- FindStat

Your data matches 331 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001195
(load all 81 compositions to match this statistic)

Mapping

St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ .

Matching statistic: St000888
(load all 2 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00003: Alternating sign matrices —rotate counterclockwise⟶ Alternating sign matrices
St000888: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 2 = 1 + 1

Description

The maximal sum of entries on a diagonal of an alternating sign matrix. For example, the sums of the diagonals of the matrix

$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)$ are

$(0,1,1,0,1,1,0)$ , so the statistic is

$1$ . This is a natural extension of [[St000887]] to alternating sign matrices.

Matching statistic: St000481

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000486

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cycles of length at least 3 of a permutation.

Matching statistic: St001085
(load all 8 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern

$213$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.

Matching statistic: St001092

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct even parts of a partition. See Section 3.3.1 of [1].

Matching statistic: St001174
(load all 6 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Gorenstein dimension of the algebra

$A/I$ when

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

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

Matching statistic: St001204
(load all 8 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Call 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. Associate to this special CNakayama algebra a Dyck path as follows: In the list L delete the first entry

$c_0$ and substract from all other entries

$n$ −1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra. The statistic gives the

$(t-1)/2$ when

$t$ is the projective dimension of the simple module

$S_{n-2}$ .

Matching statistic: St001217
(load all 2 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.

Matching statistic: St001273

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001273: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The projective dimension of the first term in an injective coresolution of the regular module. The algebra has the double centraliser property when 0 is returned and it is 1-Gorenstein in case a number < =1 is returned.

The following 321 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001948The number of augmented double ascents of a permutation. St000007The number of saliances of the permutation. St000314The number of left-to-right-maxima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000654The first descent of a permutation. St000701The protection number of a binary tree. St000783The side length of the largest staircase partition fitting into a partition. St000920The logarithmic height of a Dyck path. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001096The size of the overlap set of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001432The order dimension of the partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000451The length of the longest pattern of the form k 1 2. St000659The number of rises of length at least 2 of a Dyck path. St000402Half the size of the symmetry class of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000897The number of different multiplicities of parts of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000260The radius of a connected graph. St000929The constant term of the character polynomial of an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001737The number of descents of type 2 in a permutation. St001933The largest multiplicity of a part in an integer partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000891The number of distinct diagonal sums of a permutation matrix. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000517The Kreweras number of an integer partition. St000913The number of ways to refine the partition into singletons. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001196The global dimension of

$A$ minus the global dimension of

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

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001520The number of strict 3-descents. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001728The number of invisible descents of a permutation. St000651The maximal size of a rise in a permutation. St000741The Colin de Verdière graph invariant. St001618The cardinality of the Frattini sublattice of a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000914The sum of the values of the Möbius function of a poset. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000374The number of exclusive right-to-left minima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000768The number of peaks in an integer composition. St001617The dimension of the space of valuations of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001890The maximum magnitude of the Möbius function of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St000456The monochromatic index of a connected graph. St000871The number of very big ascents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001638The book thickness of a graph. St001820The size of the image of the pop stack sorting operator. St000893The number of distinct diagonal sums of an alternating sign matrix. St000628The balance of a binary word. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St000903The number of different parts of an integer composition. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001730The number of times the path corresponding to a binary word crosses the base line. St001960The number of descents of a permutation minus one if its first entry is not one. St000764The number of strong records in an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. 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

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001424The number of distinct squares in a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001884The number of borders of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. 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. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000100The number of linear extensions of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000635The number of strictly order preserving maps of a poset into itself. St000658The number of rises of length 2 of a Dyck path. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001139The number of occurrences of hills of size 2 in a Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St001615The number of join prime elements of a lattice. St000981The length of the longest zigzag subpath. St001812The biclique partition number of a graph. St001651The Frankl number of a lattice. St001624The breadth of a lattice. St001868The number of alignments of type NE of a signed permutation. St000647The number of big descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000661The number of rises of length 3 of a Dyck path. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001335The cardinality of a minimal cycle-isolating set of a graph. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000925The number of topologically connected components of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000670The reversal length of a permutation. St000068The number of minimal elements in a poset. St000567The sum of the products of all pairs of parts. St000648The number of 2-excedences of a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. 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. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000665The number of rafts of a permutation. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001846The number of elements which do not have a complement in the lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001394The genus of a permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000562The number of internal points of a set partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001153The number of blocks with even minimum in a set partition. St001396Number of triples of incomparable elements in a finite poset. St001895The oddness of a signed permutation. St000251The number of nonsingleton blocks of a set partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000662The staircase size of the code of a permutation. St001769The reflection length of a signed permutation. St001372The length of a longest cyclic run of ones of a binary word. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000441The number of successions of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001083The number of boxed occurrences of 132 in a permutation. St001115The number of even descents of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000031The number of cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000834The number of right outer peaks of a permutation. St001052The length of the exterior of a permutation. St000493The los statistic of a set partition. St001075The minimal size of a block of a set partition. St001863The number of weak excedances of a signed permutation.