New Graph
[3, 4, 4, 3, 2, 5], [6, 1, 2, 6, 1, 3]
π =
[3, 4, 6, 5, 2, 4]
POSSIBLE RANKS
1 x 24
2 x 12
3 x 8
4 x 6
BASE DETERMINANT
55/256, .2148437500
NullSpace of Δ
{3, 5, 6}, {1, 2, 4}
Nullspace of A
[{5, 6},{3}]
1
.
Coloring, {}
R:
[3, 4, 4, 3, 2, 5]
B:
[6, 1, 2, 6, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[0, 2, 8, 10, 4, 0]
,
[0, 4, 10, 10, 0, 0]
,
[0, 0, 10, 14, 0, 0]
,
[0, 0, 14, 10, 0, 0]
] $
[0, y1, y2, y3, y4, 0]
Omega Rank for B :
cycles:
{{1, 2, 3, 6}}
order:
4
See Matrix
$ [
[6, 6, 4, 0, 0, 8]
,
[6, 4, 8, 0, 0, 6]
,
[4, 8, 6, 0, 0, 6]
,
[8, 6, 6, 0, 0, 4]
] $
[y1, y2, y3, 0, 0, y4]
2
.
Coloring, {2}
R:
[3, 1, 4, 3, 2, 5]
B:
[6, 4, 2, 6, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
5 vs 5 |
5 vs 5 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[4, 2, 8, 6, 4, 0]
,
[2, 4, 10, 8, 0, 0]
,
[4, 0, 10, 10, 0, 0]
,
[0, 0, 14, 10, 0, 0]
,
[0, 0, 10, 14, 0, 0]
] $
[y1, y2, y3, y4, y5, 0]
Omega Rank for B :
cycles:
{{2, 3, 4, 6}}
order:
4
See Matrix
$ [
[2, 6, 4, 4, 0, 8]
,
[0, 4, 8, 6, 0, 6]
,
[0, 8, 6, 4, 0, 6]
,
[0, 6, 6, 8, 0, 4]
,
[0, 6, 4, 6, 0, 8]
] $
[y3, y2, y1, y4, 0, y5]
3
.
Coloring, {3}
R:
[3, 4, 2, 3, 2, 5]
B:
[6, 1, 4, 6, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{2, 3, 4}}
order:
3
See Matrix
$ [
[0, 8, 8, 4, 4, 0]
,
[0, 12, 4, 8, 0, 0]
,
[0, 4, 8, 12, 0, 0]
,
[0, 8, 12, 4, 0, 0]
] $
[0, y1, y2, y3, y4, 0]
Omega Rank for B :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[6, 0, 4, 6, 0, 8]
,
[0, 0, 8, 4, 0, 12]
,
[0, 0, 12, 8, 0, 4]
,
[0, 0, 4, 12, 0, 8]
] $
[y1, 0, y2, y4, 0, y3]
4
.
Coloring, {4}
R:
[3, 4, 4, 6, 2, 5]
B:
[6, 1, 2, 3, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{2, 4, 5, 6}}
order:
4
See Matrix
$ [
[0, 2, 3, 10, 4, 5]
,
[0, 4, 0, 5, 5, 10]
,
[0, 5, 0, 4, 10, 5]
,
[0, 10, 0, 5, 5, 4]
,
[0, 5, 0, 10, 4, 5]
] $
[0, y5, y4, y3, y1, y2]
Omega Rank for B :
cycles:
{{1, 2, 3, 6}}
order:
4
See Matrix
$ [
[6, 6, 9, 0, 0, 3]
,
[6, 9, 3, 0, 0, 6]
,
[9, 3, 6, 0, 0, 6]
,
[3, 6, 6, 0, 0, 9]
] $
[y1, y2, y3, 0, 0, y4]
5
.
Coloring, {5}
Ωp(Δ)=0:
p =
s 3 - 2s 4
R:
[3, 4, 4, 3, 1, 5]
B:
[6, 1, 2, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 4 |
3 vs 5 |
3 vs 5 |
3 vs 4 |
2 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[2, 0, 8, 10, 4, 0]
,
[4, 0, 12, 8, 0, 0]
,
[0, 0, 12, 12, 0, 0]
,
[0, 0, 12, 12, 0, 0]
] $
[y1, 0, y1 + y3 - y2, y3, y2, 0]
p =
s 3 - s 4
Omega Rank for B :
cycles:
{{1, 2, 3, 6}}
order:
4
See Matrix
$ [
[4, 8, 4, 0, 0, 8]
,
[8, 4, 8, 0, 0, 4]
,
[4, 8, 4, 0, 0, 8]
,
[8, 4, 8, 0, 0, 4]
] $
[y1, y2, y1, 0, 0, y2]
p' =
- s + s 3
p =
- s + s 3
M
N
$ [
[0, 0, 3, 0, 0, 0]
,
[0, 0, 0, 0, 0, 4]
,
[3, 0, 0, 3, 0, 0]
,
[0, 0, 3, 0, 2, 0]
,
[0, 0, 0, 2, 0, 0]
,
[0, 4, 0, 0, 0, 0]
] $
$ [
[0, 2, 3, 0, 3, 1]
,
[2, 0, 1, 2, 1, 3]
,
[3, 1, 0, 3, 0, 2]
,
[0, 2, 3, 0, 3, 1]
,
[3, 1, 0, 3, 0, 2]
,
[1, 3, 2, 1, 2, 0]
] $
R:
[3, 4, 4, 3, 1, 5]
B:
[6, 1, 2, 6, 2, 3]
Ranges
Action of R on ranges, [[3], [4], [3], [1]]
Action of B on ranges, [[2], [1], [2], [2]]
Cycles:
R , {{3, 4}}, B , {{1, 2, 3, 6}}
β({1, 3})
=
1/4
β({2, 6})
=
1/3
β({3, 4})
=
1/4
β({4, 5})
=
1/6
Partitions
Action of R on partitions, [[1], [1]]
Action of B on partitions, [[2], [1]]
α([{1, 4, 6}, {2, 3, 5}]) = 2/3
α([{1, 2, 4}, {3, 5, 6}]) = 1/3
b1 = {1, 2, 4}
` , ` b2 = {1, 4, 6}
` , ` b3 = {2, 3, 5}
` , ` b4 = {3, 5, 6}
Action of R and B on the blocks of the partitions:
=
[3, 3, 2, 2]
[3, 1, 4, 2]
with invariant measure
[1, 2, 2, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-2 partition graph. `
| Sandwich |
|---|
| Coloring |
{5}
|
|---|
| Rank | 2 |
|---|
| R,B |
[3, 4, 4, 3, 1, 5], [6, 1, 2, 6, 2, 3]
|
|---|
| π2 |
[0, 3, 0, 0, 0, 0, 0, 0, 4, 3, 0, 0, 2, 0, 0]
|
|---|
| u2 |
[2, 3, 0, 3, 1, 1, 2, 1, 3, 3, 0, 2, 3, 1, 2]
(dim 1) |
|---|
| wpp |
[3, 3, 3, 3, 3, 3]
|
|---|
6
.
Coloring, {6}
R:
[3, 4, 4, 3, 2, 3]
B:
[6, 1, 2, 6, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
4 vs 5 |
5 vs 5 |
3 vs 3 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[0, 2, 12, 10, 0, 0]
,
[0, 0, 10, 14, 0, 0]
,
[0, 0, 14, 10, 0, 0]
] $
[0, y3, y2, y1, 0, 0]
Omega Rank for B :
cycles:
{{1, 5, 6}}
order:
3
See Matrix
$ [
[6, 6, 0, 0, 4, 8]
,
[10, 0, 0, 0, 8, 6]
,
[8, 0, 0, 0, 6, 10]
,
[6, 0, 0, 0, 10, 8]
] $
[y1, y2, 0, 0, y3, y4]
7
.
Coloring, {2, 3}
R:
[3, 1, 2, 3, 2, 5]
B:
[6, 4, 4, 6, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[4, 8, 8, 0, 4, 0]
,
[8, 12, 4, 0, 0, 0]
,
[12, 4, 8, 0, 0, 0]
,
[4, 8, 12, 0, 0, 0]
] $
[y1, y2, y4, 0, y3, 0]
Omega Rank for B :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[2, 0, 4, 10, 0, 8]
,
[0, 0, 8, 4, 0, 12]
,
[0, 0, 12, 8, 0, 4]
,
[0, 0, 4, 12, 0, 8]
] $
[y1, 0, y2, y3, 0, y4]
8
.
Coloring, {2, 4}
R:
[3, 1, 4, 6, 2, 5]
B:
[6, 4, 2, 3, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 2, 3, 4, 5, 6}}
order:
6
See Matrix
$ [
[4, 2, 3, 6, 4, 5]
,
[2, 4, 4, 3, 5, 6]
,
[4, 5, 2, 4, 6, 3]
,
[5, 6, 4, 2, 3, 4]
,
[6, 3, 5, 4, 4, 2]
,
[3, 4, 6, 5, 2, 4]
] $
[y1, y2, y3, y4, y5, y6]
Omega Rank for B :
cycles:
{{2, 3, 4}}
order:
3
See Matrix
$ [
[2, 6, 9, 4, 0, 3]
,
[0, 9, 7, 6, 0, 2]
,
[0, 7, 8, 9, 0, 0]
,
[0, 8, 9, 7, 0, 0]
,
[0, 9, 7, 8, 0, 0]
] $
[y3, y4, y2, y1, 0, y5]
9
.
Coloring, {2, 5}
Ωp(Δ)=0:
p =
s 3 - 2s 4
R:
[3, 1, 4, 3, 1, 5]
B:
[6, 4, 2, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 4 |
3 vs 5 |
3 vs 5 |
3 vs 4 |
2 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[6, 0, 8, 6, 4, 0]
,
[4, 0, 12, 8, 0, 0]
,
[0, 0, 12, 12, 0, 0]
,
[0, 0, 12, 12, 0, 0]
] $
[y1 - y2 + y3, 0, y1, y2, y3, 0]
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{2, 3, 4, 6}}
order:
4
See Matrix
$ [
[0, 8, 4, 4, 0, 8]
,
[0, 4, 8, 8, 0, 4]
,
[0, 8, 4, 4, 0, 8]
,
[0, 4, 8, 8, 0, 4]
] $
[0, y2, y1, y1, 0, y2]
p =
- s + s 3
p' =
- s + s 3
M
N
$ [
[0, 0, 1, 0, 2, 0]
,
[0, 0, 0, 0, 0, 4]
,
[1, 0, 0, 5, 0, 0]
,
[0, 0, 5, 0, 0, 0]
,
[2, 0, 0, 0, 0, 0]
,
[0, 4, 0, 0, 0, 0]
] $
$ [
[0, 2, 3, 0, 3, 1]
,
[2, 0, 1, 2, 1, 3]
,
[3, 1, 0, 3, 0, 2]
,
[0, 2, 3, 0, 3, 1]
,
[3, 1, 0, 3, 0, 2]
,
[1, 3, 2, 1, 2, 0]
] $
R:
[3, 1, 4, 3, 1, 5]
B:
[6, 4, 2, 6, 2, 3]
Ranges
Action of R on ranges, [[4], [1], [2], [4]]
Action of B on ranges, [[3], [3], [4], [3]]
Cycles:
R , {{3, 4}}, B , {{2, 3, 4, 6}}
β({1, 3})
=
1/12
β({1, 5})
=
1/6
β({2, 6})
=
1/3
β({3, 4})
=
5/12
Partitions
Action of R on partitions, [[1], [1]]
Action of B on partitions, [[2], [1]]
α([{1, 4, 6}, {2, 3, 5}]) = 2/3
α([{1, 2, 4}, {3, 5, 6}]) = 1/3
b1 = {1, 2, 4}
` , ` b2 = {1, 4, 6}
` , ` b3 = {2, 3, 5}
` , ` b4 = {3, 5, 6}
Action of R and B on the blocks of the partitions:
=
[3, 3, 2, 2]
[3, 1, 4, 2]
with invariant measure
[1, 2, 2, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-2 partition graph. `
| Sandwich |
|---|
| Coloring |
{2, 5}
|
|---|
| Rank | 2 |
|---|
| R,B |
[3, 1, 4, 3, 1, 5], [6, 4, 2, 6, 2, 3]
|
|---|
| π2 |
[0, 1, 0, 2, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 0]
|
|---|
| u2 |
[2, 3, 0, 3, 1, 1, 2, 1, 3, 3, 0, 2, 3, 1, 2]
(dim 1) |
|---|
| wpp |
[3, 3, 3, 3, 3, 3]
|
|---|
10
.
Coloring, {2, 6}
R:
[3, 1, 4, 3, 2, 3]
B:
[6, 4, 2, 6, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
4 vs 5 |
5 vs 5 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[4, 2, 12, 6, 0, 0]
,
[2, 0, 10, 12, 0, 0]
,
[0, 0, 14, 10, 0, 0]
,
[0, 0, 10, 14, 0, 0]
] $
[y3, y1, y2, y4, 0, 0]
Omega Rank for B :
cycles:
{{1, 5, 6}}
order:
3
See Matrix
$ [
[2, 6, 0, 4, 4, 8]
,
[4, 0, 0, 6, 8, 6]
,
[8, 0, 0, 0, 6, 10]
,
[6, 0, 0, 0, 10, 8]
,
[10, 0, 0, 0, 8, 6]
] $
[y3, y1, 0, y2, y4, y5]
11
.
Coloring, {3, 4}
R:
[3, 4, 2, 6, 2, 5]
B:
[6, 1, 4, 3, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{2, 4, 5, 6}}
order:
4
See Matrix
$ [
[0, 8, 3, 4, 4, 5]
,
[0, 7, 0, 8, 5, 4]
,
[0, 5, 0, 7, 4, 8]
,
[0, 4, 0, 5, 8, 7]
,
[0, 8, 0, 4, 7, 5]
] $
[0, y1, y2, y4, y3, y5]
Omega Rank for B :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[6, 0, 9, 6, 0, 3]
,
[0, 0, 9, 9, 0, 6]
,
[0, 0, 15, 9, 0, 0]
,
[0, 0, 9, 15, 0, 0]
] $
[y1, 0, y2, y3, 0, y4]
12
.
Coloring, {3, 5}
R:
[3, 4, 2, 3, 1, 5]
B:
[6, 1, 4, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
4 vs 5 |
5 vs 5 |
5 vs 5 |
Omega Rank for R :
cycles:
{{2, 3, 4}}
order:
3
See Matrix
$ [
[2, 6, 8, 4, 4, 0]
,
[4, 8, 6, 6, 0, 0]
,
[0, 6, 10, 8, 0, 0]
,
[0, 10, 8, 6, 0, 0]
,
[0, 8, 6, 10, 0, 0]
] $
[y4, y2, y3, y1, y5, 0]
Omega Rank for B :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[4, 2, 4, 6, 0, 8]
,
[2, 0, 8, 4, 0, 10]
,
[0, 0, 10, 8, 0, 6]
,
[0, 0, 6, 10, 0, 8]
,
[0, 0, 8, 6, 0, 10]
] $
[y1, y3, y4, y5, 0, y2]
13
.
Coloring, {3, 6}
R:
[3, 4, 2, 3, 2, 3]
B:
[6, 1, 4, 6, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
3 vs 3 |
4 vs 4 |
Omega Rank for R :
cycles:
{{2, 3, 4}}
order:
3
See Matrix
$ [
[0, 8, 12, 4, 0, 0]
,
[0, 12, 4, 8, 0, 0]
,
[0, 4, 8, 12, 0, 0]
] $
[0, y2, y3, y1, 0, 0]
Omega Rank for B :
cycles:
{{1, 5, 6}}
order:
3
See Matrix
$ [
[6, 0, 0, 6, 4, 8]
,
[4, 0, 0, 0, 8, 12]
,
[8, 0, 0, 0, 12, 4]
,
[12, 0, 0, 0, 4, 8]
] $
[y1, 0, 0, y4, y3, y2]
14
.
Coloring, {4, 5}
R:
[3, 4, 4, 6, 1, 5]
B:
[6, 1, 2, 3, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[2, 0, 3, 10, 4, 5]
,
[4, 0, 2, 3, 5, 10]
,
[5, 0, 4, 2, 10, 3]
,
[10, 0, 5, 4, 3, 2]
,
[3, 0, 10, 5, 2, 4]
] $
[y4, 0, y5, y1, y2, y3]
Omega Rank for B :
cycles:
{{1, 2, 3, 6}}
order:
4
See Matrix
$ [
[4, 8, 9, 0, 0, 3]
,
[8, 9, 3, 0, 0, 4]
,
[9, 3, 4, 0, 0, 8]
,
[3, 4, 8, 0, 0, 9]
] $
[y2, y1, y4, 0, 0, y3]
15
.
Coloring, {4, 6}
R:
[3, 4, 4, 6, 2, 3]
B:
[6, 1, 2, 3, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 6 |
6 vs 6 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[0, 2, 7, 10, 0, 5]
,
[0, 0, 5, 9, 0, 10]
,
[0, 0, 10, 5, 0, 9]
,
[0, 0, 9, 10, 0, 5]
] $
[0, y2, y3, y4, 0, y1]
Omega Rank for B :
cycles:
{{1, 5, 6}}
order:
3
See Matrix
$ [
[6, 6, 5, 0, 4, 3]
,
[10, 5, 0, 0, 3, 6]
,
[8, 0, 0, 0, 6, 10]
,
[6, 0, 0, 0, 10, 8]
,
[10, 0, 0, 0, 8, 6]
] $
[y1, y2, y5, 0, y4, y3]
16
.
Coloring, {5, 6}
Ωp(Δ)=0:
p =
s 3 - 2s 4
R:
[3, 4, 4, 3, 1, 3]
B:
[6, 1, 2, 6, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 4 |
3 vs 5 |
3 vs 5 |
2 vs 3 |
2 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[2, 0, 12, 10, 0, 0]
,
[0, 0, 12, 12, 0, 0]
,
[0, 0, 12, 12, 0, 0]
] $
[y2, 0, y1, -y2 + y1, 0, 0]
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{1, 2, 5, 6}}
order:
4
See Matrix
$ [
[4, 8, 0, 0, 4, 8]
,
[8, 4, 0, 0, 8, 4]
,
[4, 8, 0, 0, 4, 8]
,
[8, 4, 0, 0, 8, 4]
] $
[y2, y1, 0, 0, y2, y1]
p' =
s - s 3
p =
s - s 3
M
N
$ [
[0, 0, 1, 0, 2, 0]
,
[0, 0, 0, 0, 0, 4]
,
[1, 0, 0, 5, 0, 0]
,
[0, 0, 5, 0, 0, 0]
,
[2, 0, 0, 0, 0, 0]
,
[0, 4, 0, 0, 0, 0]
] $
$ [
[0, 2, 3, 0, 3, 1]
,
[2, 0, 1, 2, 1, 3]
,
[3, 1, 0, 3, 0, 2]
,
[0, 2, 3, 0, 3, 1]
,
[3, 1, 0, 3, 0, 2]
,
[1, 3, 2, 1, 2, 0]
] $
R:
[3, 4, 4, 3, 1, 3]
B:
[6, 1, 2, 6, 2, 5]
Ranges
Action of R on ranges, [[4], [1], [4], [4]]
Action of B on ranges, [[3], [3], [2], [3]]
Cycles:
R , {{3, 4}}, B , {{1, 2, 5, 6}}
β({1, 3})
=
1/12
β({1, 5})
=
1/6
β({2, 6})
=
1/3
β({3, 4})
=
5/12
Partitions
Action of R on partitions, [[1], [1]]
Action of B on partitions, [[2], [1]]
α([{1, 4, 6}, {2, 3, 5}]) = 2/3
α([{1, 2, 4}, {3, 5, 6}]) = 1/3
b1 = {1, 2, 4}
` , ` b2 = {1, 4, 6}
` , ` b3 = {2, 3, 5}
` , ` b4 = {3, 5, 6}
Action of R and B on the blocks of the partitions:
=
[3, 3, 2, 2]
[3, 1, 4, 2]
with invariant measure
[1, 2, 2, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-2 partition graph. `
| Sandwich |
|---|
| Coloring |
{5, 6}
|
|---|
| Rank | 2 |
|---|
| R,B |
[3, 4, 4, 3, 1, 3], [6, 1, 2, 6, 2, 5]
|
|---|
| π2 |
[0, 1, 0, 2, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 0]
|
|---|
| u2 |
[2, 3, 0, 3, 1, 1, 2, 1, 3, 3, 0, 2, 3, 1, 2]
(dim 1) |
|---|
| wpp |
[3, 3, 3, 3, 3, 3]
|
|---|
17
.
Coloring, {2, 3, 4}
R:
[3, 1, 2, 6, 2, 5]
B:
[6, 4, 4, 3, 1, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[4, 8, 3, 0, 4, 5]
,
[8, 7, 4, 0, 5, 0]
,
[7, 9, 8, 0, 0, 0]
,
[9, 8, 7, 0, 0, 0]
,
[8, 7, 9, 0, 0, 0]
] $
[y1, y3, y2, 0, y5, y4]
Omega Rank for B :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[2, 0, 9, 10, 0, 3]
,
[0, 0, 13, 9, 0, 2]
,
[0, 0, 11, 13, 0, 0]
,
[0, 0, 13, 11, 0, 0]
] $
[y1, 0, y3, y2, 0, y4]
18
.
Coloring, {2, 3, 5}
R:
[3, 1, 2, 3, 1, 5]
B:
[6, 4, 4, 6, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
4 vs 5 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[6, 6, 8, 0, 4, 0]
,
[10, 8, 6, 0, 0, 0]
,
[8, 6, 10, 0, 0, 0]
,
[6, 10, 8, 0, 0, 0]
] $
[y4, y2, y3, 0, y1, 0]
Omega Rank for B :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[0, 2, 4, 10, 0, 8]
,
[0, 0, 8, 6, 0, 10]
,
[0, 0, 10, 8, 0, 6]
,
[0, 0, 6, 10, 0, 8]
] $
[0, y2, y1, y3, 0, y4]
19
.
Coloring, {2, 3, 6}
R:
[3, 1, 2, 3, 2, 3]
B:
[6, 4, 4, 6, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
3 vs 3 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[4, 8, 12, 0, 0, 0]
,
[8, 12, 4, 0, 0, 0]
,
[12, 4, 8, 0, 0, 0]
] $
[y1, y2, y3, 0, 0, 0]
Omega Rank for B :
cycles:
{{1, 5, 6}}
order:
3
See Matrix
$ [
[2, 0, 0, 10, 4, 8]
,
[4, 0, 0, 0, 8, 12]
,
[8, 0, 0, 0, 12, 4]
,
[12, 0, 0, 0, 4, 8]
] $
[y3, 0, 0, y2, y1, y4]
20
.
Coloring, {2, 4, 5}
R:
[3, 1, 4, 6, 1, 5]
B:
[6, 4, 2, 3, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[6, 0, 3, 6, 4, 5]
,
[4, 0, 6, 3, 5, 6]
,
[5, 0, 4, 6, 6, 3]
,
[6, 0, 5, 4, 3, 6]
,
[3, 0, 6, 5, 6, 4]
] $
[y1, 0, y3, y4, y5, y2]
Omega Rank for B :
cycles:
{{2, 3, 4}}
order:
3
See Matrix
$ [
[0, 8, 9, 4, 0, 3]
,
[0, 9, 7, 8, 0, 0]
,
[0, 7, 8, 9, 0, 0]
,
[0, 8, 9, 7, 0, 0]
] $
[0, y4, y2, y3, 0, y1]
21
.
Coloring, {2, 4, 6}
R:
[3, 1, 4, 6, 2, 3]
B:
[6, 4, 2, 3, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 6 |
6 vs 6 |
5 vs 5 |
3 vs 6 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[4, 2, 7, 6, 0, 5]
,
[2, 0, 9, 7, 0, 6]
,
[0, 0, 8, 9, 0, 7]
,
[0, 0, 7, 8, 0, 9]
,
[0, 0, 9, 7, 0, 8]
] $
[y1, y2, y3, y4, 0, y5]
Omega Rank for B :
cycles:
{{1, 5, 6}, {2, 3, 4}}
order:
3
See Matrix
$ [
[2, 6, 5, 4, 4, 3]
,
[4, 5, 4, 6, 3, 2]
,
[3, 4, 6, 5, 2, 4]
,
[2, 6, 5, 4, 4, 3]
,
[4, 5, 4, 6, 3, 2]
,
[3, 4, 6, 5, 2, 4]
] $
[9 y1 - 2 y2 - 11 y3, 2 y1 + 2 y2 - 2 y3, 2 y1,
11 y1 - 2 y2 - 13 y3, 2 y2, 2 y3]
p' =
- s 2 + s 5
p' =
- s + s 4
p' =
- 1 + s 3
22
.
Coloring, {2, 5, 6}
Ωp(Δ)=0:
p =
s 3 - 2s 4
R:
[3, 1, 4, 3, 1, 3]
B:
[6, 4, 2, 6, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 4 |
3 vs 5 |
3 vs 5 |
2 vs 3 |
2 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[6, 0, 12, 6, 0, 0]
,
[0, 0, 12, 12, 0, 0]
,
[0, 0, 12, 12, 0, 0]
] $
[y2, 0, y2 + y1, y1, 0, 0]
p =
s 2 - s 3
Omega Rank for B :
cycles:
{{2, 4, 5, 6}}
order:
4
See Matrix
$ [
[0, 8, 0, 4, 4, 8]
,
[0, 4, 0, 8, 8, 4]
,
[0, 8, 0, 4, 4, 8]
,
[0, 4, 0, 8, 8, 4]
] $
[0, y2, 0, y1, y1, y2]
p' =
s - s 3
p =
s - s 3
M
N
$ [
[0, 0, 3, 0, 0, 0]
,
[0, 0, 0, 0, 0, 4]
,
[3, 0, 0, 3, 0, 0]
,
[0, 0, 3, 0, 2, 0]
,
[0, 0, 0, 2, 0, 0]
,
[0, 4, 0, 0, 0, 0]
] $
$ [
[0, 2, 3, 0, 3, 1]
,
[2, 0, 1, 2, 1, 3]
,
[3, 1, 0, 3, 0, 2]
,
[0, 2, 3, 0, 3, 1]
,
[3, 1, 0, 3, 0, 2]
,
[1, 3, 2, 1, 2, 0]
] $
R:
[3, 1, 4, 3, 1, 3]
B:
[6, 4, 2, 6, 2, 5]
Ranges
Action of R on ranges, [[3], [1], [3], [1]]
Action of B on ranges, [[2], [4], [2], [2]]
Cycles:
R , {{3, 4}}, B , {{2, 4, 5, 6}}
β({1, 3})
=
1/4
β({2, 6})
=
1/3
β({3, 4})
=
1/4
β({4, 5})
=
1/6
Partitions
Action of R on partitions, [[1], [1]]
Action of B on partitions, [[2], [1]]
α([{1, 4, 6}, {2, 3, 5}]) = 2/3
α([{1, 2, 4}, {3, 5, 6}]) = 1/3
b1 = {1, 2, 4}
` , ` b2 = {1, 4, 6}
` , ` b3 = {2, 3, 5}
` , ` b4 = {3, 5, 6}
Action of R and B on the blocks of the partitions:
=
[3, 3, 2, 2]
[3, 1, 4, 2]
with invariant measure
[1, 2, 2, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-2 partition graph. `
| Sandwich |
|---|
| Coloring |
{2, 5, 6}
|
|---|
| Rank | 2 |
|---|
| R,B |
[3, 1, 4, 3, 1, 3], [6, 4, 2, 6, 2, 5]
|
|---|
| π2 |
[0, 3, 0, 0, 0, 0, 0, 0, 4, 3, 0, 0, 2, 0, 0]
|
|---|
| u2 |
[2, 3, 0, 3, 1, 1, 2, 1, 3, 3, 0, 2, 3, 1, 2]
(dim 1) |
|---|
| wpp |
[3, 3, 3, 3, 3, 3]
|
|---|
23
.
Coloring, {3, 4, 5}
R:
[3, 4, 2, 6, 1, 5]
B:
[6, 1, 4, 3, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
5 vs 6 |
6 vs 6 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 2, 3, 4, 5, 6}}
order:
6
See Matrix
$ [
[2, 6, 3, 4, 4, 5]
,
[4, 3, 2, 6, 5, 4]
,
[5, 2, 4, 3, 4, 6]
,
[4, 4, 5, 2, 6, 3]
,
[6, 5, 4, 4, 3, 2]
,
[3, 4, 6, 5, 2, 4]
] $
[y1, y4, y5, y6, y2, y3]
Omega Rank for B :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[4, 2, 9, 6, 0, 3]
,
[2, 0, 9, 9, 0, 4]
,
[0, 0, 13, 9, 0, 2]
,
[0, 0, 11, 13, 0, 0]
,
[0, 0, 13, 11, 0, 0]
] $
[y3, y2, y1, y5, 0, y4]
24
.
Coloring, {3, 4, 6}
R:
[3, 4, 2, 6, 2, 3]
B:
[6, 1, 4, 3, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
4 vs 5 |
Omega Rank for R :
cycles:
{{2, 3, 4, 6}}
order:
4
See Matrix
$ [
[0, 8, 7, 4, 0, 5]
,
[0, 7, 5, 8, 0, 4]
,
[0, 5, 4, 7, 0, 8]
,
[0, 4, 8, 5, 0, 7]
] $
[0, y1, y4, y2, 0, y3]
Omega Rank for B :
cycles:
{{3, 4}, {1, 5, 6}}
order:
6
See Matrix
$ [
[6, 0, 5, 6, 4, 3]
,
[4, 0, 6, 5, 3, 6]
,
[3, 0, 5, 6, 6, 4]
,
[6, 0, 6, 5, 4, 3]
,
[4, 0, 5, 6, 3, 6]
] $
[11 y3, 0, 11 y2, 11 y1, -11 y3 + 13 y2 + 13 y1 - 11 y4, 11 y4]
p =
s + s 2 - s 4 - s 5
25
.
Coloring, {3, 5, 6}
R:
[3, 4, 2, 3, 1, 3]
B:
[6, 1, 4, 6, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{2, 3, 4}}
order:
3
See Matrix
$ [
[2, 6, 12, 4, 0, 0]
,
[0, 12, 6, 6, 0, 0]
,
[0, 6, 6, 12, 0, 0]
,
[0, 6, 12, 6, 0, 0]
] $
[y1, y2, y3, y4, 0, 0]
Omega Rank for B :
cycles:
{{1, 2, 5, 6}}
order:
4
See Matrix
$ [
[4, 2, 0, 6, 4, 8]
,
[2, 4, 0, 0, 8, 10]
,
[4, 8, 0, 0, 10, 2]
,
[8, 10, 0, 0, 2, 4]
,
[10, 2, 0, 0, 4, 8]
] $
[y2, y1, 0, y5, y4, y3]
26
.
Coloring, {4, 5, 6}
R:
[3, 4, 4, 6, 1, 3]
B:
[6, 1, 2, 3, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[2, 0, 7, 10, 0, 5]
,
[0, 0, 7, 7, 0, 10]
,
[0, 0, 10, 7, 0, 7]
,
[0, 0, 7, 10, 0, 7]
] $
[y3, 0, y4, y1, 0, y2]
Omega Rank for B :
cycles:
{{1, 2, 5, 6}}
order:
4
See Matrix
$ [
[4, 8, 5, 0, 4, 3]
,
[8, 9, 0, 0, 3, 4]
,
[9, 3, 0, 0, 4, 8]
,
[3, 4, 0, 0, 8, 9]
,
[4, 8, 0, 0, 9, 3]
] $
[y4, y1, y2, 0, y3, y5]
27
.
Coloring, {2, 3, 4, 5}
R:
[3, 1, 2, 6, 1, 5]
B:
[6, 4, 4, 3, 2, 3]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
5 vs 6 |
5 vs 5 |
3 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[6, 6, 3, 0, 4, 5]
,
[10, 3, 6, 0, 5, 0]
,
[8, 6, 10, 0, 0, 0]
,
[6, 10, 8, 0, 0, 0]
,
[10, 8, 6, 0, 0, 0]
] $
[y2, y1, y4, 0, y5, y3]
Omega Rank for B :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[0, 2, 9, 10, 0, 3]
,
[0, 0, 13, 11, 0, 0]
,
[0, 0, 11, 13, 0, 0]
,
[0, 0, 13, 11, 0, 0]
] $
[0, 2 y1, 2 y2, 2 y3, 0, 3 y1]
p =
- s 2 + s 4
28
.
Coloring, {2, 3, 4, 6}
R:
[3, 1, 2, 6, 2, 3]
B:
[6, 4, 4, 3, 1, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
4 vs 5 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[4, 8, 7, 0, 0, 5]
,
[8, 7, 9, 0, 0, 0]
,
[7, 9, 8, 0, 0, 0]
,
[9, 8, 7, 0, 0, 0]
] $
[y3, y2, y1, 0, 0, y4]
Omega Rank for B :
cycles:
{{3, 4}, {1, 5, 6}}
order:
6
See Matrix
$ [
[2, 0, 5, 10, 4, 3]
,
[4, 0, 10, 5, 3, 2]
,
[3, 0, 5, 10, 2, 4]
,
[2, 0, 10, 5, 4, 3]
,
[4, 0, 5, 10, 3, 2]
] $
[3 y3, 0, 3 y4, 5 y3 - 3 y4 + 5 y1 + 5 y2, 3 y1, 3 y2]
p =
- s - s 2 + s 4 + s 5
29
.
Coloring, {2, 3, 5, 6}
R:
[3, 1, 2, 3, 1, 3]
B:
[6, 4, 4, 6, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
5 vs 5 |
5 vs 5 |
3 vs 3 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[6, 6, 12, 0, 0, 0]
,
[6, 12, 6, 0, 0, 0]
,
[12, 6, 6, 0, 0, 0]
] $
[y2, y3, y1, 0, 0, 0]
Omega Rank for B :
cycles:
{{2, 4, 5, 6}}
order:
4
See Matrix
$ [
[0, 2, 0, 10, 4, 8]
,
[0, 4, 0, 2, 8, 10]
,
[0, 8, 0, 4, 10, 2]
,
[0, 10, 0, 8, 2, 4]
] $
[0, y1, 0, y2, y3, y4]
30
.
Coloring, {2, 4, 5, 6}
R:
[3, 1, 4, 6, 1, 3]
B:
[6, 4, 2, 3, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{3, 4, 6}}
order:
3
See Matrix
$ [
[6, 0, 7, 6, 0, 5]
,
[0, 0, 11, 7, 0, 6]
,
[0, 0, 6, 11, 0, 7]
,
[0, 0, 7, 6, 0, 11]
] $
[y4, 0, y3, y2, 0, y1]
Omega Rank for B :
cycles:
{{2, 3, 4}}
order:
3
See Matrix
$ [
[0, 8, 5, 4, 4, 3]
,
[0, 9, 4, 8, 3, 0]
,
[0, 7, 8, 9, 0, 0]
,
[0, 8, 9, 7, 0, 0]
,
[0, 9, 7, 8, 0, 0]
] $
[0, y3, y4, y5, y2, y1]
31
.
Coloring, {3, 4, 5, 6}
R:
[3, 4, 2, 6, 1, 3]
B:
[6, 1, 4, 3, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
6 vs 6 |
5 vs 5 |
4 vs 6 |
Omega Rank for R :
cycles:
{{2, 3, 4, 6}}
order:
4
See Matrix
$ [
[2, 6, 7, 4, 0, 5]
,
[0, 7, 7, 6, 0, 4]
,
[0, 7, 4, 7, 0, 6]
,
[0, 4, 6, 7, 0, 7]
,
[0, 6, 7, 4, 0, 7]
] $
[y1, y2, y3, y4, 0, y5]
Omega Rank for B :
cycles:
{{1, 2, 5, 6}, {3, 4}}
order:
4
See Matrix
$ [
[4, 2, 5, 6, 4, 3]
,
[2, 4, 6, 5, 3, 4]
,
[4, 3, 5, 6, 4, 2]
,
[3, 4, 6, 5, 2, 4]
,
[4, 2, 5, 6, 4, 3]
,
[2, 4, 6, 5, 3, 4]
] $
[-23 y2 + 39 y1 - 10 y3 - 23 y4, 10 y2, 10 y1,
-11 y2 + 23 y1 - 11 y4, 10 y3, 10 y4]
p' =
- s + s 5
p' =
- 1 + s 4
32
.
Coloring, {2, 3, 4, 5, 6}
R:
[3, 1, 2, 6, 1, 3]
B:
[6, 4, 4, 3, 2, 5]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 4 vs 4 |
6 vs 6 |
5 vs 6 |
4 vs 4 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 2, 3}}
order:
3
See Matrix
$ [
[6, 6, 7, 0, 0, 5]
,
[6, 7, 11, 0, 0, 0]
,
[7, 11, 6, 0, 0, 0]
,
[11, 6, 7, 0, 0, 0]
] $
[y1, y2, y3, 0, 0, y4]
Omega Rank for B :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[0, 2, 5, 10, 4, 3]
,
[0, 4, 10, 7, 3, 0]
,
[0, 3, 7, 14, 0, 0]
,
[0, 0, 14, 10, 0, 0]
,
[0, 0, 10, 14, 0, 0]
] $
[0, y1, y2, y3, y4, y5]
| SUMMARY |
|---|
| Graph Type |
| NOT CC |
|---|
| ν(A) |
|
1
|
|---|
| ν(Δ) |
|
2
|
|---|
| π |
|
[3, 4, 6, 5, 2, 4]
|
|---|
| Dbly Stoch |
| false |
|---|
| 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 |
|---|
| 28 |
0 |
24 , 23 |
28 , 23 |
4 |
32 |
32 |