- FindStat

Your data matches 590 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000058
(load all 4 compositions to match this statistic)

Mapping

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => 3
[[1],[2],[4]]
 => [[1,2],[4]]
 => [3,1,2] => 3
[[1],[3],[4]]
 => [[1,3],[4]]
 => [3,1,2] => 3
[[2],[3],[4]]
 => [[2,3],[4]]
 => [3,1,2] => 3
[[1],[2],[5]]
 => [[1,2],[5]]
 => [3,1,2] => 3
[[1],[3],[5]]
 => [[1,3],[5]]
 => [3,1,2] => 3
[[1],[4],[5]]
 => [[1,4],[5]]
 => [3,1,2] => 3
[[2],[3],[5]]
 => [[2,3],[5]]
 => [3,1,2] => 3
[[2],[4],[5]]
 => [[2,4],[5]]
 => [3,1,2] => 3
[[3],[4],[5]]
 => [[3,4],[5]]
 => [3,1,2] => 3
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 4
[[1],[2],[6]]
 => [[1,2],[6]]
 => [3,1,2] => 3
[[1],[3],[6]]
 => [[1,3],[6]]
 => [3,1,2] => 3
[[1],[4],[6]]
 => [[1,4],[6]]
 => [3,1,2] => 3
[[1],[5],[6]]
 => [[1,5],[6]]
 => [3,1,2] => 3
[[2],[3],[6]]
 => [[2,3],[6]]
 => [3,1,2] => 3
[[2],[4],[6]]
 => [[2,4],[6]]
 => [3,1,2] => 3
[[2],[5],[6]]
 => [[2,5],[6]]
 => [3,1,2] => 3
[[3],[4],[6]]
 => [[3,4],[6]]
 => [3,1,2] => 3
[[3],[5],[6]]
 => [[3,5],[6]]
 => [3,1,2] => 3
[[4],[5],[6]]
 => [[4,5],[6]]
 => [3,1,2] => 3
[[1],[2],[3],[5]]
 => [[1,2],[3],[5]]
 => [4,3,1,2] => 4
[[1],[2],[4],[5]]
 => [[1,2],[4],[5]]
 => [4,3,1,2] => 4
[[1],[3],[4],[5]]
 => [[1,3],[4],[5]]
 => [4,3,1,2] => 4
[[2],[3],[4],[5]]
 => [[2,3],[4],[5]]
 => [4,3,1,2] => 4
[[1],[2],[7]]
 => [[1,2],[7]]
 => [3,1,2] => 3
[[1],[3],[7]]
 => [[1,3],[7]]
 => [3,1,2] => 3
[[1],[4],[7]]
 => [[1,4],[7]]
 => [3,1,2] => 3
[[1],[5],[7]]
 => [[1,5],[7]]
 => [3,1,2] => 3
[[1],[6],[7]]
 => [[1,6],[7]]
 => [3,1,2] => 3
[[2],[3],[7]]
 => [[2,3],[7]]
 => [3,1,2] => 3
[[2],[4],[7]]
 => [[2,4],[7]]
 => [3,1,2] => 3
[[2],[5],[7]]
 => [[2,5],[7]]
 => [3,1,2] => 3
[[2],[6],[7]]
 => [[2,6],[7]]
 => [3,1,2] => 3
[[3],[4],[7]]
 => [[3,4],[7]]
 => [3,1,2] => 3
[[3],[5],[7]]
 => [[3,5],[7]]
 => [3,1,2] => 3
[[3],[6],[7]]
 => [[3,6],[7]]
 => [3,1,2] => 3
[[4],[5],[7]]
 => [[4,5],[7]]
 => [3,1,2] => 3
[[4],[6],[7]]
 => [[4,6],[7]]
 => [3,1,2] => 3
[[5],[6],[7]]
 => [[5,6],[7]]
 => [3,1,2] => 3
[[1],[2],[3],[6]]
 => [[1,2],[3],[6]]
 => [4,3,1,2] => 4
[[1],[2],[4],[6]]
 => [[1,2],[4],[6]]
 => [4,3,1,2] => 4
[[1],[2],[5],[6]]
 => [[1,2],[5],[6]]
 => [4,3,1,2] => 4
[[1],[3],[4],[6]]
 => [[1,3],[4],[6]]
 => [4,3,1,2] => 4
[[1],[3],[5],[6]]
 => [[1,3],[5],[6]]
 => [4,3,1,2] => 4
[[1],[4],[5],[6]]
 => [[1,4],[5],[6]]
 => [4,3,1,2] => 4
[[2],[3],[4],[6]]
 => [[2,3],[4],[6]]
 => [4,3,1,2] => 4
[[2],[3],[5],[6]]
 => [[2,3],[5],[6]]
 => [4,3,1,2] => 4
[[2],[4],[5],[6]]
 => [[2,4],[5],[6]]
 => [4,3,1,2] => 4
[[3],[4],[5],[6]]
 => [[3,4],[5],[6]]
 => [4,3,1,2] => 4

Description

The order of a permutation.

$\operatorname{ord}(\pi)$ is given by the minimial

$k$ for which

$\pi^k$ is the identity permutation.

Matching statistic: St001004
(load all 24 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 3
[[1],[2],[4]]
 => [3,2,1] => [1,2,3] => 3
[[1],[3],[4]]
 => [3,2,1] => [1,2,3] => 3
[[2],[3],[4]]
 => [3,2,1] => [1,2,3] => 3
[[1],[2],[5]]
 => [3,2,1] => [1,2,3] => 3
[[1],[3],[5]]
 => [3,2,1] => [1,2,3] => 3
[[1],[4],[5]]
 => [3,2,1] => [1,2,3] => 3
[[2],[3],[5]]
 => [3,2,1] => [1,2,3] => 3
[[2],[4],[5]]
 => [3,2,1] => [1,2,3] => 3
[[3],[4],[5]]
 => [3,2,1] => [1,2,3] => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[2],[6]]
 => [3,2,1] => [1,2,3] => 3
[[1],[3],[6]]
 => [3,2,1] => [1,2,3] => 3
[[1],[4],[6]]
 => [3,2,1] => [1,2,3] => 3
[[1],[5],[6]]
 => [3,2,1] => [1,2,3] => 3
[[2],[3],[6]]
 => [3,2,1] => [1,2,3] => 3
[[2],[4],[6]]
 => [3,2,1] => [1,2,3] => 3
[[2],[5],[6]]
 => [3,2,1] => [1,2,3] => 3
[[3],[4],[6]]
 => [3,2,1] => [1,2,3] => 3
[[3],[5],[6]]
 => [3,2,1] => [1,2,3] => 3
[[4],[5],[6]]
 => [3,2,1] => [1,2,3] => 3
[[1],[2],[3],[5]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[2],[4],[5]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[3],[4],[5]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[2],[3],[4],[5]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[2],[7]]
 => [3,2,1] => [1,2,3] => 3
[[1],[3],[7]]
 => [3,2,1] => [1,2,3] => 3
[[1],[4],[7]]
 => [3,2,1] => [1,2,3] => 3
[[1],[5],[7]]
 => [3,2,1] => [1,2,3] => 3
[[1],[6],[7]]
 => [3,2,1] => [1,2,3] => 3
[[2],[3],[7]]
 => [3,2,1] => [1,2,3] => 3
[[2],[4],[7]]
 => [3,2,1] => [1,2,3] => 3
[[2],[5],[7]]
 => [3,2,1] => [1,2,3] => 3
[[2],[6],[7]]
 => [3,2,1] => [1,2,3] => 3
[[3],[4],[7]]
 => [3,2,1] => [1,2,3] => 3
[[3],[5],[7]]
 => [3,2,1] => [1,2,3] => 3
[[3],[6],[7]]
 => [3,2,1] => [1,2,3] => 3
[[4],[5],[7]]
 => [3,2,1] => [1,2,3] => 3
[[4],[6],[7]]
 => [3,2,1] => [1,2,3] => 3
[[5],[6],[7]]
 => [3,2,1] => [1,2,3] => 3
[[1],[2],[3],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[2],[4],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[2],[5],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[3],[4],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[3],[5],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[1],[4],[5],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[2],[3],[4],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[2],[3],[5],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[2],[4],[5],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4
[[3],[4],[5],[6]]
 => [4,3,2,1] => [2,4,1,3] => 4

Description

The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a

$321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].

Matching statistic: St001486
(load all 4 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [3,2,1] => [2,1] => 3
[[1],[2],[4]]
 => [3,2,1] => [2,1] => 3
[[1],[3],[4]]
 => [3,2,1] => [2,1] => 3
[[2],[3],[4]]
 => [3,2,1] => [2,1] => 3
[[1],[2],[5]]
 => [3,2,1] => [2,1] => 3
[[1],[3],[5]]
 => [3,2,1] => [2,1] => 3
[[1],[4],[5]]
 => [3,2,1] => [2,1] => 3
[[2],[3],[5]]
 => [3,2,1] => [2,1] => 3
[[2],[4],[5]]
 => [3,2,1] => [2,1] => 3
[[3],[4],[5]]
 => [3,2,1] => [2,1] => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[2],[6]]
 => [3,2,1] => [2,1] => 3
[[1],[3],[6]]
 => [3,2,1] => [2,1] => 3
[[1],[4],[6]]
 => [3,2,1] => [2,1] => 3
[[1],[5],[6]]
 => [3,2,1] => [2,1] => 3
[[2],[3],[6]]
 => [3,2,1] => [2,1] => 3
[[2],[4],[6]]
 => [3,2,1] => [2,1] => 3
[[2],[5],[6]]
 => [3,2,1] => [2,1] => 3
[[3],[4],[6]]
 => [3,2,1] => [2,1] => 3
[[3],[5],[6]]
 => [3,2,1] => [2,1] => 3
[[4],[5],[6]]
 => [3,2,1] => [2,1] => 3
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,2,1] => 4
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[2],[7]]
 => [3,2,1] => [2,1] => 3
[[1],[3],[7]]
 => [3,2,1] => [2,1] => 3
[[1],[4],[7]]
 => [3,2,1] => [2,1] => 3
[[1],[5],[7]]
 => [3,2,1] => [2,1] => 3
[[1],[6],[7]]
 => [3,2,1] => [2,1] => 3
[[2],[3],[7]]
 => [3,2,1] => [2,1] => 3
[[2],[4],[7]]
 => [3,2,1] => [2,1] => 3
[[2],[5],[7]]
 => [3,2,1] => [2,1] => 3
[[2],[6],[7]]
 => [3,2,1] => [2,1] => 3
[[3],[4],[7]]
 => [3,2,1] => [2,1] => 3
[[3],[5],[7]]
 => [3,2,1] => [2,1] => 3
[[3],[6],[7]]
 => [3,2,1] => [2,1] => 3
[[4],[5],[7]]
 => [3,2,1] => [2,1] => 3
[[4],[6],[7]]
 => [3,2,1] => [2,1] => 3
[[5],[6],[7]]
 => [3,2,1] => [2,1] => 3
[[1],[2],[3],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[2],[4],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[2],[5],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[3],[4],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[3],[5],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[1],[4],[5],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[2],[3],[4],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[2],[3],[5],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[2],[4],[5],[6]]
 => [4,3,2,1] => [1,2,1] => 4
[[3],[4],[5],[6]]
 => [4,3,2,1] => [1,2,1] => 4

Description

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition

$c=(c_1,\dots,c_n)$ with

$c_i$ cells in the

$i$ -th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

Matching statistic: St000339

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000339: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[5],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[3],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1

Description

The maf index of a permutation. Let

$\sigma$ be a permutation with fixed point set

$\operatorname{FIX}(\sigma)$ , and let

$\operatorname{Der}(\sigma)$ be the derangement obtained from

$\sigma$ by removing the fixed points. Then

$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$ where

$\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of

$\sigma$ .

Matching statistic: St001375
(load all 10 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[2],[4]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[3],[4]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[3],[4]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[2],[5]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[3],[5]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[4],[5]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[3],[5]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[4],[5]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[3],[4],[5]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[2],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[3],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[4],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[5],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[3],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[4],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[5],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[3],[4],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[3],[5],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[4],[5],[6]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[2],[4],[5]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[3],[4],[5]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[2],[3],[4],[5]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[2],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[3],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[4],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[5],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[6],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[3],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[4],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[5],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[2],[6],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[3],[4],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[3],[5],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[3],[6],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[4],[5],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[4],[6],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[5],[6],[7]]
 => [3,2,1] => [2,3,1] => 2 = 3 - 1
[[1],[2],[3],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[2],[4],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[2],[5],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[3],[4],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[3],[5],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[1],[4],[5],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[2],[3],[4],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[2],[3],[5],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[2],[4],[5],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1
[[3],[4],[5],[6]]
 => [4,3,2,1] => [3,4,1,2] => 3 = 4 - 1

Description

The pancake length of a permutation. This is the minimal number of pancake moves needed to generate a permutation where a pancake move is a reversal of a prefix in a permutation.

Matching statistic: St001726
(load all 2 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[5],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[3],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1

Description

The number of visible inversions of a permutation. A visible inversion of a permutation

$\pi$ is a pair

$i < j$ such that

$\pi(j) \leq \min(i, \pi(i))$ .

Matching statistic: St001761
(load all 2 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[5]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[5],[6]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[4],[5]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[3],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[3],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[4],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[3],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[5],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[4],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[5],[6],[7]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1],[2],[3],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[2],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[3],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[1],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[4],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[3],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[2],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1
[[3],[4],[5],[6]]
 => [4,3,2,1] => [4,3,1,2] => 3 = 4 - 1

Description

The maximal multiplicity of a letter in a reduced word of a permutation. For example, the permutation

$3421$ has the reduced word

$s_2 s_1 s_2 s_3 s_2$ , where

$s_2$ appears three times.

Matching statistic: St000013

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[4],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[4],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[4],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[4],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[4],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[4],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[5],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[3],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[4],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[3],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000025

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[4],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[3],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[3],[4],[5]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[3],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[4],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[3],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[4],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[2],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[4],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[3],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[4],[5],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[4],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[5],[6],[7]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[[1],[2],[3],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[4],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[1],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[3],[4],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[3],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[2],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[3],[4],[5],[6]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4

Description

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

$D$ .

Matching statistic: St000031
(load all 2 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[2],[4]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[3],[4]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[3],[4]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[2],[5]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[3],[5]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[4],[5]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[3],[5]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[4],[5]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[3],[4],[5]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[2],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[3],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[4],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[5],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[3],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[4],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[5],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[3],[4],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[3],[5],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[4],[5],[6]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[2],[3],[5]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[2],[4],[5]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[3],[4],[5]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[2],[3],[4],[5]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[2],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[3],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[4],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[5],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[6],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[3],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[4],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[5],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[2],[6],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[3],[4],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[3],[5],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[3],[6],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[4],[5],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[4],[6],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[5],[6],[7]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1],[2],[3],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[2],[4],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[2],[5],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[3],[4],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[3],[5],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[1],[4],[5],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[2],[3],[4],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[2],[3],[5],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[2],[4],[5],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4
[[3],[4],[5],[6]]
 => [4,3,2,1] => [1,3,2,4] => [1,2,3,4] => 4

Description

The number of cycles in the cycle decomposition of a permutation.

The following 580 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000056The decomposition (or block) number of a permutation. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000203The number of external nodes of a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000638The number of up-down runs of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000784The maximum of the length and the largest part of the integer partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000863The length of the first row of the shifted shape of a permutation. St000918The 2-limited packing number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000007The number of saliances of the permutation. St000024The number of double up and double down steps of a Dyck path. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000209Maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000439The position of the first down step of a Dyck path. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000868The aid statistic in the sense of Shareshian-Wachs. St000891The number of distinct diagonal sums of a permutation matrix. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001245The cyclic maximal difference between two consecutive entries of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001760The number of prefix or suffix reversals needed to sort a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000316The number of non-left-to-right-maxima of a permutation. St000539The number of odd inversions of a permutation. St000956The maximal displacement of a permutation. St001285The number of primes in the column sums of the two line notation 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. St001488The number of corners of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St000005The bounce statistic of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000135The number of lucky cars of the parking function. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001861The number of Bruhat lower covers of a permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000001The number of reduced words for a permutation. St000028The number of stack-sorts needed to sort a permutation. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000255The number of reduced Kogan faces with the permutation as type. St000335The difference of lower and upper interactions. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000446The disorder of a permutation. St000519The largest length of a factor maximising the subword complexity. St000528The height of a poset. St000672The number of minimal elements in Bruhat order not less than the permutation. St000693The modular (standard) major index of a standard tableau. St000719The number of alignments in a perfect matching. St000744The length of the path to the largest entry in a standard Young tableau. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000922The minimal number such that all substrings of this length are unique. St000942The number of critical left to right maxima of the parking functions. St000945The number of matchings in the dihedral orbit of a perfect matching. St000996The number of exclusive left-to-right maxima of a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001433The flag major index of a signed permutation. St001485The modular major index of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001622The number of join-irreducible elements of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001684The reduced word complexity of a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001770The number of facets of a certain subword complex associated with the signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001812The biclique partition number of a graph. St001817The number of flag weak exceedances of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001892The flag excedance statistic of a signed permutation. St001903The number of fixed points of a parking function. St001904The length of the initial strictly increasing segment of a parking function. St001925The minimal number of zeros in a row of an alternating sign matrix. St001937The size of the center of a parking function. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000039The number of crossings of a permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000080The rank of the poset. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000197The number of entries equal to positive one in the alternating sign matrix. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000355The number of occurrences of the pattern 21-3. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000516The number of stretching pairs of a permutation. St000731The number of double exceedences of a permutation. St000780The size of the orbit under rotation of a perfect matching. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000809The reduced reflection length of the permutation. St000837The number of ascents of distance 2 of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000881The number of short braid edges in the graph of braid moves of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000924The number of topologically connected components of a perfect matching. St000957The number of Bruhat lower covers of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001008Number of indecomposable injective modules with projective dimension 1 in 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001180Number of indecomposable injective modules with projective dimension at most 1. St001209The pmaj statistic of a parking function. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences 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. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001480The number of simple summands of the module J^2/J^3. St001535The number of cyclic alignments of a permutation. St001557The number of inversions of the second entry of a permutation. 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. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001769The reflection length of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001782The order of rowmotion on the set of order ideals of a poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001864The number of excedances of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001893The flag descent of a signed permutation. St001894The depth of a signed permutation. St001902The number of potential covers of a poset. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001948The number of augmented double ascents of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000224The sorting index of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000841The largest opener of a perfect matching. St000844The size of the largest block in the direct sum decomposition 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. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001065Number of indecomposable reflexive 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. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St000495The number of inversions of distance at most 2 of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000243The number of cyclic valleys and cyclic peaks of a permutation. St000402Half the size of the symmetry class of a permutation. St000702The number of weak deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000873The aix statistic of a permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001568The smallest positive integer that does not appear twice in the partition. St000226The convexity of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000624The normalized sum of the minimal distances to a greater element. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000963The 2-shifted major index of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001388The number of non-attacking neighbors of a permutation. St001423The number of distinct cubes in a binary word. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000444The length of the maximal rise of a Dyck path. St000489The number of cycles of a permutation of length at most 3. St000504The cardinality of the first block of a set partition. St000668The least common multiple of the parts of the partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St000823The number of unsplittable factors of the set partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001062The maximal size of a block of a set partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000060The greater neighbor of the maximum. St000061The number of nodes on the left branch of a binary tree. St000442The maximal area to the right of an up step of a Dyck path. St000458The number of permutations obtained by switching adjacencies or successions. St000472The sum of the ascent bottoms of a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000797The stat`` of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000842The breadth of a permutation. St000874The position of the last double rise in a Dyck path. St000893The number of distinct diagonal sums of an alternating sign matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000983The length of the longest alternating subword. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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

$eA$ . St001246The maximal difference between two consecutive entries of a permutation. St001346The number of parking functions that give the same permutation. St001497The position of the largest weak excedence of a permutation. St001516The number of cyclic bonds of a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000308The height of the tree associated to a permutation. St000354The number of recoils of a permutation. St000376The bounce deficit of a Dyck path. St000390The number of runs of ones in a binary word. St000461The rix statistic of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000538The number of even inversions of a permutation. St000568The hook number of a binary tree. St000570The Edelman-Greene number of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000691The number of changes of a binary word. St000694The number of affine bounded permutations that project to a given permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000829The Ulam distance of a permutation to the identity permutation. St000862The number of parts of the shifted shape of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000990The first ascent of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001052The length of the exterior of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001114The number of odd descents of a permutation. St001128The exponens consonantiae of a partition. St001162The minimum jump of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001220The width 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. St001405The number of bonds in a permutation. St001432The order dimension of the partition. St001471The magnitude of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001741The largest integer such that all patterns of this size are contained in the permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000088The row sums of the character table of the symmetric group. St000143The largest repeated part of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000219The number of occurrences of the pattern 231 in a permutation. St000253The crossing number of a set partition. St000257The number of distinct parts of a partition that occur at least twice. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000462The major index minus the number of excedences of a permutation. St000481The number of upper covers of a partition in dominance order. St000488The number of cycles of a permutation of length at most 2. St000492The rob statistic of a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000682The Grundy value of Welter's game on a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000919The number of maximal left branches of a binary tree. 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. St000944The 3-degree of an integer partition. St000961The shifted major index of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001153The number of blocks with even minimum in a set partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001394The genus of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001469The holeyness of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001665The number of pure excedances of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001729The number of visible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001839The number of excedances of a set partition. St001961The sum of the greatest common divisors of all pairs of parts. St001569The maximal modular displacement of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. 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. St001668The number of points of the poset minus the width of the poset. St001863The number of weak excedances of a signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001927Sparre Andersen's number of positives of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. 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. St001854The size of the left Kazhdan-Lusztig cell, St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001857The number of edges in the reduced word graph of a signed permutation. St001858The number of covering elements of a signed permutation in absolute order. St001621The number of atoms of a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St000043The number of crossings plus two-nestings of a perfect matching. St000094The depth of an ordered tree. St000168The number of internal nodes of an ordered tree. St000422The energy of a graph, if it is integral. St000521The number of distinct subtrees of an ordered tree. St000632The jump number of the poset. St000741The Colin de Verdière graph invariant. St000753The Grundy value for the game of Kayles on a binary word. St000850The number of 1/2-balanced pairs in a poset. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001313The number of Dyck paths above the lattice path given by a binary word. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001856The number of edges in the reduced word graph of a permutation. St001267The length of the Lyndon factorization of the binary word. St001625The Möbius invariant of a lattice. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000735The last entry on the main diagonal of a standard tableau. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000100The number of linear extensions of a poset. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St001890The maximum magnitude of the Möbius function of a poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000634The number of endomorphisms of a poset. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000045The number of linear extensions of a binary tree. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000477The weight of a partition according to Alladi. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000928The sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001408The number of maximal entries in a semistandard tableau. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001472The permanent of the Coxeter matrix of the poset. St001754The number of tolerances of a finite lattice. St000103The sum of the entries of a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St000307The number of rowmotion orbits of a poset. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001401The number of distinct entries in a semistandard tableau. St001624The breadth of a lattice. St001645The pebbling number of a connected graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St000101The cocharge of a semistandard tableau. St000102The charge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000189The number of elements in the poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001964The interval resolution global dimension of a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. 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

$K[x]/(x^n)$ . St000180The number of chains of a poset. St001909The number of interval-closed sets of a poset. St000529The number of permutations whose descent word is the given binary word.