- FindStat

Your data matches 405 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000222
(load all 26 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of alignments in the permutation.

Matching statistic: St001377
(load all 4 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001377: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The major index minus the number of inversions of a permutation. This is, the difference between [[St000004]] and [[St000018]].

Matching statistic: St001810
(load all 4 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001810: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation smaller than its largest moved point.

Matching statistic: St000530
(load all 25 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000530: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of permutations with the same descent word as the given permutation. The descent word of a permutation is the binary word given by [[Mp00109]]. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map [[Mp00109]] containing a given permutation. This statistic appears as ''up-down analysis'' in statistical applications in genetics, see [1] and the references therein.

Matching statistic: St001958
(load all 63 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the polynomial interpolating the values of a permutation. Given a permutation

$\pi\in\mathfrak S_n$ there is a polynomial

$p$ of minimal degree such that

$p(n)=\pi(n)$ for

$n\in\{1,\dots,n\}$ . This statistic records the degree of

$p$ .

Matching statistic: St000018
(load all 4 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inversions of a permutation. This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length of

$\pi$ .

Matching statistic: St000019
(load all 11 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of the support of a permutation. A permutation

$\sigma$ may be written as a product

$\sigma = s_{i_1}\dots s_{i_k}$ with

$k$ minimal, where

$s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions

$i$ and

$i+1$ . The set of indices

$\{i_1,\dots,i_k\}$ is the '''support''' of

$\sigma$ and independent of the chosen way to write

$\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of

$\sigma$ of length

$n$ is the set of indices

$1 \leq i < n$ such that

$\sigma(k) < i$ for all

$k < i$ . Thus, the connectivity set is the complement of the support.

Matching statistic: St000029
(load all 9 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The depth of a permutation. This is given by

$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3]

$\sum_i |\sigma_i-i|$ . Permutations with depth at most

$1$ are called ''almost-increasing'' in [5].

Matching statistic: St000030
(load all 10 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the descent differences of a permutations. This statistic is given by

$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$ See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.

Matching statistic: St000057
(load all 4 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.

The following 395 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000155The number of exceedances (also excedences) of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000216The absolute length of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000224The sorting index of a permutation. St000237The number of small exceedances. St000241The number of cyclical small excedances. St000293The number of inversions of a binary word. St000359The number of occurrences of the pattern 23-1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000670The reversal length of a permutation. St000682The Grundy value of Welter's game on a binary word. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000957The number of Bruhat lower covers of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001402The number of separators in a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001586The number of odd parts smaller than the largest even part in an integer partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001699The major index of a standard tableau minus the weighted size of its shape. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001911A descent variant minus the number of inversions. St000003The number of standard Young tableaux of the partition. St000047The number of standard immaculate tableaux of a given shape. St000071The number of maximal chains in a poset. St000075The orbit size of a standard tableau under promotion. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000868The aid statistic in the sense of Shareshian-Wachs. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St001090The number of pop-stack-sorts needed to sort a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001246The maximal difference between two consecutive entries of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001313The number of Dyck paths above the lattice path given by a binary word. St001510The number of self-evacuating linear extensions of a finite poset. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000058The order of a permutation. St000485The length of the longest cycle of a permutation. St001486The number of corners of the ribbon associated with an integer composition. St000004The major index of a permutation. St000021The number of descents of a permutation. St000022The number of fixed points of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000035The number of left outer peaks of a permutation. St000039The number of crossings of a permutation. St000089The absolute variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000156The Denert index of a permutation. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000217The number of occurrences of the pattern 312 in a permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000295The length of the border of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000497The lcb statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000653The last descent of a permutation. St000662The staircase size of the code of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000747A variant of the major index of a set partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000794The mak of a permutation. St000795The mad of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000836The number of descents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000956The maximal displacement of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001077The prefix exchange distance of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001082The number of boxed occurrences of 123 in a permutation. St001114The number of odd descents of a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001176The size of a partition minus its first part. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001388The number of non-attacking neighbors of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001428The number of B-inversions of a signed permutation. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001584The area statistic between a Dyck path and its bounce path. St001593This is the number of standard Young tableaux of the given shifted shape. St001644The dimension of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. 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. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001841The number of inversions of a set partition. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001910The height of the middle non-run of a Dyck path. St001928The number of non-overlapping descents in a permutation. St001962The proper pathwidth of a graph. St000001The number of reduced words for a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000060The greater neighbor of the maximum. St000110The number of permutations less than or equal to a permutation in left weak order. St000299The number of nonisomorphic vertex-induced subtrees. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000325The width of the tree associated to a permutation. St000388The number of orbits of vertices of a graph under automorphisms. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000638The number of up-down runs of a permutation. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000797The stat`` of a permutation. St000820The number of compositions obtained by rotating the composition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000844The size of the largest block in the direct sum decomposition of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001093The detour number of a graph. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001108The 2-dynamic chromatic number of a graph. St001128The exponens consonantiae of a partition. St001220The width of a permutation. St001312Number of parabolic noncrossing partitions indexed by the composition. St001352The number of internal nodes in the modular decomposition of a graph. St001375The pancake length of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001497The position of the largest weak excedence of a permutation. St001595The number of standard Young tableaux of the skew partition. St001674The number of vertices of the largest induced star graph in the graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001883The mutual visibility number of a graph. St001884The number of borders of a binary word. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000026The position of the first return of a Dyck path. St000147The largest part of an integer partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000378The diagonal inversion number of an integer partition. St000504The cardinality of the first block of a set partition. St000668The least common multiple of the parts of the partition. St000722The number of different neighbourhoods in a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St001458The rank of the adjacency matrix of a graph. St001746The coalition number of a graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000219The number of occurrences of the pattern 231 in a permutation. St000045The number of linear extensions of a binary tree. St001330The hat guessing number of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000422The energy of a graph, if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001498The normalised height of a Nakayama algebra with magnitude 1. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. 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

$eA$ . St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001811The Castelnuovo-Mumford regularity of a permutation. St000516The number of stretching pairs of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001438The number of missing boxes of a skew partition. St001535The number of cyclic alignments of a permutation. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001684The reduced word complexity of a permutation. St001769The reflection length of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001894The depth of a signed permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001822The number of alignments of a signed permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001555The order of a signed permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001623The number of doubly irreducible elements of a lattice. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St001130The number of two successive successions in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001741The largest integer such that all patterns of this size are contained in the permutation.