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Your data matches 703 different statistics following compositions of up to 3 maps.
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Matching statistic: St000873
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000873: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => 0 = 2 - 2
[[1],[2],[4]]
=> [3,2,1] => 0 = 2 - 2
[[1],[3],[4]]
=> [3,2,1] => 0 = 2 - 2
[[2],[3],[4]]
=> [3,2,1] => 0 = 2 - 2
[[1,2],[2],[3]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,3],[2],[3]]
=> [3,2,1,4] => 0 = 2 - 2
[[1],[2],[5]]
=> [3,2,1] => 0 = 2 - 2
[[1],[3],[5]]
=> [3,2,1] => 0 = 2 - 2
[[1],[4],[5]]
=> [3,2,1] => 0 = 2 - 2
[[2],[3],[5]]
=> [3,2,1] => 0 = 2 - 2
[[2],[4],[5]]
=> [3,2,1] => 0 = 2 - 2
[[3],[4],[5]]
=> [3,2,1] => 0 = 2 - 2
[[1,2],[2],[4]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,3],[2],[4]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 2 - 2
[[1,4],[2],[4]]
=> [3,2,1,4] => 0 = 2 - 2
[[1,3],[3],[4]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,4],[3],[4]]
=> [3,2,1,4] => 0 = 2 - 2
[[2,3],[3],[4]]
=> [4,2,1,3] => 0 = 2 - 2
[[2,4],[3],[4]]
=> [3,2,1,4] => 0 = 2 - 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 2 - 2
[[1],[2],[6]]
=> [3,2,1] => 0 = 2 - 2
[[1],[3],[6]]
=> [3,2,1] => 0 = 2 - 2
[[1],[4],[6]]
=> [3,2,1] => 0 = 2 - 2
[[1],[5],[6]]
=> [3,2,1] => 0 = 2 - 2
[[2],[3],[6]]
=> [3,2,1] => 0 = 2 - 2
[[2],[4],[6]]
=> [3,2,1] => 0 = 2 - 2
[[2],[5],[6]]
=> [3,2,1] => 0 = 2 - 2
[[3],[4],[6]]
=> [3,2,1] => 0 = 2 - 2
[[3],[5],[6]]
=> [3,2,1] => 0 = 2 - 2
[[4],[5],[6]]
=> [3,2,1] => 0 = 2 - 2
[[1,2],[2],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,3],[2],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,5],[2],[3]]
=> [3,2,1,4] => 0 = 2 - 2
[[1,4],[2],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,5],[2],[4]]
=> [3,2,1,4] => 0 = 2 - 2
[[1,5],[2],[5]]
=> [3,2,1,4] => 0 = 2 - 2
[[1,3],[3],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,4],[3],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,5],[3],[4]]
=> [3,2,1,4] => 0 = 2 - 2
[[1,5],[3],[5]]
=> [3,2,1,4] => 0 = 2 - 2
[[1,4],[4],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[1,5],[4],[5]]
=> [3,2,1,4] => 0 = 2 - 2
[[2,3],[3],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[2,4],[3],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[2,5],[3],[4]]
=> [3,2,1,4] => 0 = 2 - 2
[[2,5],[3],[5]]
=> [3,2,1,4] => 0 = 2 - 2
[[2,4],[4],[5]]
=> [4,2,1,3] => 0 = 2 - 2
[[2,5],[4],[5]]
=> [3,2,1,4] => 0 = 2 - 2
[[3,4],[4],[5]]
=> [4,2,1,3] => 0 = 2 - 2
Description
The aix statistic of a permutation.
According to [1], this statistic on finite strings $\pi$ of integers is given as follows: let $m$ be the leftmost occurrence of the minimal entry and let $\pi = \alpha\ m\ \beta$. Then
$$
\operatorname{aix}\pi = \begin{cases}
\operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\
1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\
0 & \text{ if } \beta = \emptyset
\end{cases}\ .
$$
Matching statistic: St001606
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001606: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> 0 = 2 - 2
[[1],[2],[4]]
=> [1,1,1]
=> 0 = 2 - 2
[[1],[3],[4]]
=> [1,1,1]
=> 0 = 2 - 2
[[2],[3],[4]]
=> [1,1,1]
=> 0 = 2 - 2
[[1,2],[2],[3]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,3],[2],[3]]
=> [2,1,1]
=> 0 = 2 - 2
[[1],[2],[5]]
=> [1,1,1]
=> 0 = 2 - 2
[[1],[3],[5]]
=> [1,1,1]
=> 0 = 2 - 2
[[1],[4],[5]]
=> [1,1,1]
=> 0 = 2 - 2
[[2],[3],[5]]
=> [1,1,1]
=> 0 = 2 - 2
[[2],[4],[5]]
=> [1,1,1]
=> 0 = 2 - 2
[[3],[4],[5]]
=> [1,1,1]
=> 0 = 2 - 2
[[1,2],[2],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,3],[2],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,4],[2],[3]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,4],[2],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,3],[3],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,4],[3],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,3],[3],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,4],[3],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> 0 = 2 - 2
[[1],[2],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[1],[3],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[1],[4],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[1],[5],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[2],[3],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[2],[4],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[2],[5],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[3],[4],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[3],[5],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[4],[5],[6]]
=> [1,1,1]
=> 0 = 2 - 2
[[1,2],[2],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,3],[2],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,5],[2],[3]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,4],[2],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,5],[2],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,5],[2],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,3],[3],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,4],[3],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,5],[3],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,5],[3],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,4],[4],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[1,5],[4],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,3],[3],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,4],[3],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,5],[3],[4]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,5],[3],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,4],[4],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[2,5],[4],[5]]
=> [2,1,1]
=> 0 = 2 - 2
[[3,4],[4],[5]]
=> [2,1,1]
=> 0 = 2 - 2
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.
Matching statistic: St001184
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[[1],[2],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[3],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[3],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[4],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,3],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,5],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,5],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,5],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,3],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,5],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,5],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,5],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,3],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,4],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,5],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,5],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,4],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2,5],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,4],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000160
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,3],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[3,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000374
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00329: Permutations —Tanimoto⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,3],[2],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,5],[2],[3]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1,4],[2],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,5],[2],[4]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1,5],[2],[5]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1,3],[3],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,4],[3],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,5],[3],[4]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1,5],[3],[5]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[1,4],[4],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[1,5],[4],[5]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[2,3],[3],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[2,4],[3],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[2,5],[3],[4]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[2,5],[3],[5]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[2,4],[4],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[[2,5],[4],[5]]
=> [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[3,4],[4],[5]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not 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 do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000990
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,5],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[2],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,5],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,5],[2],[5]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[3],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[3],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,5],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,5],[3],[5]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[4],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,5],[4],[5]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,3],[3],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,4],[3],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,5],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,5],[3],[5]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,4],[4],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,5],[4],[5]]
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[[3,4],[4],[5]]
=> [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
Description
The first ascent of a permutation.
For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$.
For the first descent, see [[St000654]].
Matching statistic: St000993
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,3],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[3,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001162
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[[1],[2],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,3],[2],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,5],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,4],[2],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,5],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,5],[2],[5]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,3],[3],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,4],[3],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,5],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[1,4],[4],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[1,5],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[2,3],[3],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[2,4],[3],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[2,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[2,5],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[2,4],[4],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[2,5],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[[3,4],[4],[5]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
Description
The minimum jump of a permutation.
This is $\min_i |\pi_{i+1}-\pi_i|$, see [1].
Matching statistic: St001289
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001289: 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
St001289: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1],[2],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3],[2],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[2],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5],[2],[4]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,5],[2],[5]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[3],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,4],[3],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5],[3],[4]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,5],[3],[5]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[4],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5],[4],[5]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,3],[3],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,4],[3],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,5],[3],[4]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,5],[3],[5]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,4],[4],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[2,5],[4],[5]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[3,4],[4],[5]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero.
This n-fold tensor product seems to be always injective.
Matching statistic: St001711
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001711: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001711: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1,1]
=> 1 = 2 - 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,3],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[1,5],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,3],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[2,5],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[[3,4],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
Description
The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation.
Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that
$$ \alpha\pi\alpha^{-1} = \pi^2.$$
The following 693 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001933The largest multiplicity of a part in an integer partition. St000143The largest repeated part of a partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000461The rix statistic of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001091The number of parts in an integer partition whose next smaller part has the same size. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001525The number of symmetric hooks on the diagonal of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000058The order of a permutation. St000147The largest part of an integer partition. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000628The balance of a binary word. St000638The number of up-down runs of a permutation. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000896The number of zeros on the main diagonal of an alternating sign matrix. 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 $c_{n-1}$. St000995The largest even part of an integer partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001060The distinguishing index of a graph. St001062The maximal size of a block of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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:
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 $eA$. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001371The length of the longest Yamanouchi prefix of a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000006The dinv of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000442The maximal area to the right of an up step of a Dyck path. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000667The greatest common divisor of the parts of the partition. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000729The minimal arc length of a set partition. St000756The sum of the positions of the left to right maxima of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000816The number of standard composition tableaux of the composition. St000820The number of compositions obtained by rotating the composition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000847The number of standard Young tableaux whose descent set is the binary word. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000909The number of maximal chains of maximal size in a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000988The orbit size of a permutation under Foata's bijection. St000989The number of final rises of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000089The absolute variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000133The "bounce" of a permutation. St000150The floored half-sum of the multiplicities of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000223The number of nestings in the permutation. St000241The number of cyclical small excedances. St000257The number of distinct parts of a partition that occur at least twice. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000297The number of leading ones in a binary word. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000403The Szeged index minus the Wiener index of a graph. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000647The number of big descents of a permutation. St000663The number of right floats of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000779The tier of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001033The normalized area 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 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. 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. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001248Sum of the even parts of a partition. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001279The sum of the parts of an integer partition that are at least two. St001310The number of induced diamond graphs in a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001584The area statistic between a Dyck path and its bounce path. St001587Half of the largest even part of an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St000154The sum of the descent bottoms of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001735The number of permutations with the same set of runs. St000570The Edelman-Greene number of a permutation. St000968We 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 $c_{n−1}$. 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 $eA$. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001549The number of restricted non-inversions between exceedances. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001948The number of augmented double ascents of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001307The number of induced stars on four vertices in a graph. St000054The first entry of the permutation. St000842The breadth of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000352The Elizalde-Pak rank of a permutation. St000618The number of self-evacuating tableaux of given shape. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000884The number of isolated descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. 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. St000022The number of fixed points of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000119The number of occurrences of the pattern 321 in a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000214The number of adjacencies of a permutation. St000225Difference between largest and smallest parts in a partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000944The 3-degree of an integer partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001947The number of ties in a parking function. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001434The number of negative sum pairs of a signed permutation. St001651The Frankl number of a lattice. St001271The competition number of a graph. St001330The hat guessing number of a graph. St000889The number of alternating sign matrices with the same antidiagonal sums. St000069The number of maximal elements of a poset. St001260The permanent of an alternating sign matrix. St000893The number of distinct diagonal sums of an alternating sign matrix. St001964The interval resolution global dimension of a poset. 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 $K[x]/(x^n)$. St001811The Castelnuovo-Mumford regularity of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. 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 $eA$. St001261The Castelnuovo-Mumford regularity of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001545The second Elser number of a connected graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000287The number of connected components of a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000710The number of big deficiencies of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000958The number of Bruhat factorizations of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001344The neighbouring number of a permutation. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001427The number of descents of a signed permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St001928The number of non-overlapping descents in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000177The number of free tiles in the pattern. St000178Number of free entries. St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000462The major index minus the number of excedences of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000545The number of parabolic double cosets with minimal element being the given permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000787The number of flips required to make a perfect matching noncrossing. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. 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. 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. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices 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. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001381The fertility of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001569The maximal modular displacement of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001637The number of (upper) dissectors of a poset. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001684The reduced word complexity of a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001555The order of a signed permutation. St001893The flag descent of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000068The number of minimal elements in a poset. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001301The first Betti number of the order complex associated with the poset. St001846The number of elements which do not have a complement in the lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001597The Frobenius rank of a skew partition. St001472The permanent of the Coxeter matrix of the poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000298The order dimension or Dushnik-Miller dimension of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000942The number of critical left to right maxima of the parking functions. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001399The distinguishing number of a poset. St001423The number of distinct cubes in a binary word. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000717The number of ordinal summands of a poset. St000850The number of 1/2-balanced pairs in a poset. St000911The number of maximal antichains of maximal size in a poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001768The number of reduced words of a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001884The number of borders of a binary word. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000295The length of the border of a binary word. St001433The flag major index of a signed permutation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001845The number of join irreducibles minus the rank of a lattice. St001851The number of Hecke atoms of a signed permutation. St001863The number of weak excedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001819The flag Denert index of a signed permutation. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St000753The Grundy value for the game of Kayles on a binary word. St001462The number of factors of a standard tableaux under concatenation. St001524The degree of symmetry of a binary word. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. 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. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St001267The length of the Lyndon factorization of the binary word. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000016The number of attacking pairs of a standard tableau. St000479The Ramsey number of a graph. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001885The number of binary words with the same proper border set. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000526The number of posets with combinatorially isomorphic order polytopes. St001618The cardinality of the Frattini sublattice of a lattice. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. 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. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001626The number of maximal proper sublattices of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000260The radius of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000635The number of strictly order preserving maps of a poset into itself. St001877Number of indecomposable injective modules with projective dimension 2. St000307The number of rowmotion orbits of a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. 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. 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. St000567The sum of the products of all pairs of parts. St000699The toughness times the least common multiple of 1,. 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. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001410The minimal entry of a semistandard tableau. St001623The number of doubly irreducible elements of a lattice. St001621The number of atoms of a lattice. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001625The Möbius invariant of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. St000084The number of subtrees. St000100The number of linear extensions of a poset. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000632The jump number of the poset. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001668The number of points of the poset minus the width of the poset. St001754The number of tolerances of a finite lattice. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St001095The number of non-isomorphic posets with precisely one further covering relation. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000189The number of elements in the poset. St000327The number of cover relations in a poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000528The height of a poset. St000849The number of 1/3-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001510The number of self-evacuating linear extensions of a finite poset. St001645The pebbling number of a connected graph. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001902The number of potential covers of a poset. St000680The Grundy value for Hackendot on posets. St001268The size of the largest ordinal summand in the poset. St001343The dimension of the reduced incidence algebra of a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000912The number of maximal antichains in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001718The number of non-empty open intervals in a poset. St000656The number of cuts of a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001533The largest coefficient of the Poincare polynomial of the poset cone. St001717The largest size of an interval in a poset. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000643The size of the largest orbit of antichains under Panyushev complementation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000524The number of posets with the same order polynomial. St000180The number of chains of a poset. St000400The path length of an ordered tree. St001779The order of promotion on the set of linear extensions of a poset. St001909The number of interval-closed sets of a poset. St000641The number of non-empty boolean intervals in a poset. St000529The number of permutations whose descent word is the given binary word. St000639The number of relations in a poset. St000416The number of inequivalent increasing trees of an ordered tree. St000525The number of posets with the same zeta polynomial. St000634The number of endomorphisms of a poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001709The number of homomorphisms to the three element chain of a poset. St000410The tree factorial of an ordered tree. St001815The number of order preserving surjections from a poset to a total order. St001813The product of the sizes of the principal order filters in a poset.
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