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Your data matches 175 different statistics following compositions of up to 3 maps.
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Matching statistic: St001557
(load all 5 compositions to match this statistic)

Mapping

St001557: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inversions of the second entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

Matching statistic: St000054
(load all 17 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation

$\pi$ of

$n$ , together with its rotations, obtained by conjugating with the long cycle

$(1,\dots,n)$ . Drawing the labels

$1$ to

$n$ in this order on a circle, and the arcs

$(i, \pi(i))$ as straight lines, the rotation of

$\pi$ is obtained by replacing each number

$i$ by

$(i\bmod n) +1$ . Then,

$\pi(1)-1$ is the number of rotations of

$\pi$ where the arc

$(1, \pi(1))$ is a deficiency. In particular, if

$O(\pi)$ is the orbit of rotations of

$\pi$ , then the number of deficiencies of

$\pi$ equals

$\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).$

Matching statistic: St000740
(load all 7 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last entry of a permutation. This statistic is undefined for the empty permutation.

Matching statistic: St001806

Mapping

Mp00252: Permutations —restriction⟶ Permutations
St001806: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The upper middle entry of a permutation. This is the entry

$\sigma(\frac{n+1}{2})$ when

$n$ is odd, and

$\sigma(\frac{n}{2}+1)$ when

$n$ is even, where

$n$ is the size of the permutation

$\sigma$ .

Matching statistic: St001807
(load all 2 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
St001807: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The lower middle entry of a permutation. This is the entry

$\sigma(\frac{n+1}{2})$ when

$n$ is odd, and

$\sigma(\frac{n}{2})$ when

$n$ is even, where

$n$ is the size of the permutation

$\sigma$ .

Matching statistic: St000051
(load all 2 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of the left subtree of a binary tree.

Matching statistic: St000133
(load all 3 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The "bounce" of a permutation.

Matching statistic: St001227
(load all 4 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

Matching statistic: St000025
(load all 4 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000066
(load all 4 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The column of the unique '1' in the first row of the alternating sign matrix. The generating function of this statistic is given by

$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$ see [2].

The following 165 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000839The largest opener of a set partition. St000971The smallest closer of a set partition. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000439The position of the first down step of a Dyck path. St000010The length of the partition. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000316The number of non-left-to-right-maxima of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001161The major index north count of a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001434The number of negative sum pairs of a signed permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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. St001497The position of the largest weak excedence of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000738The first entry in the last row of a standard tableau. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000693The modular (standard) major index of a standard tableau. St000504The cardinality of the first block of a set partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000989The number of final rises of a permutation. St001480The number of simple summands of the module J^2/J^3. St001927Sparre Andersen's number of positives of a signed permutation. St000061The number of nodes on the left branch of a binary tree. St000485The length of the longest cycle of a permutation. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001280The number of parts of an integer partition that are at least two. St001176The size of a partition minus its first part. St000225Difference between largest and smallest parts in a partition. St000519The largest length of a factor maximising the subword complexity. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001498The normalised height of a Nakayama algebra with magnitude 1. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000681The Grundy value of Chomp on Ferrers diagrams. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000260The radius of a connected graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. 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. 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. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001877Number of indecomposable injective modules with projective dimension 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001568The smallest positive integer that does not appear twice in the partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001964The interval resolution global dimension of a poset. St000741The Colin de Verdière graph invariant. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000456The monochromatic index of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000307The number of rowmotion orbits of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001624The breadth of a lattice. St001651The Frankl number of a lattice. St001556The number of inversions of the third entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001822The number of alignments of a signed permutation. St000392The length of the longest run of ones in a binary word. St000422The energy of a graph, if it is integral. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St001926Sparre Andersen's position of the maximum of a signed permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001875The number of simple modules with projective dimension at most 1. St000284The Plancherel distribution on integer partitions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. 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. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001545The second Elser number of a connected graph. St001623The number of doubly irreducible elements of a lattice. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000264The girth of a graph, which is not a tree.