- FindStat

Your data matches 715 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000122
(load all 26 compositions to match this statistic)

Mapping

St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => 0
[.,[.,.]]
 => 0
[[.,.],.]
 => 0
[.,[.,[.,.]]]
 => 0
[.,[[.,.],.]]
 => 0
[[.,.],[.,.]]
 => 0
[[.,[.,.]],.]
 => 0
[[[.,.],.],.]
 => 0
[.,[.,[.,[.,.]]]]
 => 0
[.,[.,[[.,.],.]]]
 => 1
[.,[[.,.],[.,.]]]
 => 0
[.,[[.,[.,.]],.]]
 => 0
[.,[[[.,.],.],.]]
 => 0
[[.,.],[.,[.,.]]]
 => 0
[[.,.],[[.,.],.]]
 => 0
[[.,[.,.]],[.,.]]
 => 0
[[[.,.],.],[.,.]]
 => 0
[[.,[.,[.,.]]],.]
 => 0
[[.,[[.,.],.]],.]
 => 0
[[[.,.],[.,.]],.]
 => 0
[[[.,[.,.]],.],.]
 => 0
[[[[.,.],.],.],.]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => 0
[[.,.],[.,[[.,.],.]]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => 0
[[.,.],[[.,[.,.]],.]]
 => 0
[[.,.],[[[.,.],.],.]]
 => 0
[[[.,.],.],[.,[.,.]]]
 => 0
[[[.,.],.],[[.,.],.]]
 => 0
[[[.,.],[.,.]],[.,.]]
 => 0
[[[[.,.],.],.],[.,.]]
 => 0
[[[.,.],[.,[.,.]]],.]
 => 0
[[[.,.],[[.,.],.]],.]
 => 0
[[[[.,.],.],[.,.]],.]
 => 0
[[[[.,.],[.,.]],.],.]
 => 0
[[[[[.,.],.],.],.],.]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[[.,.],.]]]}}} in a binary tree. [[oeis:A086581]] counts binary trees avoiding this pattern.

Matching statistic: St000317
(load all 22 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cycle descent number of a permutation. Let

$(i_1,\ldots,i_k)$ be a cycle of a permutation

$\pi$ such that

$i_1$ is its smallest element. A **cycle descent** of

$(i_1,\ldots,i_k)$ is an

$i_a$ for

$1 \leq a < k$ such that

$i_a > i_{a+1}$ . The **cycle descent set** of

$\pi$ is then the set of descents in all the cycles of

$\pi$ , and the **cycle descent number** is its cardinality.

Matching statistic: St001394
(load all 17 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The genus of a permutation. The genus

$g(\pi)$ of a permutation

$\pi\in\mathfrak S_n$ is defined via the relation

$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$ where

$\zeta = (1,2,\dots,n)$ is the long cycle and

$z(\cdot)$ is the number of cycles in the permutation.

Matching statistic: St001537
(load all 20 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001537: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclic crossings of a permutation. The pair

$(i,j)$ is a cyclic crossing of a permutation

$\pi$ if

$i, \pi(j), \pi(i), j$ are cyclically ordered and all distinct, see Section 5 of [1].

Matching statistic: St001549
(load all 36 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of restricted non-inversions between exceedances. This is for a permutation

$\sigma$ of length

$n$ given by

$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$

Matching statistic: St000034
(load all 10 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum defect over any reduced expression for a permutation and any subexpression.

Matching statistic: St000039
(load all 9 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of crossings of a permutation. A crossing of a permutation

$\pi$ is given by a pair

$(i,j)$ such that either

$i < j \leq \pi(i) \leq \pi(j)$ or

$\pi(i) < \pi(j) < i < j$ . Pictorially, the diagram of a permutation is obtained by writing the numbers from

$1$ to

$n$ in this order on a line, and connecting

$i$ and

$\pi(i)$ with an arc above the line if

$i\leq\pi(i)$ and with an arc below the line if

$i > \pi(i)$ . Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.

Matching statistic: St000217
(load all 12 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000217: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 312 in a permutation.

Matching statistic: St000232
(load all 10 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of crossings of a set partition. This is given by the number of

$i < i' < j < j'$ such that

$i,j$ are two consecutive entries on one block, and

$i',j'$ are consecutive entries in another block.

Matching statistic: St000358
(load all 14 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

$31\!\!-\!\!2$ .

The following 705 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000406The number of occurrences of the pattern 3241 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000664The number of right ropes of a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001310The number of induced diamond graphs in a graph. St001411The number of patterns 321 or 3412 in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000119The number of occurrences of the pattern 321 in a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000233The number of nestings of a set partition. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000648The number of 2-excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000731The number of double exceedences of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000871The number of very big ascents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001535The number of cyclic alignments of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001736The total number of cycles in a graph. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001871The number of triconnected components of a graph. St000001The number of reduced words for a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000570The Edelman-Greene number of a permutation. St000805The number of peaks of the associated bargraph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000913The number of ways to refine the partition into singletons. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000842The breadth of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000292The number of ascents of a binary word. St000348The non-inversion sum of a binary word. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000650The number of 3-rises of a permutation. St000682The Grundy value of Welter's game on a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000872The number of very big descents of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St001220The width of a permutation. St001271The competition number of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St000219The number of occurrences of the pattern 231 in a permutation. 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

$eA$ . St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000629The defect of a binary word. St001845The number of join irreducibles minus the rank of a lattice. St001330The hat guessing number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001868The number of alignments of type NE of a signed permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000445The number of rises of length 1 of a Dyck path. St000264The girth of a graph, which is not a tree. St000929The constant term of the character polynomial of an integer partition. St001964The interval resolution global dimension of a poset. St000068The number of minimal elements in a poset. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000455The second largest eigenvalue of a graph if it is integral. St001141The number of occurrences of hills of size 3 in a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000268The number of strongly connected orientations of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of a graph. St001309The number of four-cliques in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000456The monochromatic index of a connected graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001277The degeneracy of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001826The maximal number of leaves on a vertex of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St000523The number of 2-protected nodes of a rooted tree. St000700The protection number of an ordered tree. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000567The sum of the products of all pairs of parts. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. 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. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000678The number of up steps after the last double rise of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000444The length of the maximal rise of a Dyck path. 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. St000655The length of the minimal rise of a Dyck path. St000439The position of the first down step of a Dyck path. St000981The length of the longest zigzag subpath. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000451The length of the longest pattern of the form k 1 2. 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. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000069The number of maximal elements of a poset. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001625The Möbius invariant of a lattice. St000011The number of touch points (or returns) of a Dyck path. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000617The number of global maxima of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000534The number of 2-rises of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000701The protection number of a binary tree. St000883The number of longest increasing subsequences of a permutation. St000909The number of maximal chains of maximal size in a poset. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000351The determinant of the adjacency matrix of a graph. St000951The dimension of

$Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001357The maximal degree of a regular spanning subgraph of a graph. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000536The pathwidth of a graph. St000640The rank of the largest boolean interval in a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001333The cardinality of a minimal edge-isolating set of a graph. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000061The number of nodes on the left branch of a binary tree. St000469The distinguishing number of a graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St001111The weak 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001399The distinguishing number of a poset. St001716The 1-improper chromatic number of a graph. St001108The 2-dynamic chromatic number of a graph. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000090The variation of a composition. St000095The number of triangles of a graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000312The number of leaves in a graph. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St000365The number of double ascents of a permutation. St000449The number of pairs of vertices of a graph with distance 4. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000943The number of spots the most unlucky car had to go further in a parking function. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St000962The 3-shifted major index of a permutation. St000974The length of the trunk of an ordered tree. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000989The number of final rises of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001057The Grundy value of the game of creating an independent set in a graph. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001306The number of induced paths on four vertices in a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. 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. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001810The number of fixed points of a permutation smaller than its largest moved point. St001857The number of edges in the reduced word graph of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St000021The number of descents of a permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000037The sign of a permutation. St000056The decomposition (or block) number of a permutation. St000096The number of spanning trees of a graph. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000239The number of small weak excedances. St000253The crossing number of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000287The number of connected components of a graph. St000307The number of rowmotion orbits of a poset. St000310The minimal degree of a vertex of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000383The last part of an integer composition. St000450The number of edges minus the number of vertices plus 2 of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000633The size of the automorphism group of a poset. St000654The first descent of a permutation. St000667The greatest common divisor of the parts of the partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000823The number of unsplittable factors of the set partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000876The number of factors in the Catalan decomposition of a binary word. St000889The number of alternating sign matrices with the same antidiagonal sums. St000907The number of maximal antichains of minimal length in a poset. St000910The number of maximal chains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000990The first ascent of a permutation. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001393The induced matching number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. 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. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001806The upper middle entry of a permutation. St001828The Euler characteristic of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001949The rigidity index of a graph. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000273The domination number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000364The exponent of the automorphism group of a graph. St000461The rix statistic of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000916The packing number of a graph. St000991The number of right-to-left minima of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001654The monophonic hull number of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001926Sparre Andersen's position of the maximum of a signed permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St001624The breadth of a lattice. St000741The Colin de Verdière graph invariant. St000782The indicator function of whether a given perfect matching is an L & P matching. 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

$eA$ . St001645The pebbling number of a connected graph. 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. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001510The number of self-evacuating linear extensions of a finite poset. St001596The number of two-by-two squares inside a skew partition. St000680The Grundy value for Hackendot on posets. St000681The Grundy value of Chomp on Ferrers diagrams. St001597The Frobenius rank of a skew partition. St001545The second Elser number of a connected graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001631The number of simple modules

$S$ with

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

$A$ of the poset. St001783The number of odd automorphisms of a graph. St000309The number of vertices with even degree. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001734The lettericity of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000102The charge of a semistandard tableau. St000153The number of adjacent cycles of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000447The number of pairs of vertices of a graph with distance 3. St000488The number of cycles of a permutation of length at most 2. St000552The number of cut vertices of a graph. St000754The Grundy value for the game of removing nestings in a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001307The number of induced stars on four vertices in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001423The number of distinct cubes in a binary word. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001479The number of bridges of a graph. St001520The number of strict 3-descents. St001536The number of cyclic misalignments of a permutation. St001556The number of inversions of the third entry of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001948The number of augmented double ascents of a permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000028The number of stack-sorts needed to sort a permutation. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000553The number of blocks of a graph. St000756The sum of the positions of the left to right maxima of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000996The number of exclusive left-to-right maxima of a permutation. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001272The number of graphs with the same degree sequence. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001405The number of bonds in a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001569The maximal modular displacement of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001590The crossing number of a perfect matching. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000064The number of one-box pattern of a permutation. St000259The diameter of a connected graph. St000311The number of vertices of odd degree in a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000746The number of pairs with odd minimum in a perfect matching. St000778The metric dimension of a graph. St000862The number of parts of the shifted shape of a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000458The number of permutations obtained by switching adjacencies or successions. St000891The number of distinct diagonal sums of a permutation matrix. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001315The dissociation number of a graph. St001342The number of vertices in the center of a graph. St001656The monophonic position number of a graph. St000983The length of the longest alternating subword. St001110The 3-dynamic chromatic number of a graph. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.