- FindStat

Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000254
(load all 12 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => {{1},{2}}
 => 0
[1,1,0,0]
 => {{1,2}}
 => 1
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 1
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 1

Description

The nesting number of a set partition. This is the maximal number of chords in the standard representation of a set partition that mutually nest.

Matching statistic: St000253
(load all 8 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => 0
[1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => 1
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1},{2,3}}
 => 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 1
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => 1
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1},{2,3,4}}
 => 1
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 1
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 1
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 1
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1,3,4},{2}}
 => 1
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 1
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 2
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,2,4},{3}}
 => 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1},{2},{3,4,5}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1},{2,5},{3},{4}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => {{1},{2,4,5},{3}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1},{2,4},{3,5}}
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1},{2,3,5},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1},{2,3,4,5}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => {{1,2},{3,4},{5}}
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2},{3,4,5}}
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => {{1,4},{2},{3},{5}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1,4,5},{2},{3}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => {{1,3,4},{2},{5}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1,4},{2},{3,5}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1,3,4,5},{2}}
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 1
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,5,6,7,8},{3,4}}
 => {{1,2,4},{3,5,6,7,8}}
 => ? = 2
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,6,7,8},{4,5}}
 => {{1,2,3,5},{4,6,7,8}}
 => ? = 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,7,8},{5,6}}
 => {{1,2,3,4,6},{5,7,8}}
 => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => {{1,8},{2,3,4,5,6,7}}
 => {{1,3,5,7},{2,4,6,8}}
 => ? = 2

Description

The crossing number of a set partition. This is the maximal number of chords in the standard representation of a set partition, that mutually cross.

Matching statistic: St000451
(load all 2 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,1,2] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => [3,4,2,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,5,2,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,5,2,4,3] => [1,4,5,3,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,3,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => [2,4,1,5,3] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [5,2,1,4,3] => [2,4,5,3,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => [2,5,4,3,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => [1,5,4,3,2,7,6] => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,5,2,3,6,4,7] => [1,5,2,6,3,4,7] => [1,5,3,2,6,4,7] => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => [1,5,3,2,6,7,4] => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,5,2,3,7,4,6] => [1,5,2,7,3,4,6] => [1,5,3,2,7,6,4] => ? = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,2,6,3,4,7] => [1,5,6,3,2,4,7] => [1,5,2,6,4,3,7] => ? = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,5,2,6,3,7,4] => [1,5,6,7,2,3,4] => [1,5,2,6,3,7,4] => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,5,2,6,7,3,4] => [1,5,7,3,2,6,4] => [1,5,2,6,7,4,3] => ? = 2 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,5,2,7,3,4,6] => [1,5,7,3,2,4,6] => [1,5,2,7,6,4,3] => ? = 2 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,5,6,2,3,4,7] => [1,6,2,3,5,4,7] => [1,5,6,4,3,2,7] => ? = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,5,6,2,3,7,4] => [1,6,2,7,5,3,4] => [1,5,6,3,2,7,4] => ? = 2 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,5,6,2,7,3,4] => [1,6,7,3,5,2,4] => [1,5,6,2,7,4,3] => ? = 2 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,7,5] => [1,6,2,3,7,4,5] => [1,6,4,3,2,7,5] => ? = 2 + 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,6,2,3,7,4,5] => [1,6,2,7,4,3,5] => [1,6,3,2,7,5,4] => ? = 2 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6,2,7,3,4,5] => [1,6,7,3,4,2,5] => [1,6,2,7,5,4,3] => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => [2,1,3,4,7,6,5] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,6,4,7] => [2,1,3,6,5,4,7] => [2,1,3,5,6,4,7] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,1,3,5,6,7,4] => [2,1,3,7,5,6,4] => [2,1,3,5,6,7,4] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [2,1,3,5,7,4,6] => [2,1,3,7,5,4,6] => [2,1,3,5,7,6,4] => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,4,5,7] => [2,1,3,6,4,5,7] => [2,1,3,6,5,4,7] => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [2,1,3,6,4,7,5] => [2,1,3,6,7,4,5] => [2,1,3,6,4,7,5] => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [2,1,3,6,7,4,5] => [2,1,3,7,4,6,5] => [2,1,3,6,7,5,4] => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => [2,1,3,7,6,5,4] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,7,6,5] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,4,5,3,7,6] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,4,5,6,3,7] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => [2,1,4,5,6,7,3] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,4,5,7,3,6] => [2,1,7,4,5,3,6] => [2,1,4,5,7,6,3] => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [2,1,4,6,5,3,7] => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [2,1,4,6,3,7,5] => [2,1,6,4,7,3,5] => [2,1,4,6,3,7,5] => ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [2,1,4,6,7,3,5] => [2,1,7,4,3,6,5] => [2,1,4,6,7,5,3] => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => [2,1,4,7,6,5,3] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,5,4,3,6,7] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,3,4,7,6] => [2,1,5,3,4,7,6] => [2,1,5,4,3,7,6] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [2,1,5,3,6,4,7] => [2,1,5,6,3,4,7] => [2,1,5,3,6,4,7] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,1,5,3,6,7,4] => [2,1,5,7,3,6,4] => [2,1,5,3,6,7,4] => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [2,1,5,3,7,4,6] => [2,1,5,7,3,4,6] => [2,1,5,3,7,6,4] => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [2,1,5,6,3,4,7] => [2,1,6,3,5,4,7] => [2,1,5,6,4,3,7] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [2,1,5,6,3,7,4] => [2,1,6,7,5,3,4] => [2,1,5,6,3,7,4] => ? = 2 + 1

Description

The length of the longest pattern of the form k 1 2...(k-1).

Matching statistic: St001761

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,1,2] => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => [3,1,4,2] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,4,5,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,4,2,5,3] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,1,4] => [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,4,5,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,2,1,4,3] => [2,4,1,5,3] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,2,1,3,4] => [2,5,1,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,4,3,7,6,5,2] => [1,7,4,3,2,5,6] => [1,4,7,2,3,5,6] => ? = 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,4,6,5,3,7,2] => [1,7,6,4,3,5,2] => [1,4,6,7,2,3,5] => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,5,4,3,6,2,7] => [1,6,5,3,4,2,7] => [1,5,6,2,3,4,7] => ? = 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,4,3,6,7,2] => [1,7,5,3,4,6,2] => [1,5,6,2,3,7,4] => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,5,4,3,7,6,2] => [1,7,5,3,4,2,6] => [1,5,7,2,3,4,6] => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,4,6,3,2,7] => [1,6,2,5,4,3,7] => [1,5,2,6,3,4,7] => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,5,6,7,2,3,4] => ? = 3
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,5,4,6,7,3,2] => [1,7,2,5,4,6,3] => [1,5,2,6,3,7,4] => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,5,4,7,6,3,2] => [1,7,2,5,4,3,6] => [1,5,2,7,3,4,6] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,5,6,4,3,2,7] => [1,6,2,3,5,4,7] => [1,5,2,3,6,4,7] => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,5,6,4,3,7,2] => [1,7,6,3,5,4,2] => [1,5,6,2,7,3,4] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,5,6,4,7,3,2] => [1,7,2,6,5,4,3] => [1,5,2,6,7,3,4] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,5,6,7,4,3,2] => [1,7,2,3,5,6,4] => [1,5,2,3,6,7,4] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,5,7,6,4,3,2] => [1,7,2,3,5,4,6] => [1,5,2,3,7,4,6] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,6,5,4,3,7,2] => [1,7,6,3,4,5,2] => [1,6,7,2,3,4,5] => ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,6,5,4,7,3,2] => [1,7,2,6,4,5,3] => [1,6,2,7,3,4,5] => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6,5,7,4,3,2] => [1,7,2,3,6,5,4] => [1,6,2,3,7,4,5] => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,5,4,3,2] => [1,7,2,3,4,6,5] => [1,6,2,3,4,7,5] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,6,4,7] => [2,1,3,6,5,4,7] => [2,1,3,5,6,4,7] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,1,3,5,6,7,4] => [2,1,3,7,5,6,4] => [2,1,3,5,6,7,4] => ? = 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [2,1,3,5,7,6,4] => [2,1,3,7,5,4,6] => [2,1,3,5,7,4,6] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => [2,1,3,6,4,5,7] => [2,1,3,6,4,5,7] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [2,1,3,6,5,7,4] => [2,1,3,7,6,5,4] => [2,1,3,6,7,4,5] => ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [2,1,3,6,7,5,4] => [2,1,3,7,4,6,5] => [2,1,3,6,4,7,5] => ? = 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ? = 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,6,5] => [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,4,5,3,7,6] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,4,5,6,3,7] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => [2,1,4,5,6,7,3] => ? = 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,4,5,7,6,3] => [2,1,7,4,5,3,6] => [2,1,4,5,7,3,6] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,6,5,3,7] => [2,1,6,4,3,5,7] => [2,1,4,6,3,5,7] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [2,1,4,6,5,7,3] => [2,1,7,4,6,5,3] => [2,1,4,6,7,3,5] => ? = 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [2,1,4,6,7,5,3] => [2,1,7,4,3,6,5] => [2,1,4,6,3,7,5] => ? = 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [2,1,4,7,6,5,3] => [2,1,7,4,3,5,6] => [2,1,4,7,3,5,6] => ? = 1

Description

The maximal multiplicity of a letter in a reduced word of a permutation. For example, the permutation

$3421$ has the reduced word

$s_2 s_1 s_2 s_3 s_2$ , where

$s_2$ appears three times.

Matching statistic: St000454

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1,2] => ([],2)
 => ([],1)
 => 0
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ([(1,2)],3)
 => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St001665

Mapping

Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [[.,.],.]
 => [1,2] => [1,2] => 0
[1,1,0,0]
 => [.,[.,.]]
 => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => 1
[1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,1,4,2] => 1
[1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [3,4,2,1] => 1
[1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => 2
[1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [2,4,3,1] => 1
[1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [4,1,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,1,3,4,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [4,1,5,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [3,1,4,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,1,4,3,2] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [3,1,5,4,2] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [3,1,4,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [2,4,1,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [2,5,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [2,4,1,5,3] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [3,5,2,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [4,5,2,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,4,2,3,1] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [3,5,2,4,1] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [3,4,2,5,1] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [[[[[[.,.],.],.],.],[.,.]],.]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[[[[.,.],.],.],.],[[.,.],.]]
 => [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[[[[[.,.],.],.],[.,.]],.],.]
 => [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[[.,.],.],.],[.,.]],[.,.]]
 => [7,5,1,2,3,4,6] => [5,1,2,3,7,4,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [[[[[.,.],.],.],[[.,.],.]],.]
 => [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],[[[.,.],.],.]]
 => [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [[[[.,.],.],.],[[.,.],[.,.]]]
 => [7,5,6,1,2,3,4] => [6,1,2,3,7,5,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[[.,.],.],.],[.,[.,.]]],.]
 => [6,5,1,2,3,4,7] => [5,1,2,3,6,4,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [[[[.,.],.],.],[[.,[.,.]],.]]
 => [6,5,7,1,2,3,4] => [7,1,2,3,6,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [[[[.,.],.],.],[.,[[.,.],.]]]
 => [6,7,5,1,2,3,4] => [5,1,2,3,7,6,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],[.,[.,[.,.]]]]
 => [7,6,5,1,2,3,4] => [5,1,2,3,6,7,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[[[[[.,.],.],[.,.]],.],.],.]
 => [4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[[.,.],.],[.,.]],.],[.,.]]
 => [7,4,1,2,3,5,6] => [4,1,2,7,3,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [6,4,1,2,3,5,7] => [4,1,2,6,3,5,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [[[[.,.],.],[.,.]],[[.,.],.]]
 => [6,7,4,1,2,3,5] => [4,1,2,7,3,6,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [[[[.,.],.],[.,.]],[.,[.,.]]]
 => [7,6,4,1,2,3,5] => [4,1,2,6,3,7,5] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [[[[[.,.],.],[[.,.],.]],.],.]
 => [4,5,1,2,3,6,7] => [5,1,2,4,3,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [[[[.,.],.],[[.,.],.]],[.,.]]
 => [7,4,5,1,2,3,6] => [5,1,2,7,4,3,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [[[[.,.],.],[[[.,.],.],.]],.]
 => [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => [4,5,6,7,1,2,3] => [7,1,2,4,5,6,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [[[.,.],.],[[[.,.],.],[.,.]]]
 => [7,4,5,6,1,2,3] => [6,1,2,7,4,5,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [[[[.,.],.],[[.,.],[.,.]]],.]
 => [6,4,5,1,2,3,7] => [5,1,2,6,4,3,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],.],[[[.,.],[.,.]],.]]
 => [6,4,5,7,1,2,3] => [7,1,2,6,4,5,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [[[.,.],.],[[.,.],[[.,.],.]]]
 => [6,7,4,5,1,2,3] => [5,1,2,7,4,6,3] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [[[.,.],.],[[.,.],[.,[.,.]]]]
 => [7,6,4,5,1,2,3] => [5,1,2,6,4,7,3] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => [4,1,2,5,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => [7,5,4,1,2,3,6] => [4,1,2,5,7,3,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [[[[.,.],.],[[.,[.,.]],.]],.]
 => [5,4,6,1,2,3,7] => [6,1,2,5,4,3,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],.],[[[.,[.,.]],.],.]]
 => [5,4,6,7,1,2,3] => [7,1,2,5,4,6,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [[[.,.],.],[[.,[.,.]],[.,.]]]
 => [7,5,4,6,1,2,3] => [6,1,2,5,7,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [[[[.,.],.],[.,[[.,.],.]]],.]
 => [5,6,4,1,2,3,7] => [4,1,2,6,5,3,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [[[.,.],.],[[.,[[.,.],.]],.]]
 => [5,6,4,7,1,2,3] => [7,1,2,6,5,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [[[.,.],.],[.,[[[.,.],.],.]]]
 => [5,6,7,4,1,2,3] => [4,1,2,7,5,6,3] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [[[.,.],.],[.,[[.,.],[.,.]]]]
 => [7,5,6,4,1,2,3] => [4,1,2,6,7,5,3] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,[.,.]]]],.]
 => [6,5,4,1,2,3,7] => [4,1,2,5,6,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [[[.,.],.],[[.,[.,[.,.]]],.]]
 => [6,5,4,7,1,2,3] => [7,1,2,5,6,4,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [[[.,.],.],[.,[[.,[.,.]],.]]]
 => [6,5,7,4,1,2,3] => [4,1,2,7,6,5,3] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [[[.,.],.],[.,[.,[[.,.],.]]]]
 => [6,7,5,4,1,2,3] => [4,1,2,5,7,6,3] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [7,6,5,4,1,2,3] => [4,1,2,5,6,7,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[[.,.],[.,.]],.],.],.],.]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => [7,3,1,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => [3,1,6,2,4,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [[[[.,.],[.,.]],.],[[.,.],.]]
 => [6,7,3,1,2,4,5] => [3,1,7,2,4,6,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],[.,[.,.]]]
 => [7,6,3,1,2,4,5] => [3,1,6,2,4,7,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => [3,1,5,2,4,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[[.,.],[.,.]],[.,.]],[.,.]]
 => [7,5,3,1,2,4,6] => [3,1,5,2,7,4,6] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [[[[.,.],[.,.]],[[.,.],.]],.]
 => [5,6,3,1,2,4,7] => [3,1,6,2,5,4,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [[[.,.],[.,.]],[[[.,.],.],.]]
 => [5,6,7,3,1,2,4] => [3,1,7,2,5,6,4] => ? = 1

Description

The number of pure excedances of a permutation. A pure excedance of a permutation

$\pi$ is a position

$i < \pi_i$ such that there is no

$j < i$ with

$i\leq \pi_j < \pi_i$ .

Matching statistic: St001198

Mapping

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 75%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1 + 1

Description

The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001206

Mapping

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 75%

Values

[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1 + 1

Description

The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001330

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [1,2] => [2] => ([],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [3] => ([],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5] => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [6] => ([],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,2,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,2,3,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,4,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,5,3,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,5,6,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,6,3,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,3,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,6,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,1,5,6,3,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,3,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [2,3,1,4,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [2,3,1,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St000455
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 0 - 1
[1,1,0,0]
 => [1,2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 0 - 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 - 1
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 0 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 0 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => -1 = 0 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001890The maximum magnitude of the Möbius function of a poset. St000914The sum of the values of the Möbius function of a poset. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St000632The jump number of the poset. St001423The number of distinct cubes in a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000640The rank of the largest boolean interval in a poset.