New Graph
[2, 1, 5, 5, 1], [4, 3, 4, 3, 2]
π =
[1, 1, 1, 1, 1]
POSSIBLE RANKS
1 x 5
BASE DETERMINANT
725/4096, .1770019531
NullSpace of Δ
{1, 2, 3, 4, 5}
Nullspace of A
` det(A) = ` 1/16
1
.
Coloring, {}
R:
[2, 1, 5, 5, 1]
B:
[4, 3, 4, 3, 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, 2}}
order:
2
See Matrix
$ [
[2, 1, 0, 0, 2]
,
[3, 2, 0, 0, 0]
,
[2, 3, 0, 0, 0]
] $
[y1, y2, 0, 0, y3]
Omega Rank for B :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[0, 1, 2, 2, 0]
,
[0, 0, 3, 2, 0]
,
[0, 0, 2, 3, 0]
] $
[0, y1, y2, y3, 0]
2
.
Coloring, {2}
R:
[2, 3, 5, 5, 1]
B:
[4, 1, 4, 3, 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 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[1, 1, 1, 0, 2]
,
[2, 1, 1, 0, 1]
,
[1, 2, 1, 0, 1]
,
[1, 1, 2, 0, 1]
] $
[y2, y3, y1, 0, y4]
Omega Rank for B :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[1, 1, 1, 2, 0]
,
[1, 0, 2, 2, 0]
,
[0, 0, 2, 3, 0]
,
[0, 0, 3, 2, 0]
] $
[y4, y1, y2, y3, 0]
3
.
Coloring, {3}
R:
[2, 1, 4, 5, 1]
B:
[4, 3, 5, 3, 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 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2}}
order:
4
See Matrix
$ [
[2, 1, 0, 1, 1]
,
[2, 2, 0, 0, 1]
,
[3, 2, 0, 0, 0]
,
[2, 3, 0, 0, 0]
] $
[y4, y3, 0, y1, y2]
Omega Rank for B :
cycles:
{{2, 3, 5}}
order:
3
See Matrix
$ [
[0, 1, 2, 1, 1]
,
[0, 1, 2, 0, 2]
,
[0, 2, 1, 0, 2]
,
[0, 2, 2, 0, 1]
] $
[0, y1, y2, y3, y4]
4
.
Coloring, {4}
R:
[2, 1, 5, 3, 1]
B:
[4, 3, 4, 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 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2}}
order:
4
See Matrix
$ [
[2, 1, 1, 0, 1]
,
[2, 2, 0, 0, 1]
,
[3, 2, 0, 0, 0]
,
[2, 3, 0, 0, 0]
] $
[y3, y4, y2, 0, y1]
Omega Rank for B :
cycles:
{{2, 3, 4, 5}}
order:
4
See Matrix
$ [
[0, 1, 1, 2, 1]
,
[0, 1, 1, 1, 2]
,
[0, 2, 1, 1, 1]
,
[0, 1, 2, 1, 1]
] $
[0, y1, y2, y3, y4]
5
.
Coloring, {5}
R:
[2, 1, 5, 5, 2]
B:
[4, 3, 4, 3, 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:
{{1, 2}}
order:
2
See Matrix
$ [
[1, 2, 0, 0, 2]
,
[2, 3, 0, 0, 0]
,
[3, 2, 0, 0, 0]
] $
[y1, y2, 0, 0, y3]
Omega Rank for B :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[1, 0, 2, 2, 0]
,
[0, 0, 2, 3, 0]
,
[0, 0, 3, 2, 0]
] $
[y1, 0, y2, y3, 0]
6
.
Coloring, {2, 3}
Ωp(Δ)=0:
p' =
s
p' =
s 2
p' =
s 3
p =
s
R:
[2, 3, 4, 5, 1]
B:
[4, 1, 5, 3, 2]
` See graph `
` See pair graph `
| Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
| 0 vs 4 |
1 vs 5 |
1 vs 5 |
1 vs 5 |
1 vs 5 |
Omega Rank for R :
cycles:
{{1, 2, 3, 4, 5}}
order:
5
See Matrix
$ [
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
] $
[y1, y1, y1, y1, y1]
p' =
- 1 + s
p' =
- 1 + s 2
p' =
- 1 + s 3
p' =
- 1 + s 4
Omega Rank for B :
cycles:
{{1, 2, 3, 4, 5}}
order:
5
See Matrix
$ [
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1]
] $
[y1, y1, y1, y1, y1]
p' =
- s + s 3
p' =
1 - s
p' =
- s + s 4
p' =
- s + s 2
` See 5-level graph `
M
N
$ [
[0, 1, 1, 1, 1]
,
[1, 0, 1, 1, 1]
,
[1, 1, 0, 1, 1]
,
[1, 1, 1, 0, 1]
,
[1, 1, 1, 1, 0]
] $
$ [
[0, 1, 1, 1, 1]
,
[1, 0, 1, 1, 1]
,
[1, 1, 0, 1, 1]
,
[1, 1, 1, 0, 1]
,
[1, 1, 1, 1, 0]
] $
R:
[2, 3, 4, 5, 1]
B:
[4, 1, 5, 3, 2]
Ranges
Action of R on ranges, [[1]]
Action of B on ranges, [[1]]
Cycles:
R , {{1, 2, 3, 4, 5}}, B , {{1, 2, 3, 4, 5}}
β({1, 2, 3, 4, 5})
=
1/1
Partitions
α([{1}, {5}, {2}, {3}, {4}]) = 1/1
b1 = {1}
` , ` b2 = {5}
` , ` b3 = {2}
` , ` b4 = {3}
` , ` b5 = {4}
Action of R and B on the blocks of the partitions:
=
[2, 5, 1, 3, 4]
[3, 4, 2, 5, 1]
with invariant measure
[1, 1, 1, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-5 partition graph. `
| Right Group |
|---|
| Coloring |
{2, 3}
|
|---|
| Rank | 5 |
|---|
| R,B |
[2, 3, 4, 5, 1], [4, 1, 5, 3, 2]
|
|---|
| π2 |
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
|
|---|
| u2 |
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(dim 1) |
|---|
| wpp |
[1, 1, 1, 1, 1]
|
|---|
| π5 |
[1]
|
|---|
| u5 |
[1]
|
|---|
7
.
Coloring, {2, 4}
R:
[2, 3, 5, 3, 1]
B:
[4, 1, 4, 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 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[1, 1, 2, 0, 1]
,
[1, 1, 1, 0, 2]
,
[2, 1, 1, 0, 1]
,
[1, 2, 1, 0, 1]
] $
[y2, y1, y3, 0, y4]
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[1, 1, 0, 2, 1]
,
[1, 1, 0, 1, 2]
,
[1, 2, 0, 1, 1]
,
[2, 1, 0, 1, 1]
] $
[y1, y2, 0, y3, y4]
8
.
Coloring, {2, 5}
R:
[2, 3, 5, 5, 2]
B:
[4, 1, 4, 3, 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, 3, 5}}
order:
3
See Matrix
$ [
[0, 2, 1, 0, 2]
,
[0, 2, 2, 0, 1]
,
[0, 1, 2, 0, 2]
] $
[0, y1, y2, 0, y3]
Omega Rank for B :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[2, 0, 1, 2, 0]
,
[0, 0, 2, 3, 0]
,
[0, 0, 3, 2, 0]
] $
[y1, 0, y2, y3, 0]
9
.
Coloring, {3, 4}
R:
[2, 1, 4, 3, 1]
B:
[4, 3, 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 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2}, {3, 4}}
order:
2
See Matrix
$ [
[2, 1, 1, 1, 0]
,
[1, 2, 1, 1, 0]
,
[2, 1, 1, 1, 0]
,
[1, 2, 1, 1, 0]
] $
[-y2 + 3 y1, y2, y1, y1, 0]
p' =
s - s 3
p =
s - s 3
Omega Rank for B :
cycles:
{{2, 3, 5}}
order:
3
See Matrix
$ [
[0, 1, 1, 1, 2]
,
[0, 2, 1, 0, 2]
,
[0, 2, 2, 0, 1]
,
[0, 1, 2, 0, 2]
] $
[0, y2, y1, y3, y4]
10
.
Coloring, {3, 5}
R:
[2, 1, 4, 5, 2]
B:
[4, 3, 5, 3, 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 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2}}
order:
4
See Matrix
$ [
[1, 2, 0, 1, 1]
,
[2, 2, 0, 0, 1]
,
[2, 3, 0, 0, 0]
,
[3, 2, 0, 0, 0]
] $
[y2, y1, 0, y3, y4]
Omega Rank for B :
cycles:
{{1, 3, 4, 5}}
order:
4
See Matrix
$ [
[1, 0, 2, 1, 1]
,
[1, 0, 1, 1, 2]
,
[2, 0, 1, 1, 1]
,
[1, 0, 1, 2, 1]
] $
[y3, 0, y1, y2, y4]
11
.
Coloring, {4, 5}
R:
[2, 1, 5, 3, 2]
B:
[4, 3, 4, 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 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 2}}
order:
4
See Matrix
$ [
[1, 2, 1, 0, 1]
,
[2, 2, 0, 0, 1]
,
[2, 3, 0, 0, 0]
,
[3, 2, 0, 0, 0]
] $
[y3, y1, y2, 0, y4]
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[1, 0, 1, 2, 1]
,
[1, 0, 0, 2, 2]
,
[2, 0, 0, 1, 2]
,
[2, 0, 0, 2, 1]
] $
[y4, 0, y1, y2, y3]
12
.
Coloring, {2, 3, 4}
R:
[2, 3, 4, 3, 1]
B:
[4, 1, 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 |
4 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
4
See Matrix
$ [
[1, 1, 2, 1, 0]
,
[0, 1, 2, 2, 0]
,
[0, 0, 3, 2, 0]
,
[0, 0, 2, 3, 0]
] $
[y4, y3, y1, y2, 0]
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[1, 1, 0, 1, 2]
,
[1, 2, 0, 1, 1]
,
[2, 1, 0, 1, 1]
,
[1, 1, 0, 2, 1]
] $
[y1, y2, 0, y3, y4]
13
.
Coloring, {2, 3, 5}
R:
[2, 3, 4, 5, 2]
B:
[4, 1, 5, 3, 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 |
4 vs 4 |
Omega Rank for R :
cycles:
{{2, 3, 4, 5}}
order:
4
See Matrix
$ [
[0, 2, 1, 1, 1]
,
[0, 1, 2, 1, 1]
,
[0, 1, 1, 2, 1]
,
[0, 1, 1, 1, 2]
] $
[0, y1, y2, y4, y3]
Omega Rank for B :
cycles:
{{1, 3, 4, 5}}
order:
4
See Matrix
$ [
[2, 0, 1, 1, 1]
,
[1, 0, 1, 2, 1]
,
[1, 0, 2, 1, 1]
,
[1, 0, 1, 1, 2]
] $
[y1, 0, y2, y3, y4]
14
.
Coloring, {2, 4, 5}
R:
[2, 3, 5, 3, 2]
B:
[4, 1, 4, 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, 3, 5}}
order:
3
See Matrix
$ [
[0, 2, 2, 0, 1]
,
[0, 1, 2, 0, 2]
,
[0, 2, 1, 0, 2]
] $
[0, y1, y3, 0, y2]
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[2, 0, 0, 2, 1]
,
[1, 0, 0, 2, 2]
,
[2, 0, 0, 1, 2]
] $
[y1, 0, 0, y3, y2]
15
.
Coloring, {3, 4, 5}
R:
[2, 1, 4, 3, 2]
B:
[4, 3, 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 |
2 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{3, 4}, {1, 2}}
order:
2
See Matrix
$ [
[1, 2, 1, 1, 0]
,
[2, 1, 1, 1, 0]
,
[1, 2, 1, 1, 0]
,
[2, 1, 1, 1, 0]
] $
[-y2 + 3 y1, y2, y1, y1, 0]
p =
- s + s 3
p' =
- s + s 3
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[1, 0, 1, 1, 2]
,
[2, 0, 0, 1, 2]
,
[2, 0, 0, 2, 1]
,
[1, 0, 0, 2, 2]
] $
[y3, 0, y4, y1, y2]
16
.
Coloring, {2, 3, 4, 5}
R:
[2, 3, 4, 3, 2]
B:
[4, 1, 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 |
3 vs 3 |
Omega Rank for R :
cycles:
{{3, 4}}
order:
2
See Matrix
$ [
[0, 2, 2, 1, 0]
,
[0, 0, 3, 2, 0]
,
[0, 0, 2, 3, 0]
] $
[0, y3, y2, y1, 0]
Omega Rank for B :
cycles:
{{1, 4, 5}}
order:
3
See Matrix
$ [
[2, 0, 0, 1, 2]
,
[2, 0, 0, 2, 1]
,
[1, 0, 0, 2, 2]
] $
[y1, 0, 0, y2, y3]
| SUMMARY |
|---|
| Graph Type |
| NOT CC |
|---|
| ν(A) |
|
0
|
|---|
| ν(Δ) |
|
1
|
|---|
| π |
|
[1, 1, 1, 1, 1]
|
|---|
| Dbly Stoch |
| true |
|---|
| SANDWICH |
| Total
0
|
|---|
| No . | Coloring | Rank |
|---|
| RT GROUPS |
| Total
1
|
|---|
| No . | Coloring | Rank | Solv |
|---|
| 1 |
{2, 3}
|
5
|
["group", Not Solvable]
|
|---|
| Δ-RANK'D | SC'D !RK'D |
τ-RANK'D | R/B RANK'D | NOT SYNC'D |
Total Runs | 2n-1 |
|---|
| 15 |
0 |
15 , 15 |
13 , 15 |
1 |
16 |
16 |