- FindStat

Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001683
(load all 9 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001683: 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] => 1
[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] => 0
[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] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[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] => 1
[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] => 0
[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] => 0
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[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] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 3
[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] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2
[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] => 1
[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] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 2

Description

The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.

Matching statistic: St000372
(load all 4 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of mid points of increasing 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 $\pi(i) < \pi(j) < \pi(k)$. The generating function is given by [1].

Matching statistic: St001687
(load all 8 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.

Matching statistic: St001685
(load all 4 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.

Matching statistic: St000371
(load all 7 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 0
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [2,3,1] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [2,4,3,1] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [3,2,4,1] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [4,3,2,1] => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => [2,3,5,4,1] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => [2,4,3,5,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => [2,5,3,4,1] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => [3,2,4,5,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => [3,5,4,2,1] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => [4,3,2,5,1] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => [5,3,4,2,1] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => [5,4,3,2,1] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => [5,2,4,3,1] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => [5,3,2,4,1] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [1,3,4,5,2] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => [3,5,4,1,2] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => [4,3,1,5,2] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => [5,3,4,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => [5,4,3,1,2] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => [2,5,4,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => [2,3,4,6,5,7,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => [2,3,4,7,6,5,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [5,6,7,4,3,2,1] => [2,3,4,7,5,6,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => [2,3,5,4,6,7,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [6,4,7,5,3,2,1] => [2,3,5,7,6,4,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,3,2,1] => [5,4,7,6,3,2,1] => [2,3,6,5,4,7,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => [2,3,7,5,6,4,1] => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [5,6,4,7,3,2,1] => [2,3,7,6,5,4,1] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [4,5,7,6,3,2,1] => [2,3,6,4,5,7,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,2,1] => [4,6,5,7,3,2,1] => [2,3,7,4,6,5,1] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [5,4,6,7,3,2,1] => [2,3,7,5,4,6,1] => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => [2,4,3,5,6,7,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [6,3,7,5,4,2,1] => [2,4,5,7,6,3,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [5,3,7,6,4,2,1] => [2,4,6,5,3,7,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,4,2,1] => [6,5,3,7,4,2,1] => [2,4,7,5,6,3,1] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [5,6,3,7,4,2,1] => [2,4,7,6,5,3,1] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,6,4,3,5,2,1] => [4,3,7,6,5,2,1] => [2,5,4,3,6,7,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,5,2,1] => [6,4,3,7,5,2,1] => [2,5,4,7,6,3,1] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,5,4,3,6,2,1] => [5,4,3,7,6,2,1] => [2,6,4,5,3,7,1] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => [2,7,4,5,6,3,1] => ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,2,1] => [5,6,4,3,7,2,1] => [2,7,4,6,5,3,1] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [7,4,5,3,6,2,1] => [4,5,3,7,6,2,1] => [2,6,5,4,3,7,1] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,7,2,1] => [4,6,5,3,7,2,1] => [2,7,5,4,6,3,1] => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [5,4,6,3,7,2,1] => [2,7,6,5,4,3,1] => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [4,5,6,3,7,2,1] => [2,7,6,4,5,3,1] => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => [2,5,3,4,6,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [3,6,4,7,5,2,1] => [2,5,3,7,6,4,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [7,5,3,4,6,2,1] => [3,5,4,7,6,2,1] => [2,6,3,5,4,7,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => [3,6,5,4,7,2,1] => [2,7,3,5,6,4,1] => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [5,3,6,4,7,2,1] => [2,7,6,5,3,4,1] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,4,3,5,6,2,1] => [4,3,5,7,6,2,1] => [2,6,4,3,5,7,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,7,2,1] => [4,3,6,5,7,2,1] => [2,7,4,3,6,5,1] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [5,4,3,6,7,2,1] => [2,7,4,5,3,6,1] => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [4,5,3,6,7,2,1] => [2,7,5,4,3,6,1] => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => [2,6,3,4,5,7,1] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => [3,4,6,5,7,2,1] => [2,7,3,4,6,5,1] => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [3,5,4,6,7,2,1] => [2,7,3,5,4,6,1] => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [4,3,5,6,7,2,1] => [2,7,4,3,5,6,1] => ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => [2,7,3,4,5,6,1] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [6,2,7,5,4,3,1] => [3,4,5,7,6,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [5,2,7,6,4,3,1] => [3,4,6,5,2,7,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,2,3,1] => [6,5,2,7,4,3,1] => [3,4,7,5,6,2,1] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [5,6,2,7,4,3,1] => [3,4,7,6,5,2,1] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [4,2,7,6,5,3,1] => [3,5,4,2,6,7,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [6,4,2,7,5,3,1] => [3,5,4,7,6,2,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [7,5,4,6,2,3,1] => [5,4,2,7,6,3,1] => [3,6,4,5,2,7,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,2,3,1] => [6,5,4,2,7,3,1] => [3,7,4,5,6,2,1] => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [5,6,4,2,7,3,1] => [3,7,4,6,5,2,1] => ? = 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: St000316

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of non-left-to-right-maxima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.

Matching statistic: St000358

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.

Matching statistic: St001330
(load all 2 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%

Values

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

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001882
(load all 3 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of occurrences of a type-B 231 pattern in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.

Matching statistic: St001866
(load all 2 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%

Values

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

Description

The nesting alignments of a signed permutation. A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that * $-i < -j < -\pi(j) < -\pi(i)$, or * $-i < j \leq \pi(j) < -\pi(i)$, or * $i < j \leq \pi(j) < \pi(i)$.