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Your data matches 544 different statistics following compositions of up to 3 maps.
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Matching statistic: St000480
(load all 6 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for

$n\neq 2$ ) can cover is

$\frac{1}{2}(\sqrt{1+8n}-3)$ and an element which covers this number of elements is given by

$(c+i,c,c-1,\dots,3,2,1)$ , where

$1\leq i\leq c+2$ .

Matching statistic: St000651
(load all 5 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal size of a rise in a permutation. This is

$\max_i \sigma_{i+1}-\sigma_i$ , except for the permutations without rises, where it is

$0$ .

Matching statistic: St000665
(load all 6 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rafts of a permutation. Let

$\pi$ be a permutation of length

$n$ . A small ascent of

$\pi$ is an index

$i$ such that

$\pi(i+1)= \pi(i)+1$ , see [[St000441]], and a raft of

$\pi$ is a non-empty maximal sequence of consecutive small ascents.

Matching statistic: St000834
(load all 33 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of right outer peaks of a permutation. A right outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$n$ if

$w_n > w_{n-1}$ . In other words, it is a peak in the word

$[w_1,..., w_n,0]$ .

Matching statistic: St001280
(load all 3 compositions to match this statistic)

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St001784
(load all 2 compositions to match this statistic)

Mapping

Mp00284: Standard tableaux —rows⟶ Set partitions
St001784: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => {{1}}
 => 1 = -1 + 2
[[1,2]]
 => {{1,2}}
 => 2 = 0 + 2
[[1],[2]]
 => {{1},{2}}
 => 1 = -1 + 2
[[1,2,3]]
 => {{1,2,3}}
 => 2 = 0 + 2
[[1,2],[3]]
 => {{1,2},{3}}
 => 2 = 0 + 2
[[1],[2],[3]]
 => {{1},{2},{3}}
 => 1 = -1 + 2
[[1,2,3,4]]
 => {{1,2,3,4}}
 => 2 = 0 + 2
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2 = 0 + 2
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 2 = 0 + 2
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 1 = -1 + 2
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 2 = 0 + 2
[[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 2 = 0 + 2
[[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2 = 0 + 2
[[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 2 = 0 + 2
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 1 = -1 + 2
[[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 2 = 0 + 2
[[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 2 = 0 + 2
[[1,2,3,4],[5],[6]]
 => {{1,2,3,4},{5},{6}}
 => 2 = 0 + 2
[[1,2,3],[4],[5],[6]]
 => {{1,2,3},{4},{5},{6}}
 => 2 = 0 + 2
[[1,2],[3],[4],[5],[6]]
 => {{1,2},{3},{4},{5},{6}}
 => 2 = 0 + 2
[[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 1 = -1 + 2

Description

The minimum of the smallest closer and the second element of the block containing 1 in a set partition. A closer of a set partition is the maximal element of a non-singleton block. This statistic is defined as

$1$ if

$\{1\}$ is a singleton block, and otherwise the minimum of the smallest closer and the second element of the block containing

$1$ .

Matching statistic: St000021
(load all 15 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000028
(load all 16 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of stack-sorts needed to sort a permutation. A permutation is (West)

$t$ -stack sortable if it is sortable using

$t$ stacks in series. Let

$W_t(n,k)$ be the number of permutations of size

$n$ with

$k$ descents which are

$t$ -stack sortable. Then the polynomials

$W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have

$W_{n,1}(x) = A_n(x)$ , the Eulerian polynomials. One can show that

$W_{n,1}(x)$ and

$W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern

$231$ have statistic at most

$1$ , see [3]. These are counted by

$\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern

$2341$ and the barred pattern

$3\bar 5241$ have statistic at most

$2$ , see [4]. These are counted by

$\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).

Matching statistic: St000035
(load all 33 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left outer peaks of a permutation. A left outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ . In other words, it is a peak in the word

$[0,w_1,..., w_n]$ . This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

Matching statistic: St000141
(load all 6 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

The following 534 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000290The major index of a binary word. St000291The number of descents of a binary word. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000481The number of upper covers of a partition in dominance order. St000546The number of global descents of a permutation. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001375The pancake length of a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000007The number of saliances of the permutation. St000010The length of the partition. St000054The first entry of the permutation. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000638The number of up-down runs of a permutation. St000740The last entry of a permutation. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001642The Prague dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000055The inversion sum of a permutation. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000142The number of even parts of a partition. St000147The largest part of an integer partition. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000169The cocharge of a standard tableau. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000246The number of non-inversions of a permutation. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000331The number of upper interactions of a Dyck path. St000336The leg major index of a standard tableau. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000341The non-inversion sum of a permutation. St000347The inversion sum of a binary word. St000362The size of a minimal vertex cover of a graph. St000378The diagonal inversion number of an integer partition. St000387The matching number of a graph. St000390The number of runs of ones in a binary word. St000441The number of successions of a permutation. St000446The disorder of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St000660The number of rises of length at least 3 of a Dyck path. St000691The number of changes of a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000761The number of ascents in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001092The number of distinct even parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

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

$A$ corresponding to the Dyck path. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001412Number of minimal entries in the Bruhat order matrix of 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. St001427The number of descents of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. 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. St001587Half of the largest even part of an integer partition. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001671Haglund's hag of a permutation. St001697The shifted natural comajor index of a standard Young tableau. St001726The number of visible inversions of a permutation. St001743The discrepancy of a graph. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001812The biclique partition number of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001931The weak major index of an integer composition regarded as a word. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000056The decomposition (or block) number of a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. 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. St000239The number of small weak excedances. St000259The diameter of a connected graph. St000286The number of connected components of the complement of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000381The largest part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000397The Strahler number of a rooted tree. St000443The number of long tunnels of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000528The height of a poset. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000764The number of strong records in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000847The number of standard Young tableaux whose descent set is the binary word. St000883The number of longest increasing subsequences of a permutation. St000903The number of different parts of an integer composition. St000908The length of the shortest maximal antichain in a poset. St000912The number of maximal antichains in a poset. St000935The number of ordered refinements of an integer partition. St000983The length of the longest alternating subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001389The number of partitions of the same length below the given integer partition. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001717The largest size of an interval in a poset. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001883The mutual visibility number of a graph. St001884The number of borders of a binary word. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000472The sum of the ascent bottoms 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. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000640The rank of the largest boolean interval in a poset. St000659The number of rises of length at least 2 of a Dyck path. St000753The Grundy value for the game of Kayles on a binary word. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001062The maximal size of a block of a set partition. St001346The number of parking functions that give the same permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000289The decimal representation of a binary word. St000297The number of leading ones in a binary word. 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. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. 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. St000653The last descent of a permutation. St000658The number of rises of length 2 of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000792The Grundy value for the game of ruler on a binary word. St000794The mak of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000919The number of maximal left branches of a binary tree. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000989The number of final rises of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. 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. St001592The maximal number of simple paths between any two different vertices of a graph. St000061The number of nodes on the left branch of a binary tree. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000914The sum of the values of the Möbius function of a poset. St000933The number of multipartitions of sizes given by an integer partition. St000990The first ascent of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000023The number of inner peaks of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001352The number of internal nodes in the modular decomposition of a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000699The toughness times the least common multiple of 1,. 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. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001684The reduced word complexity of a permutation. 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. 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. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000545The number of parabolic double cosets with minimal element being the given permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. 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

$eA$ . St001500The global dimension of magnitude 1 Nakayama algebras. St001734The lettericity of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001569The maximal modular displacement of a permutation. St001668The number of points of the poset minus the width of the poset. St001896The number of right descents of a signed permutations. St001948The number of augmented double ascents of a permutation. St001555The order 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. St000454The largest eigenvalue of a graph if it is integral. St001870The number of positive entries followed by a negative entry in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000806The semiperimeter of the associated bargraph. St000379The number of Hamiltonian cycles in a graph. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000510The number of invariant oriented cycles 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. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St001118The acyclic chromatic index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. 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

$K[x]/(x^n)$ . St001281The normalized isoperimetric number of a graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. 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. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001905The number of preferred parking spots in a parking function less than the index of the car. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001935The number of ascents in a parking function. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001946The number of descents in a parking function. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000464The Schultz index of a connected graph. 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. 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

$eA$ . St001545The second Elser number of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001937The size of the center of a parking function. St000264The girth of a graph, which is not a tree. St000762The sum of the positions of the weak records of an integer composition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001060The distinguishing index of a graph. St000567The sum of the products of all pairs of parts. St001095The number of non-isomorphic posets with precisely one further covering relation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000993The multiplicity of the largest part of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001890The maximum magnitude of the Möbius function of a poset. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000145The Dyson rank of a partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. 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.