New Graph
[5, 5, 1, 5, 2], [3, 4, 5, 2, 1]
π =
[2, 2, 1, 1, 3]
POSSIBLE RANKS
1 x 9
3 x 3
BASE DETERMINANT
351/4096, .8569335938e-1
NullSpace of Δ
{1, 2, 3, 4, 5}
Nullspace of A
` det(A) = ` -1/16
1
.
Coloring, {}
R:
[5, 5, 1, 5, 2]
B:
[3, 4, 5, 2, 1]
` 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 5 |
Omega Rank for R :
cycles:
{{2, 5}}
order:
2
See Matrix
$ [
[1, 3, 0, 0, 5]
,
[0, 5, 0, 0, 4]
,
[0, 4, 0, 0, 5]
] $
[y1, y2, 0, 0, y3]
Omega Rank for B :
cycles:
{{1, 3, 5}, {2, 4}}
See Matrix
$ [
[3, 1, 2, 2, 1]
,
[1, 2, 3, 1, 2]
,
[2, 1, 1, 2, 3]
,
[3, 2, 2, 1, 1]
,
[1, 1, 3, 2, 2]
] $
[y4, y3, y2, y1, -y4 + 2 y3 - y2 + 2 y1]
p' =
- 1 - s + s 3 + s 4
2
.
Coloring, {2}
Ωp(Δ)=0:
p =
s + 4s 3
p' =
s + 4s 3
R:
[5, 4, 1, 5, 2]
B:
[3, 5, 5, 2, 1]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 2 vs 4 |
3 vs 5 |
3 vs 5 |
2 vs 4 |
2 vs 4 |
Omega Rank for R :
cycles:
{{2, 4, 5}}
order:
3
See Matrix
$ [
[1, 3, 0, 2, 3]
,
[0, 3, 0, 3, 3]
,
[0, 3, 0, 3, 3]
,
[0, 3, 0, 3, 3]
] $
[y2 - y1, y2, 0, y1, y2]
p =
- s 2 + s 3
p =
- s 2 + s 4
Omega Rank for B :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[3, 1, 2, 0, 3]
,
[3, 0, 3, 0, 3]
,
[3, 0, 3, 0, 3]
,
[3, 0, 3, 0, 3]
] $
[y2, y1, y2 - y1, 0, y2]
p' =
s 2 - s 3
p =
s 2 - s 4
` See 3-level graph `
M
N
$ [
[0, 1, 1, 0, 2]
,
[1, 0, 0, 1, 2]
,
[1, 0, 0, 0, 1]
,
[0, 1, 0, 0, 1]
,
[2, 2, 1, 1, 0]
] $
$ [
[0, 1, 1, 0, 1]
,
[1, 0, 0, 1, 1]
,
[1, 0, 0, 1, 1]
,
[0, 1, 1, 0, 1]
,
[1, 1, 1, 1, 0]
] $
R:
[5, 4, 1, 5, 2]
B:
[3, 5, 5, 2, 1]
Ranges
Action of R on ranges, [[3], [1], [3]]
Action of B on ranges, [[2], [2], [1]]
Cycles:
R , {{2, 4, 5}}, B , {{1, 3, 5}}
β({1, 2, 5})
=
1/3
β({1, 3, 5})
=
1/3
β({2, 4, 5})
=
1/3
Partitions
α([{5}, {1, 4}, {2, 3}]) = 1/1
b1 = {5}
` , ` b2 = {1, 4}
` , ` b3 = {2, 3}
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}
|
|---|
| Rank | 3 |
|---|
| R,B |
[5, 4, 1, 5, 2], [3, 5, 5, 2, 1]
|
|---|
| π2 |
[1, 1, 0, 2, 0, 1, 2, 0, 1, 1]
|
|---|
| u2 |
[1, 1, 0, 1, 0, 1, 1, 1, 1, 1]
(dim 1) |
|---|
| wpp |
[2, 2, 2, 2, 1]
|
|---|
| π3 |
[0, 0, 1, 0, 1, 0, 0, 0, 1, 0]
|
|---|
| u3 |
[0, 0, 1, 0, 1, 0, 0, 0, 1, 1]
|
|---|
3
.
Coloring, {3}
Ωp(Δ)=0:
p =
s + 2s 2 - 4s 3 - 8s 4
R:
[5, 5, 5, 5, 2]
B:
[3, 4, 1, 2, 1]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 4 |
4 vs 5 |
4 vs 5 |
2 vs 2 |
2 vs 4 |
Omega Rank for R :
cycles:
{{2, 5}}
order:
2
See Matrix
$ [
[0, 3, 0, 0, 6]
,
[0, 6, 0, 0, 3]
] $
[0, y1, 0, 0, y2]
Omega Rank for B :
cycles:
{{2, 4}, {1, 3}}
order:
2
See Matrix
$ [
[4, 1, 2, 2, 0]
,
[2, 2, 4, 1, 0]
,
[4, 1, 2, 2, 0]
,
[2, 2, 4, 1, 0]
] $
[2 y1, y2, 2 y2, y1, 0]
p' =
s - s 3
p =
s - s 3
4
.
Coloring, {4}
R:
[5, 5, 1, 2, 2]
B:
[3, 4, 5, 5, 1]
` 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, 5}}
order:
2
See Matrix
$ [
[1, 4, 0, 0, 4]
,
[0, 4, 0, 0, 5]
,
[0, 5, 0, 0, 4]
] $
[y2, y3, 0, 0, y1]
Omega Rank for B :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[3, 0, 2, 2, 2]
,
[2, 0, 3, 0, 4]
,
[4, 0, 2, 0, 3]
,
[3, 0, 4, 0, 2]
] $
[y4, 0, y3, y2, y1]
5
.
Coloring, {5}
R:
[5, 5, 1, 5, 1]
B:
[3, 4, 5, 2, 2]
` 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 |
2 vs 2 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 5}}
order:
2
See Matrix
$ [
[4, 0, 0, 0, 5]
,
[5, 0, 0, 0, 4]
] $
[y1, 0, 0, 0, y2]
Omega Rank for B :
cycles:
{{2, 4}}
order:
4
See Matrix
$ [
[0, 4, 2, 2, 1]
,
[0, 3, 0, 4, 2]
,
[0, 6, 0, 3, 0]
,
[0, 3, 0, 6, 0]
] $
[0, y1, y2, y3, y4]
6
.
Coloring, {2, 3}
R:
[5, 4, 5, 5, 2]
B:
[3, 5, 1, 2, 1]
` 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, 4, 5}}
order:
3
See Matrix
$ [
[0, 3, 0, 2, 4]
,
[0, 4, 0, 3, 2]
,
[0, 2, 0, 4, 3]
] $
[0, y3, 0, y1, y2]
Omega Rank for B :
cycles:
{{1, 3}}
order:
4
See Matrix
$ [
[4, 1, 2, 0, 2]
,
[4, 0, 4, 0, 1]
,
[5, 0, 4, 0, 0]
,
[4, 0, 5, 0, 0]
] $
[y1, y2, y3, 0, y4]
7
.
Coloring, {2, 4}
R:
[5, 4, 1, 2, 2]
B:
[3, 5, 5, 5, 1]
` 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 |
3 vs 3 |
Omega Rank for R :
cycles:
{{2, 4}}
order:
4
See Matrix
$ [
[1, 4, 0, 2, 2]
,
[0, 4, 0, 4, 1]
,
[0, 5, 0, 4, 0]
,
[0, 4, 0, 5, 0]
] $
[y1, y2, 0, y3, y4]
Omega Rank for B :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[3, 0, 2, 0, 4]
,
[4, 0, 3, 0, 2]
,
[2, 0, 4, 0, 3]
] $
[y1, 0, y2, 0, y3]
8
.
Coloring, {2, 5}
R:
[5, 4, 1, 5, 1]
B:
[3, 5, 5, 2, 2]
` 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 |
3 vs 3 |
Omega Rank for R :
cycles:
{{1, 5}}
order:
2
See Matrix
$ [
[4, 0, 0, 2, 3]
,
[3, 0, 0, 0, 6]
,
[6, 0, 0, 0, 3]
] $
[y3, 0, 0, y2, y1]
Omega Rank for B :
cycles:
{{2, 5}}
order:
2
See Matrix
$ [
[0, 4, 2, 0, 3]
,
[0, 3, 0, 0, 6]
,
[0, 6, 0, 0, 3]
] $
[0, y3, y1, 0, y2]
9
.
Coloring, {3, 4}
R:
[5, 5, 5, 2, 2]
B:
[3, 4, 1, 5, 1]
` 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 |
2 vs 2 |
4 vs 4 |
Omega Rank for R :
cycles:
{{2, 5}}
order:
2
See Matrix
$ [
[0, 4, 0, 0, 5]
,
[0, 5, 0, 0, 4]
] $
[0, y1, 0, 0, y2]
Omega Rank for B :
cycles:
{{1, 3}}
order:
4
See Matrix
$ [
[4, 0, 2, 2, 1]
,
[3, 0, 4, 0, 2]
,
[6, 0, 3, 0, 0]
,
[3, 0, 6, 0, 0]
] $
[y4, 0, y1, y2, y3]
10
.
Coloring, {3, 5}
Ωp(Δ)=0:
p =
s + 2s 2 - 4s 3 - 8s 4
R:
[5, 5, 5, 5, 1]
B:
[3, 4, 1, 2, 2]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 3 vs 4 |
4 vs 5 |
4 vs 5 |
2 vs 2 |
2 vs 4 |
Omega Rank for R :
cycles:
{{1, 5}}
order:
2
See Matrix
$ [
[3, 0, 0, 0, 6]
,
[6, 0, 0, 0, 3]
] $
[y1, 0, 0, 0, y2]
Omega Rank for B :
cycles:
{{1, 3}, {2, 4}}
order:
2
See Matrix
$ [
[1, 4, 2, 2, 0]
,
[2, 2, 1, 4, 0]
,
[1, 4, 2, 2, 0]
,
[2, 2, 1, 4, 0]
] $
[y2, 2 y1, y1, 2 y2, 0]
p =
- s + s 3
p' =
s - s 3
11
.
Coloring, {4, 5}
R:
[5, 5, 1, 2, 1]
B:
[3, 4, 5, 5, 2]
` 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, 5}}
order:
2
See Matrix
$ [
[4, 1, 0, 0, 4]
,
[4, 0, 0, 0, 5]
,
[5, 0, 0, 0, 4]
] $
[y2, y3, 0, 0, y1]
Omega Rank for B :
cycles:
{{2, 4, 5}}
order:
3
See Matrix
$ [
[0, 3, 2, 2, 2]
,
[0, 2, 0, 3, 4]
,
[0, 4, 0, 2, 3]
,
[0, 3, 0, 4, 2]
] $
[0, y3, y4, y2, y1]
12
.
Coloring, {2, 3, 4}
R:
[5, 4, 5, 2, 2]
B:
[3, 5, 1, 5, 1]
` 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 |
3 vs 3 |
Omega Rank for R :
cycles:
{{2, 4}}
order:
2
See Matrix
$ [
[0, 4, 0, 2, 3]
,
[0, 5, 0, 4, 0]
,
[0, 4, 0, 5, 0]
] $
[0, y3, 0, y2, y1]
Omega Rank for B :
cycles:
{{1, 3}}
order:
2
See Matrix
$ [
[4, 0, 2, 0, 3]
,
[5, 0, 4, 0, 0]
,
[4, 0, 5, 0, 0]
] $
[y1, 0, y2, 0, y3]
13
.
Coloring, {2, 3, 5}
R:
[5, 4, 5, 5, 1]
B:
[3, 5, 1, 2, 2]
` 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 |
2 vs 4 |
Omega Rank for R :
cycles:
{{1, 5}}
order:
2
See Matrix
$ [
[3, 0, 0, 2, 4]
,
[4, 0, 0, 0, 5]
,
[5, 0, 0, 0, 4]
] $
[y3, 0, 0, y1, y2]
Omega Rank for B :
cycles:
{{1, 3}, {2, 5}}
order:
2
See Matrix
$ [
[1, 4, 2, 0, 2]
,
[2, 2, 1, 0, 4]
,
[1, 4, 2, 0, 2]
,
[2, 2, 1, 0, 4]
] $
[y1, 2 y2, y2, 0, 2 y1]
p' =
- s + s 3
p =
- s + s 3
14
.
Coloring, {2, 4, 5}
R:
[5, 4, 1, 2, 1]
B:
[3, 5, 5, 5, 2]
` 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 |
2 vs 4 |
3 vs 3 |
Omega Rank for R :
cycles:
{{1, 5}, {2, 4}}
order:
2
See Matrix
$ [
[4, 1, 0, 2, 2]
,
[2, 2, 0, 1, 4]
,
[4, 1, 0, 2, 2]
,
[2, 2, 0, 1, 4]
] $
[2 y2, y1, 0, y2, 2 y1]
p =
s - s 3
p' =
s - s 3
Omega Rank for B :
cycles:
{{2, 5}}
order:
2
See Matrix
$ [
[0, 3, 2, 0, 4]
,
[0, 4, 0, 0, 5]
,
[0, 5, 0, 0, 4]
] $
[0, y1, y2, 0, y3]
15
.
Coloring, {3, 4, 5}
R:
[5, 5, 5, 2, 1]
B:
[3, 4, 1, 5, 2]
` 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 5 |
Omega Rank for R :
cycles:
{{1, 5}}
order:
2
See Matrix
$ [
[3, 1, 0, 0, 5]
,
[5, 0, 0, 0, 4]
,
[4, 0, 0, 0, 5]
] $
[y1, y2, 0, 0, y3]
Omega Rank for B :
cycles:
{{1, 3}, {2, 4, 5}}
See Matrix
$ [
[1, 3, 2, 2, 1]
,
[2, 1, 1, 3, 2]
,
[1, 2, 2, 1, 3]
,
[2, 3, 1, 2, 1]
,
[1, 1, 2, 3, 2]
] $
[y1, 2 y1 + 2 y2 - y4 - y3, y2, y4, y3]
p' =
- 1 - s + s 3 + s 4
16
.
Coloring, {2, 3, 4, 5}
Ωp(Δ)=0:
p =
s - 4s 3
p' =
s - 4s 3
R:
[5, 4, 5, 2, 1]
B:
[3, 5, 1, 5, 2]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 2 vs 4 |
3 vs 5 |
3 vs 5 |
2 vs 4 |
2 vs 4 |
Omega Rank for R :
cycles:
{{1, 5}, {2, 4}}
order:
2
See Matrix
$ [
[3, 1, 0, 2, 3]
,
[3, 2, 0, 1, 3]
,
[3, 1, 0, 2, 3]
,
[3, 2, 0, 1, 3]
] $
[y2, y1, 0, y2 - y1, y2]
p =
- s + s 3
p' =
- s + s 3
Omega Rank for B :
cycles:
{{1, 3}, {2, 5}}
order:
2
See Matrix
$ [
[1, 3, 2, 0, 3]
,
[2, 3, 1, 0, 3]
,
[1, 3, 2, 0, 3]
,
[2, 3, 1, 0, 3]
] $
[y2 - y1, y2, y1, 0, y2]
p =
s - s 3
p' =
- s + s 3
` See 3-level graph `
M
N
$ [
[0, 1, 0, 1, 2]
,
[1, 0, 1, 0, 2]
,
[0, 1, 0, 0, 1]
,
[1, 0, 0, 0, 1]
,
[2, 2, 1, 1, 0]
] $
$ [
[0, 1, 0, 1, 1]
,
[1, 0, 1, 0, 1]
,
[0, 1, 0, 1, 1]
,
[1, 0, 1, 0, 1]
,
[1, 1, 1, 1, 0]
] $
R:
[5, 4, 5, 2, 1]
B:
[3, 5, 1, 5, 2]
Ranges
Action of R on ranges, [[2], [1], [2]]
Action of B on ranges, [[3], [3], [1]]
Cycles:
R , {{1, 5}, {2, 4}}, B , {{1, 3}, {2, 5}}
β({1, 2, 5})
=
1/3
β({1, 4, 5})
=
1/3
β({2, 3, 5})
=
1/3
Partitions
α([{1, 3}, {5}, {2, 4}]) = 1/1
b1 = {1, 3}
` , ` b2 = {5}
` , ` b3 = {2, 4}
Action of R and B on the blocks of the partitions:
=
[2, 1, 3]
[1, 3, 2]
with invariant measure
[1, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-3 partition graph. `
| Right Group |
|---|
| Coloring |
{2, 3, 4, 5}
|
|---|
| Rank | 3 |
|---|
| R,B |
[5, 4, 5, 2, 1], [3, 5, 1, 5, 2]
|
|---|
| π2 |
[1, 0, 1, 2, 1, 0, 2, 0, 1, 1]
|
|---|
| u2 |
[1, 0, 1, 1, 1, 0, 1, 1, 1, 1]
(dim 1) |
|---|
| wpp |
[2, 2, 2, 2, 1]
|
|---|
| π3 |
[0, 0, 1, 0, 0, 1, 0, 1, 0, 0]
|
|---|
| u3 |
[0, 0, 1, 0, 0, 1, 0, 1, 0, 1]
|
|---|
| SUMMARY |
|---|
| Graph Type |
| NOT CC |
|---|
| ν(A) |
|
0
|
|---|
| ν(Δ) |
|
1
|
|---|
| π |
|
[2, 2, 1, 1, 3]
|
|---|
| Dbly Stoch |
| false |
|---|
| SANDWICH |
| Total
0
|
|---|
| No . | Coloring | Rank |
|---|
| RT GROUPS |
| Total
2
|
|---|
| No . | Coloring | Rank | Solv |
|---|
| 1 |
{2}
|
3
|
Not Solvable
|
|---|
| 2 |
{2, 3, 4, 5}
|
3
|
Not Solvable
|
|---|
| Δ-RANK'D | SC'D !RK'D |
τ-RANK'D | R/B RANK'D | NOT SYNC'D |
Total Runs | 2n-1 |
|---|
| 12 |
0 |
12 , 12 |
13 , 9 |
2 |
16 |
16 |