- FindStat

Your data matches 237 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001514
(load all 16 compositions to match this statistic)

Mapping

St001514: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.

Matching statistic: St000836
(load all 22 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of distance 2 of a permutation. This is,

$\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$ .

Matching statistic: St000837
(load all 29 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of distance 2 of a permutation. This is,

$\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$ .

Matching statistic: St001315

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dissociation number of a graph.

Matching statistic: St000331

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''

$A^{k}$

$B^{k}$ ''' for any '''

${k ≥ 1}$ ''', where '''

${A}$ ''' is a down-step and '''

${B}$ ''' is a up-step.

Matching statistic: St000672

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$ for some

$(r,a,b)$ . This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

Matching statistic: St000483

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000483: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of times a permutation switches from increasing to decreasing or decreasing to increasing. This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]

Matching statistic: St001087
(load all 6 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the vincular pattern |12-3 in a permutation. This is the number of occurrences of the pattern

$123$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.

Matching statistic: St001687
(load all 3 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001388
(load all 15 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001388: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-attacking neighbors of a permutation. For a permutation

$\sigma$ , the indices

$i$ and

$i+1$ are attacking if

$|\sigma(i)-\sigma(i+1)| = 1$ . Visually, this is, for

$\sigma$ considered as a placement of kings on a chessboard, if the kings placed in columns

$i$ and

$i+1$ are non-attacking.

The following 227 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000619The number of cyclic descents of a permutation. St000638The number of up-down runs of a permutation. St000863The length of the first row of the shifted shape of a permutation. St001554The number of distinct nonempty subtrees of a binary tree. St000454The largest eigenvalue of a graph if it is integral. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000670The reversal length of a permutation. St001638The book thickness of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000538The number of even inversions of a permutation. St000632The jump number of the poset. St000871The number of very big ascents of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001637The number of (upper) dissectors of a poset. St000092The number of outer peaks of a permutation. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001712The number of natural descents of a standard Young tableau. St001822The number of alignments of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000628The balance of 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

$K[x]/(x^n)$ . St000307The number of rowmotion orbits of a poset. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000891The number of distinct diagonal sums of a permutation matrix. St000640The rank of the largest boolean interval in a poset. St000741The Colin de Verdière graph invariant. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001812The biclique partition number of a graph. St001624The breadth of a lattice. St001083The number of boxed occurrences of 132 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001488The number of corners of a skew partition. St000862The number of parts of the shifted shape of a permutation. St000647The number of big descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000075The orbit size of a standard tableau under promotion. St001863The number of weak excedances of a signed permutation. St000402Half the size of the symmetry class of a permutation. St001864The number of excedances of a signed permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001060The distinguishing index of a graph. St000665The number of rafts of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001487The number of inner corners of a skew partition. St001896The number of right descents of a signed permutations. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001964The interval resolution global dimension of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000834The number of right outer peaks of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000562The number of internal points of a set partition. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St001645The pebbling number of a connected graph. St001866The nesting alignments of a signed permutation. St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St000422The energy of a graph, if it is integral. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000937The number of positive values of the symmetric group character corresponding to the partition. St001176The size of a partition minus its first part. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000356The number of occurrences of the pattern 13-2. St001394The genus of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000630The length of the shortest palindromic decomposition of a binary word. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001566The length of the longest arithmetic progression in a permutation. St001642The Prague dimension of a graph. St001672The restrained domination number of a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000291The number of descents of a binary word. St000354The number of recoils of a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St000570The Edelman-Greene number of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000829The Ulam distance of a permutation to the identity permutation. St000886The number of permutations with the same antidiagonal sums. St000899The maximal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001405The number of bonds in a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001423The number of distinct cubes in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001768The number of reduced words of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001823The Stasinski-Voll length of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001935The number of ascents in a parking function. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000407The number of occurrences of the pattern 2143 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001403The number of vertical separators in a permutation. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001811The Castelnuovo-Mumford regularity of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000824The sum of the number of descents and the number of recoils of a permutation. St000907The number of maximal antichains of minimal length in a poset. St001516The number of cyclic bonds of a permutation. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000906The length of the shortest maximal chain in a poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001948The number of augmented double ascents of a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset.