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Your data matches 771 different statistics following compositions of up to 3 maps.
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Matching statistic: St000205
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 0 = 1 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 0 = 1 - 1
[2,1]
=> 0 = 1 - 1
[1,1,1]
=> 0 = 1 - 1
[4]
=> 0 = 1 - 1
[3,1]
=> 0 = 1 - 1
[2,2]
=> 0 = 1 - 1
[2,1,1]
=> 0 = 1 - 1
[1,1,1,1]
=> 0 = 1 - 1
[5]
=> 0 = 1 - 1
[4,1]
=> 0 = 1 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 0 = 1 - 1
[2,2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> 0 = 1 - 1
[5,1]
=> 0 = 1 - 1
[4,2]
=> 1 = 2 - 1
[4,1,1]
=> 0 = 1 - 1
[3,3]
=> 1 = 2 - 1
[3,2,1]
=> 2 = 3 - 1
[3,1,1,1]
=> 0 = 1 - 1
[2,2,2]
=> 1 = 2 - 1
[2,2,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> 0 = 1 - 1
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000243
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,3,2] => 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,4,3,2] => 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,4,3] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,5,4,3] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,5,1,4,2] => 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,5,4] => 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,6,5,4,3,2] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,6,1,5,4,2] => 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,6,5,4] => 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,6,5,1,3,2] => 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [4,2,6,1,5,3] => 3
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,6,5] => 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [5,3,2,6,1,4] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
Matching statistic: St000570
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [2,1] => [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,2] => [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => [3,2,1] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,1,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,2,1,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,4,2,1] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,4,3,1] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,5,2,4,1] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,2,1,4] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,4,3] => 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,5,2,4,1,6] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,5,1,4,2] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [3,5,2,1,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,3,1,4] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => [3,5,2,6,4,1] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [2,5,3,1,4] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,4,2,6,5,1] => 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [2,4,3,6,5,1] => 2
Description
The Edelman-Greene number of a permutation.
This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.
Matching statistic: St001064
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 1
Description
Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.
Matching statistic: St000039
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => [3,1,2] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,3,2,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 0 = 1 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,2,4,3] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,4,3,2,5] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,1,4,2,5] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,1,3] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [3,2,1,4,5] => 0 = 1 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,3,2,5,4] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,5,4,3,2,6] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [4,1,5,3,2,6] => 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [2,3,1,4,5] => 1 = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [4,3,1,5,2,6] => 2 = 3 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4,1,2,3] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,5,2,1,4] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [4,3,2,1,5,6] => 0 = 1 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,2,3,1] => 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,2,5,4,1] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [4,3,2,1,6,5] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [1,4,3,2,6,5] => 0 = 1 - 1
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000052
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
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: St000436
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,2,3] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,3,2,4] => 0 = 1 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,3,5,2,4] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 1 = 2 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [1,3,6,2,4,5] => 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,3,2,4,5] => 0 = 1 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [1,6,2,4,3,5] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => [1,5,2,4,3,6] => 1 = 2 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,5,3] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [1,5,2,3,4,6] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => 0 = 1 - 1
Description
The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.
Matching statistic: St000481
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[3]
=> []
=> []
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[4]
=> []
=> []
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[5]
=> []
=> []
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 2 - 1
[6]
=> []
=> []
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2 = 3 - 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000513
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000513: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000513: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[3]
=> []
=> []
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[4]
=> []
=> []
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[5]
=> []
=> []
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 2 - 1
[6]
=> []
=> []
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> [1,0]
=> []
=> 0 = 1 - 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> 0 = 1 - 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1 = 2 - 1
Description
The number of invariant subsets of size 2 when acting with a permutation of given cycle type.
Matching statistic: St000687
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules.
This statistic was discussed in [Theorem 5.7, 1].
The following 761 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000710The number of big deficiencies of a permutation. St000872The number of very big descents of a permutation. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001130The number of two successive successions in a permutation. St001214The aft of an integer partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001578The minimal number of edges to add or remove to make a graph a line graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001715The number of non-records in a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001469The holeyness of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. 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 $eA$. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000824The sum of the number of descents and the number of recoils of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils 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. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000845The maximal number of elements covered by an element in a poset. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive 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 $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001220The width of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001737The number of descents of type 2 in a permutation. St001812The biclique partition number of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000252The number of nodes of degree 3 of a binary tree. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000358The number of occurrences of the pattern 31-2. St000366The number of double descents of a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000650The number of 3-rises of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000779The tier of a permutation. 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. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001394The genus of a permutation. St001471The magnitude of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001537The number of cyclic crossings of a permutation. St001642The Prague dimension of a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St001902The number of potential covers of a poset. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000880The number of connected components 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. St000035The number of left outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000619The number of cyclic descents of a permutation. St000703The number of deficiencies of a permutation. St000862The number of parts of the shifted shape of a permutation. St001964The interval resolution global dimension of a poset. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001432The order dimension of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000876The number of factors in the Catalan decomposition of a binary word. St000920The logarithmic height of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001151The number of blocks with odd minimum. St001344The neighbouring number of a permutation. St001405The number of bonds in a permutation. St001424The number of distinct squares in a binary word. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for a permutation. 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. St000408The number of occurrences of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000732The number of double deficiencies of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000711The number of big exceedences of a permutation. 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 $K[x]/(x^n)$. St000406The number of occurrences of the pattern 3241 in 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. 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 $eA$. St001520The number of strict 3-descents. St001960The number of descents of a permutation minus one if its first entry is not one. St000023The number of inner peaks of a permutation. St000245The number of ascents of a permutation. St000335The difference of lower and upper interactions. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000834The number of right outer peaks of a permutation. St000958The number of Bruhat factorizations of a permutation. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. 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 $c_{n−1}$. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension 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. St001162The minimum jump of a permutation. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001274The number of indecomposable injective modules with projective dimension equal to two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001399The distinguishing number of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001470The cyclic holeyness of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001728The number of invisible descents of a permutation. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001890The maximum magnitude of the Möbius function of a poset. St000117The number of centered tunnels of a Dyck path. St000241The number of cyclical small excedances. St000407The number of occurrences of the pattern 2143 in 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. St000623The number of occurrences of the pattern 52341 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000702The number of weak deficiencies of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle 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. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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. St001556The number of inversions of the third entry of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000923The minimal number with no two order isomorphic substrings of this length in a permutation. 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. 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. St000454The largest eigenvalue of a graph if it is integral. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000781The number of proper colouring schemes of a Ferrers diagram. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001568The smallest positive integer that does not appear twice in the partition. St001712The number of natural descents of a standard Young tableau. St001095The number of non-isomorphic posets with precisely one further covering relation. St001868The number of alignments of type NE of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000629The defect of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001128The exponens consonantiae of a partition. St000060The greater neighbor of the maximum. St000100The number of linear extensions of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000626The minimal period of a binary word. St000744The length of the path to the largest entry in a standard Young tableau. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001052The length of the exterior of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000022The number of fixed points of a permutation. St000297The number of leading ones in a binary word. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001330The hat guessing number of a graph. St001434The number of negative sum pairs of a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000891The number of distinct diagonal sums of a permutation matrix. St000741The Colin de Verdière graph invariant. St000764The number of strong records in an integer composition. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000259The diameter of a connected graph. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000627The exponent of a binary word. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001881The number of factors of a lattice as a Cartesian product of lattices. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001616The number of neutral elements in a lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000878The number of ones minus the number of zeros of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001846The number of elements which do not have a complement in the lattice. 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. St000667The greatest common divisor of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000026The position of the first return of a Dyck path. St000056The decomposition (or block) number of a permutation. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000260The radius of a connected graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000286The number of connected components of the complement of a graph. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000657The smallest part of an integer composition. St000686The finitistic dominant dimension of a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000763The sum of the positions of the strong records of an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000913The number of ways to refine the partition into singletons. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra 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. 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:
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. 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001564The value of the forgotten symmetric functions when all variables set to 1. St001590The crossing number of a perfect matching. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001621The number of atoms of a lattice. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001721The degree of a binary word. St001733The number of weak left to right maxima of a Dyck path. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001884The number of borders of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St000096The number of spanning trees of a graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. 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. St000455The second largest eigenvalue of a graph if it is integral. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000761The number of ascents in an integer composition. St000787The number of flips required to make a perfect matching noncrossing. St000894The trace of an alternating sign matrix. St000905The number of different multiplicities of parts of an integer composition. St000943The number of spots the most unlucky car had to go further in a parking function. 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. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001185The number of indecomposable injective modules of grade at least 2 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. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001625The Möbius invariant of a lattice. St001734The lettericity of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001867The number of alignments of type EN of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001527The cyclic permutation representation number 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. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000058The order of a permutation. St000153The number of adjacent cycles of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St000767The number of runs in an integer composition. St001566The length of the longest arithmetic progression in a permutation. St000045The number of linear extensions of a binary tree. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001763The Hurwitz number of an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001118The acyclic chromatic index of a graph. St001410The minimal entry of a semistandard tableau. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St000068The number of minimal elements in a poset. St000706The product of the factorials of the multiplicities of an integer partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000731The number of double exceedences of a permutation. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. 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. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000678The number of up steps after the last double rise of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000929The constant term of the character polynomial of an integer partition. St000990The first ascent of a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St000084The number of subtrees. St000091The descent variation of a composition. St000105The number of blocks in the set partition. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000251The number of nonsingleton blocks of a set partition. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000462The major index minus the number of excedences of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000504The cardinality of the first block of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000823The number of unsplittable factors of the set partition. St000836The number of descents of distance 2 of a permutation. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001903The number of fixed points of a parking function. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000546The number of global descents of a permutation. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001875The number of simple modules with projective dimension at most 1. St000567The sum of the products of all pairs of parts. 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. 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. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000886The number of permutations with the same antidiagonal sums. St000925The number of topologically connected components of a set partition. St001060The distinguishing index of a graph. St001114The number of odd descents of a permutation. St001423The number of distinct cubes in a binary word. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001839The number of excedances of a set partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000338The number of pixed points of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000649The number of 3-excedences of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001388The number of non-attacking neighbors of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001735The number of permutations with the same set of runs. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St000004The major index of a permutation. St000037The sign of a permutation. St000075The orbit size of a standard tableau under promotion. St000136The dinv of a parking function. St000166The depth minus 1 of an ordered tree. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000211The rank of the set partition. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000461The rix statistic of a permutation. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000653The last descent of a permutation. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000956The maximal displacement of a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001497The position of the largest weak excedence of a permutation. St001731The factorization defect of a permutation. St001769The reflection length of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000116The major index of a semistandard tableau obtained by standardizing. St000135The number of lucky cars of the parking function. St000503The maximal difference between two elements in a common block. St000539The number of odd inversions of a permutation. St000863The length of the first row of the shifted shape of a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001439The number of even weak deficiencies and of odd weak exceedences. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001927Sparre Andersen's number of positives of a signed permutation. St000030The sum of the descent differences of a permutations. St000044The number of vertices of the unicellular map given by a perfect matching. St000680The Grundy value for Hackendot on posets. St001160The number of proper blocks (or intervals) of a permutations. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001958The degree of the polynomial interpolating the values of a permutation. St000017The number of inversions of a standard tableau. St000064The number of one-box pattern of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000156The Denert index of a permutation. St000222The number of alignments in the permutation. St000494The number of inversions of distance at most 3 of a permutation. St000501The size of the first part in the decomposition of a permutation. St000540The sum of the entries of a parking function minus its length. St000673The number of non-fixed points of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001519The pinnacle sum of a permutation. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001807The lower middle entry of a permutation. St000795The mad of a permutation. St001077The prefix exchange distance of a permutation. St001468The smallest fixpoint of a permutation. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000543The size of the conjugacy class of a binary word. St000625The sum of the minimal distances to a greater element. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000797The stat`` of a permutation. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St000165The sum of the entries of a parking function. St001858The number of covering elements of a signed permutation in absolute order. St001865The number of alignments of a signed permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St000690The size of the conjugacy class of a permutation. St000016The number of attacking pairs of a standard tableau. St000530The number of permutations with the same descent word as the given permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001852The size of the conjugacy class of the signed permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001885The number of binary words with the same proper border set. St000324The shape of the tree associated to a permutation. St000529The number of permutations whose descent word is the given binary word. St001528The number of permutations such that the product with the permutation has the same number of fixed points.
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