New Graph
[3, 3, 5, 5, 1, 1], [2, 4, 6, 6, 4, 2]
π =
[1, 1, 1, 1, 1, 1]
POSSIBLE RANKS
1 x 6
2 x 3
BASE DETERMINANT
91/512, .1777343750
NullSpace of Δ
{1, 2, 3, 4}, {5, 6}
Nullspace of A
[{6},{5}]
`,` [{2, 4},{1, 3}]
1
.
Coloring, {}
Ωp(Δ)=0:
p' =
s 3
p =
s 2
p' =
s 2
R:
[3, 3, 5, 5, 1, 1]
B:
[2, 4, 6, 6, 4, 2]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
1 vs 4 |
1 vs 4 |
1 vs 4 |
1 vs 3 |
1 vs 3 |
Omega Rank for R :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
] $
[y1, 0, y1, 0, y1, 0]
p =
- s + s 3
p =
- s + s 2
Omega Rank for B :
cycles:
{{2, 4, 6}}
order:
3
See Matrix
$ [
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
] $
[0, y1, 0, y1, 0, y1]
p =
s - s 3
p' =
s - s 2
` See 3-level graph `
M
N
$ [
[0, 0, 1, 0, 1, 0]
,
[0, 0, 0, 1, 0, 1]
,
[1, 0, 0, 0, 1, 0]
,
[0, 1, 0, 0, 0, 1]
,
[1, 0, 1, 0, 0, 0]
,
[0, 1, 0, 1, 0, 0]
] $
$ [
[0, 1, 2, 2, 2, 1]
,
[1, 0, 2, 2, 1, 2]
,
[2, 2, 0, 0, 2, 2]
,
[2, 2, 0, 0, 2, 2]
,
[2, 1, 2, 2, 0, 1]
,
[1, 2, 2, 2, 1, 0]
] $
τ=
12
, r'=
2/3
R:
[3, 3, 5, 5, 1, 1]
B:
[2, 4, 6, 6, 4, 2]
Ranges
Action of R on ranges, [[1], [1]]
Action of B on ranges, [[2], [2]]
Cycles:
R , {{1, 3, 5}}, B , {{2, 4, 6}}
β({1, 3, 5})
=
1/2
β({2, 4, 6})
=
1/2
Partitions
Action of R on partitions, [[1], [1]]
Action of B on partitions, [[2], [2]]
α([{1, 2}, {5, 6}, {3, 4}]) = 1/2
α([{3, 4}, {2, 5}, {1, 6}]) = 1/2
b1 = {1, 2}
` , ` b2 = {5, 6}
` , ` b3 = {3, 4}
` , ` b4 = {2, 5}
` , ` b5 = {1, 6}
Action of R and B on the blocks of the partitions:
=
[2, 3, 1, 3, 2]
[5, 3, 4, 5, 3]
with invariant measure
[1, 1, 2, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-3 partition graph. `
Sandwich |
Coloring |
{}
|
Rank | 3 |
R,B |
[3, 3, 5, 5, 1, 1], [2, 4, 6, 6, 4, 2]
|
π2 |
[0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0]
|
u2 |
[1, 2, 2, 2, 1, 2, 2, 1, 2, 0, 2, 2, 2, 2, 1]
(dim 1) |
wpp |
[2, 2, 2, 2, 2, 2]
|
π3 |
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
|
u3 |
[1, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 1, 2, 1, 2, 0, 0, 0, 1, 1]
|
2
.
Coloring, {2}
Ωp(Δ)=0:
p =
s 3
p' =
s 3
R:
[3, 4, 5, 5, 1, 1]
B:
[2, 3, 6, 6, 4, 2]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
2 vs 4 |
2 vs 5 |
2 vs 5 |
2 vs 4 |
2 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[2, 0, 1, 1, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
] $
[y1, 0, y1 - y2, y2, y1, 0]
p =
- s 2 + s 4
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{2, 3, 6}}
order:
3
See Matrix
$ [
[0, 2, 1, 1, 0, 2]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
] $
[0, y2, -y1 + y2, y1, 0, y2]
p =
- s 2 + s 3
p =
- s 2 + s 4
` See 3-level graph `
M
N
$ [
[0, 0, 1, 1, 2, 0]
,
[0, 0, 1, 1, 0, 2]
,
[1, 1, 0, 0, 1, 1]
,
[1, 1, 0, 0, 1, 1]
,
[2, 0, 1, 1, 0, 0]
,
[0, 2, 1, 1, 0, 0]
] $
$ [
[0, 1, 2, 2, 2, 1]
,
[1, 0, 2, 2, 1, 2]
,
[2, 2, 0, 0, 2, 2]
,
[2, 2, 0, 0, 2, 2]
,
[2, 1, 2, 2, 0, 1]
,
[1, 2, 2, 2, 1, 0]
] $
τ=
12
, r'=
2/3
R:
[3, 4, 5, 5, 1, 1]
B:
[2, 3, 6, 6, 4, 2]
Ranges
Action of R on ranges, [[1], [1], [2], [2]]
Action of B on ranges, [[4], [4], [3], [3]]
Cycles:
R , {{1, 3, 5}}, B , {{2, 3, 6}}
β({1, 3, 5})
=
1/4
β({1, 4, 5})
=
1/4
β({2, 3, 6})
=
1/4
β({2, 4, 6})
=
1/4
Partitions
Action of R on partitions, [[2], [2]]
Action of B on partitions, [[1], [1]]
α([{3, 4}, {2, 5}, {1, 6}]) = 1/2
α([{1, 2}, {5, 6}, {3, 4}]) = 1/2
b1 = {1, 2}
` , ` b2 = {5, 6}
` , ` b3 = {3, 4}
` , ` b4 = {2, 5}
` , ` b5 = {1, 6}
Action of R and B on the blocks of the partitions:
=
[2, 3, 1, 3, 2]
[5, 3, 4, 5, 3]
with invariant measure
[1, 1, 2, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-3 partition graph. `
Sandwich |
Coloring |
{2}
|
Rank | 3 |
R,B |
[3, 4, 5, 5, 1, 1], [2, 3, 6, 6, 4, 2]
|
π2 |
[0, 1, 1, 2, 0, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0]
|
u2 |
[1, 2, 2, 2, 1, 2, 2, 1, 2, 0, 2, 2, 2, 2, 1]
(dim 1) |
wpp |
[2, 2, 2, 2, 2, 2]
|
π3 |
[0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0]
|
u3 |
[1, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 1, 2, 1, 2, 0, 0, 0, 1, 1]
|
3
.
Coloring, {3}
R:
[3, 3, 6, 5, 1, 1]
B:
[2, 4, 5, 6, 4, 2]
` 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 |
2 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 6}}
order:
3
See Matrix
$ [
[2, 0, 2, 0, 1, 1]
,
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
] $
[y1, 0, y1, 0, y1 - y2, y2]
p =
- s 2 + s 4
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{2, 4, 6}}
order:
3
See Matrix
$ [
[0, 2, 0, 2, 1, 1]
,
[0, 1, 0, 3, 0, 2]
,
[0, 2, 0, 1, 0, 3]
,
[0, 3, 0, 2, 0, 1]
] $
[0, y4, 0, y3, y2, y1]
4
.
Coloring, {4}
R:
[3, 3, 5, 6, 1, 1]
B:
[2, 4, 6, 5, 4, 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 4 |
Omega Rank for R :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[2, 0, 2, 0, 1, 1]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
] $
[y2, 0, y2, 0, y2 - y1, y1]
p =
- s 2 + s 3
p =
- s 2 + s 4
Omega Rank for B :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[0, 2, 0, 2, 1, 1]
,
[0, 1, 0, 3, 2, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[0, y1 - y2 + y3, 0, y1, y2, y3]
p =
- s 3 + s 4
5
.
Coloring, {5}
R:
[3, 3, 5, 5, 4, 1]
B:
[2, 4, 6, 6, 1, 2]
` 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 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[1, 0, 2, 1, 2, 0]
,
[0, 0, 1, 2, 3, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[y1, 0, y1 - y3 + y2, y3, y2, 0]
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{2, 4, 6}}
order:
3
See Matrix
$ [
[1, 2, 0, 1, 0, 2]
,
[0, 3, 0, 2, 0, 1]
,
[0, 1, 0, 3, 0, 2]
,
[0, 2, 0, 1, 0, 3]
] $
[y4, y3, 0, y1, 0, y2]
6
.
Coloring, {6}
R:
[3, 3, 5, 5, 1, 2]
B:
[2, 4, 6, 6, 4, 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 |
3 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[1, 1, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
] $
[-y1 + y2, y1, y2, 0, y2, 0]
p =
- s 2 + s 4
p =
- s 2 + s 3
Omega Rank for B :
cycles:
{{1, 2, 4, 6}}
order:
4
See Matrix
$ [
[1, 1, 0, 2, 0, 2]
,
[2, 1, 0, 1, 0, 2]
,
[2, 2, 0, 1, 0, 1]
,
[1, 2, 0, 2, 0, 1]
] $
[y2, y3, 0, -y2 + y3 + y1, 0, y1]
p =
- s + s 2 - s 3 + s 4
7
.
Coloring, {2, 3}
R:
[3, 4, 6, 5, 1, 1]
B:
[2, 3, 5, 6, 4, 2]
` 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 |
3 vs 5 |
5 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 6}}
order:
3
See Matrix
$ [
[2, 0, 1, 1, 1, 1]
,
[2, 0, 2, 0, 1, 1]
,
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
] $
[y2 + y3, 0, y2 + y3 - y1, y1, y2, y3]
p =
s 3 - s 4
p' =
- s 3 + s 4
Omega Rank for B :
cycles:
{{2, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[0, 2, 1, 1, 1, 1]
,
[0, 1, 2, 1, 1, 1]
,
[0, 1, 1, 1, 2, 1]
,
[0, 1, 1, 2, 1, 1]
,
[0, 1, 1, 1, 1, 2]
] $
[0, y5, y4, y2, y3, y1]
8
.
Coloring, {2, 4}
R:
[3, 4, 5, 6, 1, 1]
B:
[2, 3, 6, 5, 4, 2]
` 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 |
3 vs 5 |
3 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[2, 0, 1, 1, 1, 1]
,
[2, 0, 2, 0, 1, 1]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
] $
[y3 + y2, 0, y3 + y2 - y1, y1, y3, y2]
p =
- s 3 + s 5
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{4, 5}, {2, 3, 6}}
order:
6
See Matrix
$ [
[0, 2, 1, 1, 1, 1]
,
[0, 1, 2, 1, 1, 1]
,
[0, 1, 1, 1, 1, 2]
,
[0, 2, 1, 1, 1, 1]
,
[0, 1, 2, 1, 1, 1]
] $
[0, -y2 + 4 y1 - y3, y2, y1, y1, y3]
p' =
- s + s 4
p =
s - s 4
9
.
Coloring, {2, 5}
R:
[3, 4, 5, 5, 4, 1]
B:
[2, 3, 6, 6, 1, 2]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
4 vs 4 |
3 vs 4 |
4 vs 4 |
3 vs 4 |
4 vs 4 |
Omega Rank for R :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[1, 0, 1, 2, 2, 0]
,
[0, 0, 1, 2, 3, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[y3, 0, y3 - y1 + y2, y1, y2, 0]
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{2, 3, 6}}
order:
3
See Matrix
$ [
[1, 2, 1, 0, 0, 2]
,
[0, 3, 2, 0, 0, 1]
,
[0, 1, 3, 0, 0, 2]
,
[0, 2, 1, 0, 0, 3]
] $
[y2, y1, y3, 0, 0, y4]
10
.
Coloring, {2, 6}
R:
[3, 4, 5, 5, 1, 2]
B:
[2, 3, 6, 6, 4, 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 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[1, 1, 1, 1, 2, 0]
,
[2, 0, 1, 1, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
] $
[-y1 + y3, y1, -y2 + y3, y2, y3, 0]
p =
- s 3 + s 5
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{1, 2, 3, 6}}
order:
4
See Matrix
$ [
[1, 1, 1, 1, 0, 2]
,
[2, 1, 1, 0, 0, 2]
,
[2, 2, 1, 0, 0, 1]
,
[1, 2, 2, 0, 0, 1]
,
[1, 1, 2, 0, 0, 2]
] $
[y1, y1 + y3 + y2 - y4, y3, y2, 0, y4]
p =
- s 2 + s 3 - s 4 + s 5
11
.
Coloring, {3, 4}
Ωp(Δ)=0:
p =
s 2 - 4s 4
R:
[3, 3, 6, 6, 1, 1]
B:
[2, 4, 5, 5, 4, 2]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
3 vs 4 |
4 vs 4 |
4 vs 4 |
1 vs 3 |
3 vs 3 |
Omega Rank for R :
cycles:
{{1, 3, 6}}
order:
3
See Matrix
$ [
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
] $
[y1, 0, y1, 0, 0, y1]
p =
- s + s 2
p =
- s + s 3
Omega Rank for B :
cycles:
{{4, 5}}
order:
2
See Matrix
$ [
[0, 2, 0, 2, 2, 0]
,
[0, 0, 0, 4, 2, 0]
,
[0, 0, 0, 2, 4, 0]
] $
[0, y1, 0, y3, y2, 0]
12
.
Coloring, {3, 5}
R:
[3, 3, 6, 5, 4, 1]
B:
[2, 4, 5, 6, 1, 2]
` 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 |
3 vs 5 |
3 vs 5 |
Omega Rank for R :
cycles:
{{4, 5}, {1, 3, 6}}
order:
6
See Matrix
$ [
[1, 0, 2, 1, 1, 1]
,
[1, 0, 1, 1, 1, 2]
,
[2, 0, 1, 1, 1, 1]
,
[1, 0, 2, 1, 1, 1]
,
[1, 0, 1, 1, 1, 2]
] $
[-y1 + 4 y2 - y3, 0, y1, y2, y2, y3]
p' =
s - s 4
p =
s - s 4
Omega Rank for B :
cycles:
{{2, 4, 6}}
order:
3
See Matrix
$ [
[1, 2, 0, 1, 1, 1]
,
[1, 2, 0, 2, 0, 1]
,
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
] $
[y1, y1 + y3, 0, y1 + y3 - y2, y2, y3]
p' =
s 3 - s 4
p =
s 3 - s 5
13
.
Coloring, {3, 6}
R:
[3, 3, 6, 5, 1, 2]
B:
[2, 4, 5, 6, 4, 1]
` 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 |
3 vs 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{2, 3, 6}}
order:
3
See Matrix
$ [
[1, 1, 2, 0, 1, 1]
,
[1, 1, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
] $
[-y1 + y2 + y3, y1, y2 + y3, 0, y2, y3]
p =
- s 3 + s 4
p =
- s 3 + s 5
Omega Rank for B :
cycles:
{{1, 2, 4, 6}}
order:
4
See Matrix
$ [
[1, 1, 0, 2, 1, 1]
,
[1, 1, 0, 2, 0, 2]
,
[2, 1, 0, 1, 0, 2]
,
[2, 2, 0, 1, 0, 1]
,
[1, 2, 0, 2, 0, 1]
] $
[y1 - y2 + y3 + y4, y1, 0, y2, y3, y4]
p =
- s 2 + s 3 - s 4 + s 5
14
.
Coloring, {4, 5}
R:
[3, 3, 5, 6, 4, 1]
B:
[2, 4, 6, 5, 1, 2]
` 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 5 |
Omega Rank for R :
cycles:
{{1, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[1, 0, 2, 1, 1, 1]
,
[1, 0, 1, 1, 2, 1]
,
[1, 0, 1, 2, 1, 1]
,
[1, 0, 1, 1, 1, 2]
,
[2, 0, 1, 1, 1, 1]
] $
[y5, 0, y4, y3, y1, y2]
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[1, 2, 0, 1, 1, 1]
,
[1, 2, 0, 2, 1, 0]
,
[1, 1, 0, 2, 2, 0]
,
[2, 1, 0, 1, 2, 0]
,
[2, 2, 0, 1, 1, 0]
] $
[y1 - y2 + y3 - y4, y1, 0, y2, y3, y4]
p =
- s 2 + s 3 - s 4 + s 5
15
.
Coloring, {4, 6}
R:
[3, 3, 5, 6, 1, 2]
B:
[2, 4, 6, 5, 4, 1]
` 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 |
3 vs 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 5}}
order:
3
See Matrix
$ [
[1, 1, 2, 0, 1, 1]
,
[1, 1, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
,
[2, 0, 2, 0, 2, 0]
] $
[y1, -y1 + y3 + y2, y3 + y2, 0, y3, y2]
p' =
s 3 - s 4
p =
s 3 - s 5
Omega Rank for B :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[1, 1, 0, 2, 1, 1]
,
[1, 1, 0, 2, 2, 0]
,
[0, 1, 0, 3, 2, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[y1 - y4 + y2 + y3, y1, 0, y4, y2, y3]
p =
- s 4 + s 5
16
.
Coloring, {5, 6}
Ωp(Δ)=0:
p =
s 2 - 4s 4
R:
[3, 3, 5, 5, 4, 2]
B:
[2, 4, 6, 6, 1, 1]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
3 vs 4 |
4 vs 4 |
4 vs 4 |
3 vs 4 |
3 vs 4 |
Omega Rank for R :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[0, 1, 2, 1, 2, 0]
,
[0, 0, 1, 2, 3, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[0, y1, y2, y3, -y1 + y2 + y3, 0]
p =
- s 3 + s 4
Omega Rank for B :
cycles:
{{1, 2, 4, 6}}
order:
4
See Matrix
$ [
[2, 1, 0, 1, 0, 2]
,
[2, 2, 0, 1, 0, 1]
,
[1, 2, 0, 2, 0, 1]
,
[1, 1, 0, 2, 0, 2]
] $
[y3, y2, 0, -y3 + y2 + y1, 0, y1]
p =
- s + s 2 - s 3 + s 4
17
.
Coloring, {2, 3, 4}
R:
[3, 4, 6, 6, 1, 1]
B:
[2, 3, 5, 5, 4, 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, 3, 6}}
order:
3
See Matrix
$ [
[2, 0, 1, 1, 0, 2]
,
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
,
[2, 0, 2, 0, 0, 2]
] $
[y2, 0, -y1 + y2, y1, 0, y2]
p =
s 2 - s 4
p' =
s 2 - s 3
Omega Rank for B :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[0, 2, 1, 1, 2, 0]
,
[0, 0, 2, 2, 2, 0]
,
[0, 0, 0, 2, 4, 0]
,
[0, 0, 0, 4, 2, 0]
] $
[0, y4, y3, y2, y1, 0]
18
.
Coloring, {2, 3, 5}
Ωp(Δ)=0:
p =
s + 3s 2 + 4s 3 + 4s 4
R:
[3, 4, 6, 5, 4, 1]
B:
[2, 3, 5, 6, 1, 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 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{4, 5}, {1, 3, 6}}
order:
6
See Matrix
$ [
[1, 0, 1, 2, 1, 1]
,
[1, 0, 1, 1, 2, 1]
,
[1, 0, 1, 2, 1, 1]
,
[1, 0, 1, 1, 2, 1]
,
[1, 0, 1, 2, 1, 1]
] $
[y2, 0, y2, 3 y2 - y1, y1, y2]
p =
s - s 5
p' =
s 2 - s 4
p' =
- s + s 3
Omega Rank for B :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[1, 2, 1, 0, 1, 1]
,
[1, 2, 2, 0, 1, 0]
,
[1, 1, 2, 0, 2, 0]
,
[2, 1, 1, 0, 2, 0]
,
[2, 2, 1, 0, 1, 0]
] $
[y4 - y3 + y2 - y1, y4, y3, 0, y2, y1]
p =
s 2 - s 3 + s 4 - s 5
M
N
$ [
[0, 1, 0, 1, 1, 0]
,
[1, 0, 1, 0, 0, 1]
,
[0, 1, 0, 1, 1, 0]
,
[1, 0, 1, 0, 0, 1]
,
[1, 0, 1, 0, 0, 1]
,
[0, 1, 0, 1, 1, 0]
] $
$ [
[0, 1, 0, 1, 1, 0]
,
[1, 0, 1, 0, 0, 1]
,
[0, 1, 0, 1, 1, 0]
,
[1, 0, 1, 0, 0, 1]
,
[1, 0, 1, 0, 0, 1]
,
[0, 1, 0, 1, 1, 0]
] $
τ=
18
, r'=
1/2
R:
[3, 4, 6, 5, 4, 1]
B:
[2, 3, 5, 6, 1, 2]
Ranges
Action of R on ranges, [[6], [7], [6], [8], [2], [9], [8], [3], [2]]
Action of B on ranges, [[4], [5], [1], [7], [4], [9], [3], [5], [1]]
Cycles:
R , {{4, 5}, {1, 3, 6}}, B , {{1, 2, 3, 5}}
β({1, 2})
=
1/9
β({1, 4})
=
1/9
β({1, 5})
=
1/9
β({2, 3})
=
1/9
β({2, 6})
=
1/9
β({3, 4})
=
1/9
β({3, 5})
=
1/9
β({4, 6})
=
1/9
β({5, 6})
=
1/9
Partitions
α([{1, 3, 6}, {2, 4, 5}]) = 1/1
b1 = {1, 3, 6}
` , ` b2 = {2, 4, 5}
Action of R and B on the blocks of the partitions:
=
[1, 2]
[2, 1]
with invariant measure
[1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-2 partition graph. `
Right Group |
Coloring |
{2, 3, 5}
|
Rank | 2 |
R,B |
[3, 4, 6, 5, 4, 1], [2, 3, 5, 6, 1, 2]
|
π2 |
[1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1]
|
u2 |
[1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1]
(dim 1) |
wpp |
[3, 3, 3, 3, 3, 3]
|
19
.
Coloring, {2, 3, 6}
Ωp(Δ)=0:
p =
s
p' =
s
p' =
s 2
p' =
s 3
R:
[3, 4, 6, 5, 1, 2]
B:
[2, 3, 5, 6, 4, 1]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
0 vs 4 |
1 vs 6 |
1 vs 6 |
1 vs 6 |
1 vs 6 |
Omega Rank for R :
cycles:
{{1, 2, 3, 4, 5, 6}}
order:
6
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, 1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1, 1]
] $
[y1, y1, y1, y1, y1, y1]
p' =
- 1 + s 4
p' =
- 1 + s
p' =
- 1 + s 3
p' =
- 1 + s 5
p' =
- 1 + s 2
Omega Rank for B :
cycles:
{{1, 2, 3, 4, 5, 6}}
order:
6
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, 1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1, 1]
] $
[y1, y1, y1, y1, y1, y1]
p' =
1 - s
p' =
- s + s 5
p' =
- s + s 4
p' =
- s + s 3
p' =
- s + s 2
` See 6-level graph `
M
N
$ [
[0, 1, 1, 1, 1, 1]
,
[1, 0, 1, 1, 1, 1]
,
[1, 1, 0, 1, 1, 1]
,
[1, 1, 1, 0, 1, 1]
,
[1, 1, 1, 1, 0, 1]
,
[1, 1, 1, 1, 1, 0]
] $
$ [
[0, 1, 1, 1, 1, 1]
,
[1, 0, 1, 1, 1, 1]
,
[1, 1, 0, 1, 1, 1]
,
[1, 1, 1, 0, 1, 1]
,
[1, 1, 1, 1, 0, 1]
,
[1, 1, 1, 1, 1, 0]
] $
τ=
6
, r'=
5/6
R:
[3, 4, 6, 5, 1, 2]
B:
[2, 3, 5, 6, 4, 1]
Ranges
Action of R on ranges, [[1]]
Action of B on ranges, [[1]]
Cycles:
R , {{1, 2, 3, 4, 5, 6}}, B , {{1, 2, 3, 4, 5, 6}}
β({1, 2, 3, 4, 5, 6})
=
1/1
Partitions
α([{2}, {1}, {5}, {6}, {3}, {4}]) = 1/1
b1 = {2}
` , ` b2 = {1}
` , ` b3 = {5}
` , ` b4 = {6}
` , ` b5 = {3}
` , ` b6 = {4}
Action of R and B on the blocks of the partitions:
=
[4, 3, 6, 5, 2, 1]
[2, 4, 5, 6, 1, 3]
with invariant measure
[1, 1, 1, 1, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-6 partition graph. `
Right Group |
Coloring |
{2, 3, 6}
|
Rank | 6 |
R,B |
[3, 4, 6, 5, 1, 2], [2, 3, 5, 6, 4, 1]
|
π2 |
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
|
u2 |
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(dim 1) |
wpp |
[1, 1, 1, 1, 1, 1]
|
π6 |
[1]
|
u6 |
[1]
|
20
.
Coloring, {2, 4, 5}
R:
[3, 4, 5, 6, 4, 1]
B:
[2, 3, 6, 5, 1, 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 |
5 vs 5 |
3 vs 5 |
Omega Rank for R :
cycles:
{{1, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[1, 0, 1, 2, 1, 1]
,
[1, 0, 1, 1, 1, 2]
,
[2, 0, 1, 1, 1, 1]
,
[1, 0, 2, 1, 1, 1]
,
[1, 0, 1, 1, 2, 1]
] $
[y3, 0, y1, y2, y4, y5]
Omega Rank for B :
cycles:
{{2, 3, 6}}
order:
3
See Matrix
$ [
[1, 2, 1, 0, 1, 1]
,
[1, 2, 2, 0, 0, 1]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
] $
[y2 + y3 - y1, y2 + y3, y2, 0, y3, y1]
p =
- s 3 + s 4
p =
- s 3 + s 5
21
.
Coloring, {2, 4, 6}
Ωp(Δ)=0:
p' =
s 2
p' =
s 3
p' =
s
p =
s
R:
[3, 4, 5, 6, 1, 2]
B:
[2, 3, 6, 5, 4, 1]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
0 vs 4 |
1 vs 6 |
1 vs 6 |
1 vs 6 |
1 vs 6 |
Omega Rank for R :
cycles:
{{1, 3, 5}, {2, 4, 6}}
order:
3
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, 1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1, 1]
] $
[y1, y1, y1, y1, y1, y1]
p' =
- s 4 + s 5
p' =
1 - s 4
p' =
s - s 4
p' =
s 2 - s 4
p' =
s 3 - s 4
Omega Rank for B :
cycles:
{{1, 2, 3, 6}, {4, 5}}
order:
4
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, 1, 1, 1, 1, 1]
,
[1, 1, 1, 1, 1, 1]
] $
[y1, y1, y1, y1, y1, y1]
p' =
- 1 + s
p' =
- 1 + s 2
p' =
- 1 + s 3
p' =
- 1 + s 4
p' =
- 1 + s 5
` See 6-level graph `
M
N
$ [
[0, 1, 1, 1, 1, 1]
,
[1, 0, 1, 1, 1, 1]
,
[1, 1, 0, 1, 1, 1]
,
[1, 1, 1, 0, 1, 1]
,
[1, 1, 1, 1, 0, 1]
,
[1, 1, 1, 1, 1, 0]
] $
$ [
[0, 1, 1, 1, 1, 1]
,
[1, 0, 1, 1, 1, 1]
,
[1, 1, 0, 1, 1, 1]
,
[1, 1, 1, 0, 1, 1]
,
[1, 1, 1, 1, 0, 1]
,
[1, 1, 1, 1, 1, 0]
] $
τ=
6
, r'=
5/6
R:
[3, 4, 5, 6, 1, 2]
B:
[2, 3, 6, 5, 4, 1]
Ranges
Action of R on ranges, [[1]]
Action of B on ranges, [[1]]
Cycles:
R , {{1, 3, 5}, {2, 4, 6}}, B , {{1, 2, 3, 6}, {4, 5}}
β({1, 2, 3, 4, 5, 6})
=
1/1
Partitions
α([{2}, {1}, {5}, {6}, {3}, {4}]) = 1/1
b1 = {2}
` , ` b2 = {1}
` , ` b3 = {5}
` , ` b4 = {6}
` , ` b5 = {3}
` , ` b6 = {4}
Action of R and B on the blocks of the partitions:
=
[4, 3, 5, 6, 2, 1]
[2, 4, 6, 5, 1, 3]
with invariant measure
[1, 1, 1, 1, 1, 1]
N by blocks,
check:
true
.
` See partition graph. `
` See level-6 partition graph. `
Right Group |
Coloring |
{2, 4, 6}
|
Rank | 6 |
R,B |
[3, 4, 5, 6, 1, 2], [2, 3, 6, 5, 4, 1]
|
π2 |
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
|
u2 |
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(dim 2) |
wpp |
[1, 1, 1, 1, 1, 1]
|
π6 |
[1]
|
u6 |
[1]
|
22
.
Coloring, {2, 5, 6}
R:
[3, 4, 5, 5, 4, 2]
B:
[2, 3, 6, 6, 1, 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 |
3 vs 4 |
Omega Rank for R :
cycles:
{{4, 5}}
order:
2
See Matrix
$ [
[0, 1, 1, 2, 2, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[0, y1, y1, y2, y2, 0]
p' =
s 2 - s 3
p =
s 2 - s 4
Omega Rank for B :
cycles:
{{1, 2, 3, 6}}
order:
4
See Matrix
$ [
[2, 1, 1, 0, 0, 2]
,
[2, 2, 1, 0, 0, 1]
,
[1, 2, 2, 0, 0, 1]
,
[1, 1, 2, 0, 0, 2]
] $
[y1 - y3 + y2, y1, y3, 0, 0, y2]
p =
- s + s 2 - s 3 + s 4
23
.
Coloring, {3, 4, 5}
R:
[3, 3, 6, 6, 4, 1]
B:
[2, 4, 5, 5, 1, 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 |
2 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 6}}
order:
3
See Matrix
$ [
[1, 0, 2, 1, 0, 2]
,
[2, 0, 1, 0, 0, 3]
,
[3, 0, 2, 0, 0, 1]
,
[1, 0, 3, 0, 0, 2]
] $
[y4, 0, y3, y2, 0, y1]
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[1, 2, 0, 1, 2, 0]
,
[2, 1, 0, 2, 1, 0]
,
[1, 2, 0, 1, 2, 0]
,
[2, 1, 0, 2, 1, 0]
] $
[y1, y2, 0, y1, y2, 0]
p' =
s - s 3
p =
s - s 3
24
.
Coloring, {3, 4, 6}
R:
[3, 3, 6, 6, 1, 2]
B:
[2, 4, 5, 5, 4, 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 |
3 vs 4 |
Omega Rank for R :
cycles:
{{2, 3, 6}}
order:
3
See Matrix
$ [
[1, 1, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
,
[0, 2, 2, 0, 0, 2]
] $
[-y1 + y2, y1, y2, 0, 0, y2]
p' =
s 2 - s 3
p =
s 2 - s 4
Omega Rank for B :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[1, 1, 0, 2, 2, 0]
,
[0, 1, 0, 3, 2, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[y1 - y2 + y3, y1, 0, y2, y3, 0]
p =
- s 3 + s 4
25
.
Coloring, {3, 5, 6}
R:
[3, 3, 6, 5, 4, 2]
B:
[2, 4, 5, 6, 1, 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 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{2, 3, 6}, {4, 5}}
order:
6
See Matrix
$ [
[0, 1, 2, 1, 1, 1]
,
[0, 1, 1, 1, 1, 2]
,
[0, 2, 1, 1, 1, 1]
,
[0, 1, 2, 1, 1, 1]
,
[0, 1, 1, 1, 1, 2]
] $
[0, -y3 + 4 y2 - y1, y3, y2, y2, y1]
p =
- s + s 4
p' =
- s + s 4
Omega Rank for B :
cycles:
{{1, 2, 4, 6}}
order:
4
See Matrix
$ [
[2, 1, 0, 1, 1, 1]
,
[2, 2, 0, 1, 0, 1]
,
[1, 2, 0, 2, 0, 1]
,
[1, 1, 0, 2, 0, 2]
,
[2, 1, 0, 1, 0, 2]
] $
[y1 - y2 + y3 + y4, y1, 0, y2, y3, y4]
p =
- s 2 + s 3 - s 4 + s 5
26
.
Coloring, {4, 5, 6}
Ωp(Δ)=0:
p' =
s 2 + 2s 3
p =
s 2 - 4s 4
R:
[3, 3, 5, 6, 4, 2]
B:
[2, 4, 6, 5, 1, 1]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
2 vs 4 |
5 vs 5 |
5 vs 5 |
5 vs 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{2, 3, 4, 5, 6}}
order:
5
See Matrix
$ [
[0, 1, 2, 1, 1, 1]
,
[0, 1, 1, 1, 2, 1]
,
[0, 1, 1, 2, 1, 1]
,
[0, 1, 1, 1, 1, 2]
,
[0, 2, 1, 1, 1, 1]
] $
[0, y5, y4, y1, y2, y3]
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[2, 1, 0, 1, 1, 1]
,
[2, 2, 0, 1, 1, 0]
,
[1, 2, 0, 2, 1, 0]
,
[1, 1, 0, 2, 2, 0]
,
[2, 1, 0, 1, 2, 0]
] $
[y1, y1 + y4 - y3 - y2, 0, y4, y3, y2]
p =
s 2 - s 3 + s 4 - s 5
27
.
Coloring, {2, 3, 4, 5}
R:
[3, 4, 6, 6, 4, 1]
B:
[2, 3, 5, 5, 1, 2]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
4 vs 4 |
4 vs 4 |
4 vs 4 |
4 vs 4 |
2 vs 4 |
Omega Rank for R :
cycles:
{{1, 3, 6}}
order:
3
See Matrix
$ [
[1, 0, 1, 2, 0, 2]
,
[2, 0, 1, 0, 0, 3]
,
[3, 0, 2, 0, 0, 1]
,
[1, 0, 3, 0, 0, 2]
] $
[y1, 0, y2, y4, 0, y3]
Omega Rank for B :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[1, 2, 1, 0, 2, 0]
,
[2, 1, 2, 0, 1, 0]
,
[1, 2, 1, 0, 2, 0]
,
[2, 1, 2, 0, 1, 0]
] $
[y2, y1, y2, 0, y1, 0]
p =
s - s 3
p' =
s - s 3
28
.
Coloring, {2, 3, 4, 6}
R:
[3, 4, 6, 6, 1, 2]
B:
[2, 3, 5, 5, 4, 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 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{2, 4, 6}}
order:
3
See Matrix
$ [
[1, 1, 1, 1, 0, 2]
,
[0, 2, 1, 1, 0, 2]
,
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
] $
[-y1 + y3, y1, -y2 + y3, y2, 0, y3]
p =
- s 3 + s 4
p =
- s 3 + s 5
Omega Rank for B :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[1, 1, 1, 1, 2, 0]
,
[0, 1, 1, 2, 2, 0]
,
[0, 0, 1, 2, 3, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[y1 - y2 - y3 + y4, y1, y2, y3, y4, 0]
p =
s 4 - s 5
29
.
Coloring, {2, 3, 5, 6}
R:
[3, 4, 6, 5, 4, 2]
B:
[2, 3, 5, 6, 1, 1]
` 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 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{4, 5}}
order:
4
See Matrix
$ [
[0, 1, 1, 2, 1, 1]
,
[0, 1, 0, 2, 2, 1]
,
[0, 1, 0, 3, 2, 0]
,
[0, 0, 0, 3, 3, 0]
,
[0, 0, 0, 3, 3, 0]
] $
[0, y4, -y4 + y1 - y2 + y3, y1, y2, y3]
p =
- s 4 + s 5
Omega Rank for B :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[2, 1, 1, 0, 1, 1]
,
[2, 2, 1, 0, 1, 0]
,
[1, 2, 2, 0, 1, 0]
,
[1, 1, 2, 0, 2, 0]
,
[2, 1, 1, 0, 2, 0]
] $
[y1 - y3 + y2 + y4, y1, y3, 0, y2, y4]
p =
- s 2 + s 3 - s 4 + s 5
30
.
Coloring, {2, 4, 5, 6}
Ωp(Δ)=0:
p =
s 2 - 4s 4
p' =
s 2 - 2s 3
R:
[3, 4, 5, 6, 4, 2]
B:
[2, 3, 6, 5, 1, 1]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
2 vs 4 |
6 vs 6 |
6 vs 6 |
3 vs 5 |
4 vs 5 |
Omega Rank for R :
cycles:
{{2, 4, 6}}
order:
3
See Matrix
$ [
[0, 1, 1, 2, 1, 1]
,
[0, 1, 0, 2, 1, 2]
,
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
,
[0, 2, 0, 2, 0, 2]
] $
[0, y1 - y2, y1 - y3, y1, y2, y3]
p' =
- s 3 + s 4
p =
s 3 - s 4
Omega Rank for B :
cycles:
{{1, 2, 3, 6}}
order:
4
See Matrix
$ [
[2, 1, 1, 0, 1, 1]
,
[2, 2, 1, 0, 0, 1]
,
[1, 2, 2, 0, 0, 1]
,
[1, 1, 2, 0, 0, 2]
,
[2, 1, 1, 0, 0, 2]
] $
[y2, y3, y1, 0, y2 - y3 + y1 - y4, y4]
p =
- s 2 + s 3 - s 4 + s 5
31
.
Coloring, {3, 4, 5, 6}
Ωp(Δ)=0:
p =
s 2 + 4s 4
R:
[3, 3, 6, 6, 4, 2]
B:
[2, 4, 5, 5, 1, 1]
` See graph `
` See pair graph `
Δ-Rank | A+(1/2)Δ |
A-(1/2)Δ | R | B |
3 vs 4 |
4 vs 4 |
4 vs 4 |
4 vs 4 |
3 vs 4 |
Omega Rank for R :
cycles:
{{2, 3, 6}}
order:
3
See Matrix
$ [
[0, 1, 2, 1, 0, 2]
,
[0, 2, 1, 0, 0, 3]
,
[0, 3, 2, 0, 0, 1]
,
[0, 1, 3, 0, 0, 2]
] $
[0, y1, y2, y3, 0, y4]
Omega Rank for B :
cycles:
{{1, 2, 4, 5}}
order:
4
See Matrix
$ [
[2, 1, 0, 1, 2, 0]
,
[2, 2, 0, 1, 1, 0]
,
[1, 2, 0, 2, 1, 0]
,
[1, 1, 0, 2, 2, 0]
] $
[y2, y2 + y1 - y3, 0, y1, y3, 0]
p =
- s + s 2 - s 3 + s 4
32
.
Coloring, {2, 3, 4, 5, 6}
R:
[3, 4, 6, 6, 4, 2]
B:
[2, 3, 5, 5, 1, 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 4 |
Omega Rank for R :
cycles:
{{2, 4, 6}}
order:
3
See Matrix
$ [
[0, 1, 1, 2, 0, 2]
,
[0, 2, 0, 1, 0, 3]
,
[0, 3, 0, 2, 0, 1]
,
[0, 1, 0, 3, 0, 2]
] $
[0, y4, y3, y2, 0, y1]
Omega Rank for B :
cycles:
{{1, 2, 3, 5}}
order:
4
See Matrix
$ [
[2, 1, 1, 0, 2, 0]
,
[2, 2, 1, 0, 1, 0]
,
[1, 2, 2, 0, 1, 0]
,
[1, 1, 2, 0, 2, 0]
] $
[y3, y1, y2, 0, y3 - y1 + y2, 0]
p =
- s + s 2 - s 3 + s 4
SUMMARY |
Graph Type |
| CC |
ν(A) |
|
2
|
ν(Δ) |
|
2
|
π |
|
[1, 1, 1, 1, 1, 1]
|
Dbly Stoch |
| true |
SANDWICH |
| Total
2
|
No . | Coloring | Rank |
1 |
{}
|
3
|
2 |
{2}
|
3
|
RT GROUPS |
| Total
3
|
No . | Coloring | Rank | Solv |
1 |
{2, 3, 5}
|
2
|
Not Solvable
|
2 |
{2, 4, 6}
|
6
|
["group", Not Solvable]
|
3 |
{2, 3, 6}
|
6
|
["group", Not Solvable]
|
CC Colorings |
| Total
1
|
No . | Coloring | Sandwich,Rank |
1 |
{}
|
true, 3
|
Δ-RANK'D | SC'D !RK'D |
τ-RANK'D | R/B RANK'D | NOT SYNC'D |
Total Runs | 2n-1 |
---|
22 |
0 |
24 , 27 |
7 , 6 |
5 |
32 |
32 |