- FindStat

Your data matches 107 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000366
(load all 43 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of double descents of a permutation. A double descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i) > \pi(i+1) > \pi(i+2)$ .

Matching statistic: St000371
(load all 4 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
St000371: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 0
[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] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [3,1,2] => [3,1,2] => [3,2,1] => 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] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => [3,1,4,2] => 0
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [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] => 0
[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] => 0
[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] => 0
[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] => 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] => 0
[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] => 0
[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] => 0
[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] => 0
[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] => 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] => 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] => 0
[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] => 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
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[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] => 0
[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] => 0
[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] => 0
[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] => 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] => 0
[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] => 0
[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] => 0
[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] => 0
[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] => 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] => 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] => 0
[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] => 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
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,5,2,7,3,4,6] => [1,5,7,3,2,4,6] => [1,5,2,7,6,4,3] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,5,6,2,7,3,4] => [1,6,7,3,5,2,4] => [1,5,6,2,7,4,3] => ? = 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
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,6,2,3,7,4,5] => [1,6,2,7,4,3,5] => [1,6,3,2,7,5,4] => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6,2,7,3,4,5] => [1,6,7,3,4,2,5] => [1,6,2,7,5,4,3] => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,2] => [3,7,1,4,5,6,2] => [3,1,4,5,6,7,2] => ? = 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [3,4,1,5,6,7,2] => [4,7,3,1,5,6,2] => [3,4,1,5,6,7,2] => ? = 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [3,4,5,1,6,7,2] => [5,7,3,4,1,6,2] => [3,4,5,1,6,7,2] => ? = 0
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,6,1,7,2] => [6,7,3,4,5,1,2] => [3,4,5,6,1,7,2] => ? = 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [4,1,2,5,6,7,3] => [4,1,7,2,5,6,3] => [4,2,1,5,6,7,3] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [4,1,5,2,6,7,3] => [4,5,7,1,2,6,3] => [4,1,5,2,6,7,3] => ? = 0
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [4,1,6,2,3,5,7] => [4,6,2,1,3,5,7] => [4,1,6,5,3,2,7] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [4,5,1,6,2,3,7] => [5,6,2,4,1,3,7] => [4,5,1,6,3,2,7] => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [5,1,2,3,6,4,7] => [5,1,2,6,3,4,7] => [5,3,2,1,6,4,7] => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [5,1,2,6,3,4,7] => [5,1,6,3,2,4,7] => [5,2,1,6,4,3,7] => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,1,6,2,3,4,7] => [5,6,2,3,1,4,7] => [5,1,6,4,3,2,7] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,4,2,6,3,8,5,7] => [1,4,6,2,8,3,5,7] => [1,4,2,6,3,8,7,5] => ? = 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,4,2,8,3,5,6,7] => [1,4,8,2,3,5,6,7] => [1,4,2,8,7,6,5,3] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,6,2,3,4,8,5,7] => [1,6,2,3,8,4,5,7] => [1,6,4,3,2,8,7,5] => ? = 3
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,7,2,3,8,4,5,6] => [1,7,2,8,4,5,3,6] => [1,7,3,2,8,6,5,4] => ? = 3
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,7,2,8,3,4,5,6] => [1,7,8,3,4,5,2,6] => [1,7,2,8,6,5,4,3] => ? = 3
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,3,7,5,8,6] => [4,2,1,3,7,8,5,6] => [2,4,3,1,7,5,8,6] => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,2,6,4,8,5,7] => [3,1,2,6,8,4,5,7] => [3,2,1,6,4,8,7,5] => ? = 2
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [3,1,4,2,6,8,5,7] => [3,4,1,2,8,6,5,7] => [3,1,4,2,6,8,7,5] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,8,2] => [3,8,1,4,5,6,7,2] => [3,1,4,5,6,7,8,2] => ? = 0
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [3,1,5,2,4,8,6,7] => [3,5,1,2,4,8,6,7] => [3,1,5,4,2,8,7,6] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [3,1,5,2,7,4,6,8] => [3,5,1,7,2,4,6,8] => [3,1,5,2,7,6,4,8] => ? = 1
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [3,1,6,2,4,8,5,7] => [3,6,1,2,8,4,5,7] => [3,1,6,4,2,8,7,5] => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,7,2,4,5,6,8] => [3,7,1,2,4,5,6,8] => [3,1,7,6,5,4,2,8] => ? = 3
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,4,1,5,6,7,8,2] => [4,8,3,1,5,6,7,2] => [3,4,1,5,6,7,8,2] => ? = 0
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [3,4,5,1,6,7,8,2] => [5,8,3,4,1,6,7,2] => [3,4,5,1,6,7,8,2] => ? = 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [4,1,2,5,6,7,8,3] => [4,1,8,2,5,6,7,3] => [4,2,1,5,6,7,8,3] => ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => [4,1,6,2,8,3,5,7] => [4,6,8,1,3,2,5,7] => [4,1,6,2,8,7,5,3] => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => [5,1,2,7,3,4,6,8] => [5,3,2,1,7,6,4,8] => ? = 3
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [6,1,2,3,7,4,8,5] => [6,1,2,7,8,3,4,5] => [6,3,2,1,7,4,8,5] => ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [6,1,2,7,3,4,5,8] => [6,1,7,3,4,2,5,8] => [6,2,1,7,5,4,3,8] => ? = 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [6,1,7,2,3,4,5,8] => [6,7,2,3,4,1,5,8] => [6,1,7,5,4,3,2,8] => ? = 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [6,1,7,2,3,4,8,5] => [6,7,2,3,8,1,4,5] => [6,1,7,4,3,2,8,5] => ? = 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [6,1,8,2,3,4,5,7] => [6,8,2,3,4,1,5,7] => [6,1,8,7,5,4,3,2] => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [7,1,2,3,4,5,8,6] => [7,1,2,3,4,8,5,6] => [7,5,4,3,2,1,8,6] => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [7,1,2,3,8,4,5,6] => [7,1,2,8,4,5,3,6] => [7,3,2,1,8,6,5,4] => ? = 4
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [7,1,8,2,3,4,5,6,9] => [7,8,2,3,4,5,1,6,9] => [7,1,8,6,5,4,3,2,9] => ? = 4
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8,2,9,3,4,5,6,7] => [1,8,9,3,4,5,6,2,7] => [1,8,2,9,7,6,5,4,3] => ? = 4
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,8,9,2] => [3,9,1,4,5,6,7,8,2] => [3,1,4,5,6,7,8,9,2] => ? = 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,1,5,6,7,8,9,2] => [4,9,3,1,5,6,7,8,2] => ? => ? = 0
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,8,9,10,2] => [3,10,1,4,5,6,7,8,9,2] => [3,1,4,5,6,7,8,9,10,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,3,4,5,10,7,8,9] => [2,1,6,3,4,5,10,7,8,9] => ? => ? = 4
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [2,1,8,3,4,5,6,7,10,9] => [2,1,8,3,4,5,6,7,10,9] => ? => ? = 4
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,1,2,3,6,5,10,7,8,9] => [4,1,2,3,6,5,10,7,8,9] => ? => ? = 4
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,1,2,3,8,5,6,7,10,9] => [4,1,2,3,8,5,6,7,10,9] => ? => ? = 4

Description

The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) > \pi(j) > \pi(k)$ . In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].

Matching statistic: St000931
(load all 37 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 83%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1] => [.,.]
 => [1,0]
 => ? = 0
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,4,8,1,2] => ?
 => ?
 => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,1,2,3] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,1,2,3] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,1,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,1,2,6] => ?
 => ?
 => ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,4,1,2,8] => ?
 => ?
 => ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,6,1,2,8] => ?
 => ?
 => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,1,2,3,4] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,1,2,3,8] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,1,2,4,8] => ?
 => ?
 => ? = 0
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,1,2,6,8] => ?
 => ?
 => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [8,7,6,1,2,3,4,5] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [6,7,8,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [8,5,6,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 6
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 6
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 7
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 5
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 5
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 7
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 8
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => [[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => ?
 => ? = 6
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]
 => ?
 => ? = 4
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 5
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => ?
 => ? = 4
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 5
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 6
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
 => ?
 => ? = 8
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [9,8,1,2,3,4,5,6,7] => [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => [9,10,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
 => ?
 => ? = 6
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,6,7,8,9,10,1,2] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ?
 => ? = 6
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [10,9,1,2,3,4,5,6,7,8] => [[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 6
[]
 => [] => .
 => ?
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,1,2,9] => [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
 => ?
 => ? = 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,2,1,3,9] => [[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0

Description

The number of occurrences of the pattern UUU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

Matching statistic: St000052

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

Matching statistic: St000356
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$13\!\!-\!\!2$ .

Matching statistic: St000373
(load all 6 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 75%●distinct values known / distinct values provided: 56%

Values

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

Description

The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j \geq j$ and there exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000463

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of admissible inversions of a permutation. Let

$w = w_1,w_2,\dots,w_k$ be a word of length

$k$ with distinct letters from

$[n]$ . An admissible inversion of

$w$ is a pair

$(w_i,w_j)$ such that

$1\leq i < j\leq k$ and

$w_i > w_j$ that satisfies either of the following conditions:

$1 < i$ and

$w_{i−1} < w_i$ or there is some

$l$ such that

$i < l < j$ and

$w_i < w_l$ .

Matching statistic: St001083
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern

$132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.

Matching statistic: St000731
(load all 20 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of double exceedences of a permutation. A double exceedence is an index

$\sigma(i)$ such that

$i < \sigma(i) < \sigma(\sigma(i))$ .

Matching statistic: St001066
(load all 39 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of simple reflexive modules in the corresponding Nakayama algebra.

The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000496The rcs statistic of a set partition. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001781The interlacing number of a set partition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001489The maximum of the number of descents and the number of inverse descents. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St000619The number of cyclic descents of a permutation. St000019The cardinality of the support of a permutation. St000538The number of even inversions of a permutation. St000358The number of occurrences of the pattern 31-2. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000214The number of adjacencies of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000365The number of double ascents of a permutation. St000223The number of nestings in the permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000359The number of occurrences of the pattern 23-1. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000932The number of occurrences of the pattern UDU in a Dyck path. St000039The number of crossings of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000317The cycle descent number of a permutation. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000355The number of occurrences of the pattern 21-3. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000325The width of the tree associated to a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St001866The nesting alignments of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000983The length of the longest alternating subword. St001862The number of crossings of a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St000982The length of the longest constant subword. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001948The number of augmented double ascents of a permutation. St001060The distinguishing index of a graph. St000381The largest part of an integer composition. St000392The length of the longest run of ones in a binary word. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .