- FindStat

Your data matches 314 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000807
(load all 29 compositions to match this statistic)

Mapping

St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the heights of the valleys of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.

Matching statistic: St000805
(load all 29 compositions to match this statistic)

Mapping

St000805: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of peaks of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.

Matching statistic: St000132
(load all 6 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[[.,.],[.,[[.,.],.]]]}}} in a binary tree. [[oeis:A159773]] counts binary trees avoiding this pattern.

Matching statistic: St000664
(load all 9 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of right ropes of a permutation. Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$. See Definition 3.10 and Example 3.11 in [1].

Matching statistic: St001292
(load all 11 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Here $A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].

Matching statistic: St000091

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The descent variation of a composition. Defined in [1].

Matching statistic: St000121
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000121: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,.]]]]}}} in a binary tree. [[oeis:A036765]] counts binary trees avoiding this pattern.

Matching statistic: St000125

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000125: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree. [[oeis:A005773]] counts binary trees avoiding this pattern.

Matching statistic: St000127
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
St000127: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,[[.,.],.]]]]}}} in a binary tree. [[oeis:A159768]] counts binary trees avoiding this pattern.

Matching statistic: St000130
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000130: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[[.,.],[[.,.],.]]]}}} in a binary tree. [[oeis:A159771]] counts binary trees avoiding this pattern.

The following 304 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000761The number of ascents in an integer composition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. 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. 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. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001715The number of non-records in a permutation. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000390The number of runs of ones in a binary word. St000781The number of proper colouring schemes of a Ferrers diagram. St000785The number of distinct colouring schemes of a graph. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001282The number of graphs with the same chromatic polynomial. St001344The neighbouring number of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001722The number of minimal chains with small intervals between a binary word and the top element. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000562The number of internal points of a set partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000570The Edelman-Greene number of 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 $eA$. 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. St001175The size of a partition minus the hook length of the base cell. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001490The number of connected components of a skew partition. St001845The number of join irreducibles minus the rank of a lattice. 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. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001964The interval resolution global dimension of a poset. St001487The number of inner corners of a skew partition. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000297The number of leading ones in a binary word. 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. St000068The number of minimal elements in a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000260The radius of a connected graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. 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 $K[x]/(x^n)$. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000629The defect of a binary word. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000699The toughness times the least common multiple of 1,. St000787The number of flips required to make a perfect matching noncrossing. St000894The trace of an alternating sign matrix. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001163The number of simple modules with dominant dimension at least three 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. 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. St000056The decomposition (or block) number of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000900The minimal number of repetitions of a part in an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001260The permanent of an alternating sign matrix. 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. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect 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 $eA$. 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 $Af$. St000373The number of weak exceedences of a permutation that are also 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. 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. St000534The number of 2-rises of a permutation. St000731The number of double exceedences of a permutation. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000806The semiperimeter of the associated bargraph. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000478Another weight of a partition according to Alladi. St000486The number of cycles of length at least 3 of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000516The number of stretching pairs of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 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. St000872The number of very big descents of a permutation. St000934The 2-degree of an integer partition. 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. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001705The number of occurrences of the pattern 2413 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001847The number of occurrences of the pattern 1432 in 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. St000021The number of descents of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000284The Plancherel distribution on integer partitions. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000729The minimal arc length of a set 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001162The minimum jump 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. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001806The upper middle entry of 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. St000058The order of a permutation. St000084The number of subtrees. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. 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. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001060The distinguishing index of a graph. 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. St000219The number of occurrences of the pattern 231 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. 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. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000618The number of self-evacuating tableaux of given shape. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. 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. St001645The pebbling number of a connected graph. 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. 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. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation.