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Your data matches 389 different statistics following compositions of up to 3 maps.
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Matching statistic: St000375
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001513
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
St001513: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St001549
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 1
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St000232
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 0
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 0
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => {{1,2,3}}
=> 0
[3,1,2] => {{1,3},{2}}
=> 0
[3,2,1] => {{1,3},{2}}
=> 0
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => {{1},{2,4},{3}}
=> 0
[1,4,3,2] => {{1},{2,4},{3}}
=> 0
[2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => {{1,2,3,4}}
=> 0
[2,4,1,3] => {{1,2,4},{3}}
=> 0
[2,4,3,1] => {{1,2,4},{3}}
=> 0
[3,1,2,4] => {{1,3},{2},{4}}
=> 0
[3,1,4,2] => {{1,3,4},{2}}
=> 0
[3,2,1,4] => {{1,3},{2},{4}}
=> 0
[3,2,4,1] => {{1,3,4},{2}}
=> 0
[3,4,1,2] => {{1,3},{2,4}}
=> 1
[3,4,2,1] => {{1,3},{2,4}}
=> 1
[4,1,2,3] => {{1,4},{2},{3}}
=> 0
[4,1,3,2] => {{1,4},{2},{3}}
=> 0
[4,2,1,3] => {{1,4},{2},{3}}
=> 0
[4,2,3,1] => {{1,4},{2},{3}}
=> 0
[4,3,1,2] => {{1,4},{2,3}}
=> 0
[4,3,2,1] => {{1,4},{2,3}}
=> 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 0
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 0
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 0
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 0
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 0
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 1
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000233
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 0
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 0
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => {{1,2,3}}
=> 0
[3,1,2] => {{1,3},{2}}
=> 0
[3,2,1] => {{1,3},{2}}
=> 0
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => {{1},{2,4},{3}}
=> 0
[1,4,3,2] => {{1},{2,4},{3}}
=> 0
[2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => {{1,2,3,4}}
=> 0
[2,4,1,3] => {{1,2,4},{3}}
=> 0
[2,4,3,1] => {{1,2,4},{3}}
=> 0
[3,1,2,4] => {{1,3},{2},{4}}
=> 0
[3,1,4,2] => {{1,3,4},{2}}
=> 0
[3,2,1,4] => {{1,3},{2},{4}}
=> 0
[3,2,4,1] => {{1,3,4},{2}}
=> 0
[3,4,1,2] => {{1,3},{2,4}}
=> 0
[3,4,2,1] => {{1,3},{2,4}}
=> 0
[4,1,2,3] => {{1,4},{2},{3}}
=> 0
[4,1,3,2] => {{1,4},{2},{3}}
=> 0
[4,2,1,3] => {{1,4},{2},{3}}
=> 0
[4,2,3,1] => {{1,4},{2},{3}}
=> 0
[4,3,1,2] => {{1,4},{2,3}}
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 0
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 0
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 0
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 0
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 0
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 0
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000373
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => 0
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000036
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,3,2] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 1 = 0 + 1
[2,1,3] => [2,1,3] => 1 = 0 + 1
[2,3,1] => [2,3,1] => 1 = 0 + 1
[3,1,2] => [3,1,2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => 1 = 0 + 1
[2,3,1,4] => [2,4,1,3] => 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => 1 = 0 + 1
[3,1,2,4] => [3,1,4,2] => 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => 1 = 0 + 1
[3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[3,2,4,1] => [3,2,4,1] => 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,4,2,1] => 1 = 0 + 1
[4,1,2,3] => [4,1,3,2] => 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => 1 = 0 + 1
[4,2,1,3] => [4,2,1,3] => 1 = 0 + 1
[4,2,3,1] => [4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => 1 = 0 + 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
Matching statistic: St000052
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 0
[1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 0
[2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 0
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 0
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 0
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,2,1,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 0
[4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,3,1,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000123
Mp00223: Permutations —runsort⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 0
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 0
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => 0
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 0
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,3,5,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => 0
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => 0
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1
Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Matching statistic: St000223
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,2,4,3] => 0
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,2,4,3] => 0
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,3,5,4] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,3,5,4] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0
Description
The number of nestings in the permutation.
The following 379 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000317The cycle descent number of a permutation. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000648The number of 2-excedences of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001394The genus of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000214The number of adjacencies of a permutation. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000647The number of big descents of a permutation. St000663The number of right floats of a permutation. St000731The number of double exceedences of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001323The independence gap of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001728The number of invisible descents of a permutation. St001866The nesting alignments of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000160The multiplicity of the smallest part of a partition. St000883The number of longest increasing subsequences of a permutation. St001066The number of simple reflexive modules 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. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. 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 $c_{n-1}$. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000497The lcb statistic of a set partition. St000516The number of stretching pairs of a permutation. St000562The number of internal points of a set partition. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001638The book thickness of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000570The Edelman-Greene number of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000886The number of permutations with the same antidiagonal sums. St000031The number of cycles in the cycle decomposition of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000225Difference between largest and smallest parts in a partition. St000377The dinv defect of an integer partition. 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. St001172The number of 1-rises at odd height of a Dyck path. St001214The aft of an integer partition. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001964The interval resolution global dimension of a poset. 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 $eA$. St000552The number of cut vertices of a graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001280The number of 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. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001864The number of excedances of a signed permutation. St000379The number of Hamiltonian cycles in a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001271The competition number of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St000914The sum of the values of the Möbius function of a poset. St000117The number of centered tunnels of a Dyck path. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. 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. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000348The non-inversion sum of a binary word. St000481The number of upper covers of a partition in dominance order. St000547The number of even non-empty partial sums of an integer partition. St000628The balance of a binary word. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001175The size of a partition minus the hook length of the base cell. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001414Half the length of the longest odd length palindromic prefix 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. St001423The number of distinct cubes in a binary word. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001910The height of the middle non-run of a Dyck path. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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. 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. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000455The second largest eigenvalue of a graph if it is integral. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St000897The number of different multiplicities of parts of an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001862The number of crossings of a signed permutation. St001961The sum of the greatest common divisors of all pairs of parts. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001570The minimal number of edges to add to make a graph Hamiltonian. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000936The number of even values of the symmetric group character corresponding to the partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000068The number of minimal elements in a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001868The number of alignments of type NE of a signed permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001621The number of atoms of a lattice. St001330The hat guessing number of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000699The toughness times the least common multiple of 1,. St000454The largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St000908The length of the shortest maximal antichain in a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001820The size of the image of the pop stack sorting operator. St001867The number of alignments of type EN of a signed permutation. St000629The defect of a binary word. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000449The number of pairs of vertices of a graph with distance 4. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001282The number of graphs with the same chromatic polynomial. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001768The number of reduced words of a signed permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. 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 $K[x]/(x^n)$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000916The packing number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001739The number of graphs with the same edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001281The normalized isoperimetric number of a 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 $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St000264The girth of a graph, which is not a tree. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000022The number of fixed points of a permutation. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000907The number of maximal antichains of minimal length in a poset. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000948The chromatic discriminant of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000069The number of maximal elements of a poset. St001347The number of pairs of vertices of a graph having the same neighbourhood. St000097The order of the largest clique of the graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000917The open packing number of a graph. St001654The monophonic hull number of a graph. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St000981The length of the longest zigzag subpath.
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