- FindStat

Your data matches 376 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001208
(load all 253 compositions to match this statistic)

Mapping

St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The 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

$K[x]/(x^n)$ .

Matching statistic: St000664
(load all 33 compositions to match this statistic)

Mapping

Mp00277: Permutations —catalanization⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => 0
[3,1,2] => [2,3,1] => 0
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,3,4,2] => 0
[1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [2,3,1,4] => 0
[2,3,4,1] => [2,3,4,1] => 0
[2,4,1,3] => [4,3,1,2] => 0
[2,4,3,1] => [2,4,3,1] => 0
[3,1,2,4] => [2,3,1,4] => 0
[3,1,4,2] => [2,3,4,1] => 0
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [3,2,4,1] => 0
[3,4,1,2] => [4,3,2,1] => 0
[3,4,2,1] => [3,4,2,1] => 0
[4,1,2,3] => [2,3,4,1] => 0
[4,1,3,2] => [2,4,3,1] => 0
[4,2,1,3] => [3,2,4,1] => 0
[4,2,3,1] => [3,4,2,1] => 0
[4,3,1,2] => [3,4,2,1] => 0
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => 0
[1,3,5,2,4] => [1,5,4,2,3] => 0
[1,3,5,4,2] => [1,3,5,4,2] => 0
[1,4,2,3,5] => [1,3,4,2,5] => 0
[1,4,2,5,3] => [1,3,4,5,2] => 0
[1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,4,3,5,2] => 0
[1,4,5,2,3] => [1,5,4,3,2] => 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: St000122
(load all 18 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000185

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The weighted size of a partition. Let

$\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of

$\lambda$ is

$\sum_{i=0}^m i \cdot \lambda_i.$ This is also the sum of the leg lengths of the cells in

$\lambda$ , or

$\sum_i \binom{\lambda^{\prime}_i}{2}$ where

$\lambda^{\prime}$ is the conjugate partition of

$\lambda$ . This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2]. This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape

$\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$ , obtained uniquely by placing

$i-1$ in all the cells of the

$i$ th row of

$\lambda$ , see [2, eq.7.103].

Matching statistic: St000407
(load all 4 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000407: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 2143 in a permutation. A permutation

$\pi$ avoids this pattern if and only if it is ''vexillary'' as introduced in [1].

Matching statistic: St000481

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of upper covers of a partition in dominance order.

Matching statistic: St001025
(load all 17 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001175
(load all 39 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001682
(load all 3 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001682: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.

Matching statistic: St001906
(load all 28 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. Let

$\pi$ be a permutation. Its total displacement [[St000830]] is

$D(\pi) = \sum_i |\pi(i) - i|$ , and its absolute length [[St000216]] is the minimal number

$T(\pi)$ of transpositions whose product is

$\pi$ . Finally, let

$I(\pi)$ be the number of inversions [[St000018]] of

$\pi$ . This statistic equals

$\left(D(\pi)-T(\pi)-I(\pi)\right)/2$ . Diaconis and Graham [1] proved that this statistic is always nonnegative.

The following 366 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000183The side length of the Durfee square of an integer partition. St000346The number of coarsenings of a partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000017The number of inversions of a standard tableau. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000057The Shynar inversion number of a standard tableau. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000142The number of even parts of a partition. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000317The cycle descent number of a permutation. St000365The number of double ascents of 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

$3$ . St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000660The number of rises of length at least 3 of a Dyck path. St000666The number of right tethers of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000768The number of peaks in an integer composition. 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. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. 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. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama 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. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001394The genus of a permutation. St001513The number of nested exceedences 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. St001596The number of two-by-two squares inside a skew partition. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000352The Elizalde-Pak rank of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000628The balance of a binary word. St000701The protection number of a binary tree. St000775The multiplicity of the largest eigenvalue in a graph. St000805The number of peaks of the associated bargraph. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001282The number of graphs with the same chromatic polynomial. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001393The induced matching number of a graph. St001399The distinguishing number of a poset. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001597The Frobenius rank of a skew partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001732The number of peaks visible from the left. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000007The number of saliances of the permutation. St000542The number of left-to-right-minima of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000779The tier of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001162The minimum jump of a permutation. St000842The breadth of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000563The number of overlapping pairs of blocks of a set partition. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000872The number of very big descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000963The 2-shifted major index of a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001730The number of times the path corresponding to a binary word crosses the base line. St000031The number of cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St001220The width of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001271The competition number of a graph. 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. St000934The 2-degree of an integer partition. 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. St001626The number of maximal proper sublattices of a lattice. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001793The difference between the clique number and the chromatic number of a graph. St000323The minimal crossing number of a graph. St000552The number of cut vertices of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a 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. St001797The number of overfull subgraphs of a graph. St000627The exponent of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. 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

$eA$ . St001570The minimal number of edges to add to make a graph Hamiltonian. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. 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. St000322The skewness of a graph. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001281The normalized isoperimetric number of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St000914The sum of the values of the Möbius function of a poset. St001307The number of induced stars on four vertices in a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000095The number of triangles of a graph. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions 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

$4$ . St000347The inversion sum of a binary word. St000567The sum of the products of all pairs of parts. St000629The defect of a binary word. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000995The largest even part of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. 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. 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. 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. 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. St001172The number of 1-rises at odd height of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

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

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. 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. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. 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. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001871The number of triconnected components of a graph. St000478Another weight of a partition according to Alladi. St001141The number of occurrences of hills of size 3 in a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St000781The number of proper colouring schemes of a Ferrers diagram. St000759The smallest missing part in an integer partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000897The number of different multiplicities of parts of an integer partition. St000475The number of parts equal to 1 in a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001964The interval resolution global dimension of a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000850The number of 1/2-balanced pairs in a poset. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001877Number of indecomposable injective modules with projective dimension 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000068The number of minimal elements in a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St000069The number of maximal elements of a poset. St001621The number of atoms of a lattice. St001330The hat guessing number of a graph. St000941The number of characters of the symmetric group whose value on the partition is even. St001568The smallest positive integer that does not appear twice in the partition. St001868The number of alignments of type NE of a signed permutation. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000405The number of occurrences of the pattern 1324 in a permutation. St001862The number of crossings of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001323The independence gap of a 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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001811The Castelnuovo-Mumford regularity of a permutation. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. 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

$c_{n−1}$ . St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001344The neighbouring number of a permutation. St001363The Euler characteristic of a graph according to Knill. St001739The number of graphs with the same edge polytope as the given graph. St001768The number of reduced words of a signed permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St000259The diameter of a connected graph. St000260The radius 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. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001268The size of the largest ordinal summand in the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St000264The girth of a graph, which is not a tree. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000907The number of maximal antichains of minimal length in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000948The chromatic discriminant of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001347The number of pairs of vertices of a graph having the same neighbourhood. St000097The order of the largest clique of the graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000917The open packing number of a graph. St001654The monophonic hull number of a graph. St001139The number of occurrences of hills of size 2 in a Dyck path. St000981The length of the longest zigzag subpath.