- FindStat

Your data matches 540 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000004
(load all 21 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The major index of a permutation. This is the sum of the positions of its descents,

$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$ Its generating function is

$[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for

$[k]_q = 1 + q + q^2 + \dots q^{k-1}$ . A statistic equidistributed with the major index is called '''Mahonian statistic'''.

Matching statistic: St000018
(load all 13 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000018: 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]
 => [1,0,1,0]
 => [1,2] => 0
[1,1,0,0]
 => [1,1,0,0]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 0
[]
 => []
 => [] => 0

Description

The number of inversions of a permutation. This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length of

$\pi$ .

Matching statistic: St000147
(load all 9 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000330
(load all 3 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The (standard) major index of a standard tableau. A descent of a standard tableau

$T$ is an index

$i$ such that

$i+1$ appears in a row strictly below the row of

$i$ . The (standard) major index is the the sum of the descents.

Matching statistic: St001033
(load all 8 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].

Matching statistic: St001090
(load all 10 compositions to match this statistic)

Mapping

Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001090: 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]
 => [1,0,1,0]
 => [1,2] => 0
[1,1,0,0]
 => [1,1,0,0]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 0
[]
 => []
 => [] => 0

Description

The number of pop-stack-sorts needed to sort a permutation. The pop-stack sorting operator is defined as follows. Process the permutation

$\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order. A permutation is

$t$ -pop-stack sortable if it is sortable using

$t$ pop-stacks in series.

Matching statistic: St001504
(load all 38 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000009

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The charge of a standard tableau.

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000012

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences'''

$(a_1,\ldots,a_n)$ such that

$a_1 = 0, a_{k+1} \leq a_k + 1$ . 2. The generating function

$\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence

$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of

$q,t$ -Catalan numbers.

The following 530 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000065The number of entries equal to -1 in an alternating sign matrix. St000089The absolute variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000141The maximum drop size of a permutation. St000169The cocharge of a standard tableau. St000204The number of internal nodes of a binary tree. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000245The number of ascents of a permutation. St000246The number of non-inversions of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000496The rcs statistic of a set partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000676The number of odd rises of a Dyck path. St000703The number of deficiencies of a permutation. St000845The maximal number of elements covered by an element in a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000921The number of internal inversions of a binary word. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001161The major index north count of a Dyck path. St001377The major index minus the number of inversions of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001843The Z-index of a set partition. St000054The first entry of the permutation. St000105The number of blocks in the set partition. St000345The number of refinements of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000839The largest opener of a set partition. St000935The number of ordered refinements of an integer partition. St000305The inverse major index of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000740The last entry of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001497The position of the largest weak excedence of a permutation. St000005The bounce statistic of a Dyck path. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000067The inversion number of the alternating sign matrix. St000156The Denert index of a permutation. St000224The sorting index of a permutation. St000293The number of inversions of a binary word. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000670The reversal length of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St000240The number of indices that are not small excedances. St000638The number of up-down runs of a permutation. St000738The first entry in the last row of a standard tableau. St000008The major index of the composition. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000225Difference between largest and smallest parts in a partition. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000290The major index of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000354The number of recoils of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000446The disorder of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000651The maximal size of a rise in a permutation. St000653The last descent of a permutation. St000682The Grundy value of Welter's game on a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000794The mak of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001411The number of patterns 321 or 3412 in a permutation. St001428The number of B-inversions of a signed permutation. St001479The number of bridges of a graph. St001485The modular major index of a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001759The Rajchgot index of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001869The maximum cut size of a graph. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000086The number of subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000494The number of inversions of distance at most 3 of a permutation. St000501The size of the first part in the decomposition of a permutation. St000722The number of different neighbourhoods in a graph. St000728The dimension of a set partition. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001463The number of distinct columns in the nullspace of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St000708The product of the parts of an integer partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001814The number of partitions interlacing the given partition. St001480The number of simple summands of the module J^2/J^3. St000216The absolute length of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000693The modular (standard) major index of a standard tableau. St000840The number of closers smaller than the largest opener in a perfect matching. St000956The maximal displacement of a permutation. 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. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000289The decimal representation of a binary word. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000391The sum of the positions of the ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000539The number of odd inversions of a permutation. St000730The maximal arc length of a set partition. St000792The Grundy value for the game of ruler on a binary word. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000849The number of 1/3-balanced pairs in a poset. St000874The position of the last double rise in a Dyck path. St000961The shifted major index of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001077The prefix exchange distance of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001742The difference of the maximal and the minimal degree in a graph. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000673The number of non-fixed points of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001346The number of parking functions that give the same permutation. St001959The product of the heights of the peaks of a Dyck path. St000327The number of cover relations in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001877Number of indecomposable injective modules with projective dimension 2. St000516The number of stretching pairs of a permutation. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001118The acyclic chromatic index of a graph. St001209The pmaj statistic of a parking function. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001498The normalised height of a Nakayama algebra with magnitude 1. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St000143The largest repeated part of a partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000340The number of non-final maximal constant sub-paths of length greater than one. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001280The number of parts of an integer partition that are at least two. 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. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St000668The least common multiple of the parts of the partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000741The Colin de Verdière graph invariant. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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

$eA$ . St000806The semiperimeter of the associated bargraph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001645The pebbling number of a connected graph. St001733The number of weak left to right maxima of a Dyck path. St000259The diameter of a connected graph. St000707The product of the factorials of the parts. 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. St000933The number of multipartitions of sizes given by an integer partition. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000934The 2-degree of an integer partition. St000565The major index of a set partition. St001822The number of alignments of a signed permutation. St000742The number of big ascents of a permutation after prepending zero. St001778The largest greatest common divisor of an element and its image in a permutation. St000173The segment statistic of a semistandard tableau. St000233The number of nestings of a set partition. St000365The number of double ascents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000562The number of internal points of a set partition. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000836The number of descents of distance 2 of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St001806The upper middle entry of a permutation. St001060The distinguishing index of a graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. 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. 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. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000456The monochromatic index of a connected graph. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000893The number of distinct diagonal sums of an alternating sign matrix. St000264The girth of a graph, which is not a tree. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St000993The multiplicity of the largest part of an integer partition. St001087The number of occurrences of the vincular pattern |12-3 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

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

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001857The number of edges in the reduced word graph of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000079The number of alternating sign matrices for a given Dyck path. St000100The number of linear extensions of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000284The Plancherel distribution on integer partitions. St000315The number of isolated vertices of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000527The width of the poset. St000567The sum of the products of all pairs of parts. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St000909The number of maximal chains of maximal size in a poset. St000929The constant term of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to 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. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001114The number of odd descents of a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001151The number of blocks with odd minimum. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001413Half the length of the longest even length palindromic prefix of a binary word. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001933The largest multiplicity of a part in an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000927The alternating 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001566The length of the longest arithmetic progression in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001961The sum of the greatest common divisors of all pairs of parts. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St000736The last entry in the first row of a semistandard tableau. St001569The maximal modular displacement of a permutation. St000632The jump number of the poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000879The number of long braid edges in the graph of braid moves of a permutation. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001684The reduced word complexity of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000846The maximal number of elements covering an element of a poset. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001487The number of inner corners of a skew partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001571The Cartan determinant of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001651The Frankl number of a lattice. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001823The Stasinski-Voll length of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001937The size of the center of a parking function. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000739The first entry in the last row of a semistandard tableau. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001330The hat guessing number of a graph. 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. St001541The Gini index of an integer partition. St001657The number of twos in an integer partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St000474Dyson's crank of a partition.