0, 5, 3, 4, 9]
,
[9, 0, 4, 6, 5, 3]
,
[3, 0, 5, 9, 4, 6]
,
[6, 0, 4, 3, 5, 9]
] $
[2 y1 - y2 + 2 y3 - y4, 0, y1, y2, y3, y4]
p =
s + s 2 - s 4 - s 5
Omega Rank for B :
cycles:
{{1, 2, 5}}
order:
3
See Matrix
$ [
[9, 6, 6, 3, 3, 0]
,
[9, 9, 3, 0, 6, 0]
,
[9, 9, 0, 0, 9, 0]
,
[9, 9, 0, 0, 9, 0]
,
[9, 9, 0, 0, 9, 0]
] $
[y1 + y2, y1 - y3 + y2, y1, y3, y2, 0]
p =
- s 3 + s 5
p =
- s 3 + s 4
| SUMMARY |
|---|
| Graph Type |
| NOT CC |
|---|
| ν(A) |
|
0
|
|---|
| ν(Δ) |
|
1
|
|---|
| π |
|
[6, 3, 5, 6, 4, 3]
|
|---|
| Dbly Stoch |
| false |
|---|
| SANDWICH |
| Total
1
|
|---|
| No . | Coloring | Rank |
|---|
| 1 |
{}
|
3
|
|---|
| RT GROUPS |
| Total
0
|
|---|
| No . | Coloring | Rank | Solv |
|---|
| Δ-RANK'D | SC'D !RK'D |
τ-RANK'D | R/B RANK'D | NOT SYNC'D |
Total Runs | 2n-1 |
|---|
| 31 |
0 |
31 , 31 |
27 , 24 |
1 |
32 |
32 |