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Your data matches 527 different statistics following compositions of up to 3 maps.
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Matching statistic: St001353
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St001353: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001353: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 0
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 0
[2,1,3] => ([(1,2)],3)
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 0
[1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,1,3,4] => ([(2,3)],4)
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
Description
The number of prime nodes in the modular decomposition of a graph.
Matching statistic: St000215
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 0
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => 0
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => 0
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 0
Description
The number of adjacencies of a permutation, zero appended.
An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended.
$(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000241
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 0
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => 0
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => 0
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 0
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St001122
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,2] => ([(0,1)],2)
=> [2]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> [3]
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[3,2,1] => ([(0,2),(2,1)],3)
=> [3]
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> 0
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> [4,2,1,1]
=> 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> [4,2,1,1]
=> 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> [4,2,1]
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001283
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [1,2] => [2]
=> 0
[2,1] => [1,2] => [2]
=> 0
[1,2,3] => [1,2,3] => [3]
=> 0
[1,3,2] => [1,2,3] => [3]
=> 0
[2,1,3] => [1,2,3] => [3]
=> 0
[2,3,1] => [1,2,3] => [3]
=> 0
[3,1,2] => [1,3,2] => [2,1]
=> 0
[3,2,1] => [1,3,2] => [2,1]
=> 0
[1,2,3,4] => [1,2,3,4] => [4]
=> 0
[1,2,4,3] => [1,2,3,4] => [4]
=> 0
[1,3,2,4] => [1,2,3,4] => [4]
=> 0
[1,3,4,2] => [1,2,3,4] => [4]
=> 0
[1,4,2,3] => [1,2,4,3] => [3,1]
=> 0
[1,4,3,2] => [1,2,4,3] => [3,1]
=> 0
[2,1,3,4] => [1,2,3,4] => [4]
=> 0
[2,1,4,3] => [1,2,3,4] => [4]
=> 0
[2,3,1,4] => [1,2,3,4] => [4]
=> 0
[2,3,4,1] => [1,2,3,4] => [4]
=> 0
[2,4,1,3] => [1,2,4,3] => [3,1]
=> 0
[2,4,3,1] => [1,2,4,3] => [3,1]
=> 0
[3,1,2,4] => [1,3,2,4] => [3,1]
=> 0
[3,1,4,2] => [1,3,4,2] => [3,1]
=> 0
[3,2,1,4] => [1,3,2,4] => [3,1]
=> 0
[3,2,4,1] => [1,3,4,2] => [3,1]
=> 0
[3,4,1,2] => [1,3,2,4] => [3,1]
=> 0
[3,4,2,1] => [1,3,2,4] => [3,1]
=> 0
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> 0
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> 0
[4,3,1,2] => [1,4,2,3] => [3,1]
=> 0
[4,3,2,1] => [1,4,2,3] => [3,1]
=> 0
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St001284
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [1,2] => [2]
=> 0
[2,1] => [1,2] => [2]
=> 0
[1,2,3] => [1,2,3] => [3]
=> 0
[1,3,2] => [1,2,3] => [3]
=> 0
[2,1,3] => [1,2,3] => [3]
=> 0
[2,3,1] => [1,2,3] => [3]
=> 0
[3,1,2] => [1,3,2] => [2,1]
=> 0
[3,2,1] => [1,3,2] => [2,1]
=> 0
[1,2,3,4] => [1,2,3,4] => [4]
=> 0
[1,2,4,3] => [1,2,3,4] => [4]
=> 0
[1,3,2,4] => [1,2,3,4] => [4]
=> 0
[1,3,4,2] => [1,2,3,4] => [4]
=> 0
[1,4,2,3] => [1,2,4,3] => [3,1]
=> 0
[1,4,3,2] => [1,2,4,3] => [3,1]
=> 0
[2,1,3,4] => [1,2,3,4] => [4]
=> 0
[2,1,4,3] => [1,2,3,4] => [4]
=> 0
[2,3,1,4] => [1,2,3,4] => [4]
=> 0
[2,3,4,1] => [1,2,3,4] => [4]
=> 0
[2,4,1,3] => [1,2,4,3] => [3,1]
=> 0
[2,4,3,1] => [1,2,4,3] => [3,1]
=> 0
[3,1,2,4] => [1,3,2,4] => [3,1]
=> 0
[3,1,4,2] => [1,3,4,2] => [3,1]
=> 0
[3,2,1,4] => [1,3,2,4] => [3,1]
=> 0
[3,2,4,1] => [1,3,4,2] => [3,1]
=> 0
[3,4,1,2] => [1,3,2,4] => [3,1]
=> 0
[3,4,2,1] => [1,3,2,4] => [3,1]
=> 0
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> 0
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> 0
[4,3,1,2] => [1,4,2,3] => [3,1]
=> 0
[4,3,2,1] => [1,4,2,3] => [3,1]
=> 0
Description
The number of finite groups that are realised by the given partition over the complex numbers.
A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
Matching statistic: St001795
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[2,1] => ([],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[3,2,1] => ([],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[4,3,2,1] => ([],4)
=> ([],4)
=> 0
Description
The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1).
Matching statistic: St000022
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => [4,1,2,3] => 0
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => [4,1,2,3] => 0
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [4,2,1,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [4,2,1,3] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000221
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 0
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => [1,4,2,3] => 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => [1,4,2,3] => 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [4,2,1,3] => 0
[4,1,3,2] => [1,4,2,3] => [3,4,1,2] => [4,1,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [4,2,1,3] => 0
[4,2,3,1] => [1,4,2,3] => [3,4,1,2] => [4,1,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [3,4,1,2] => [4,1,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [3,4,1,2] => [4,1,2,3] => 0
Description
The number of strong fixed points of a permutation.
$i$ is called a strong fixed point of $\pi$ if
1. $j < i$ implies $\pi_j < \pi_i$, and
2. $j > i$ implies $\pi_j > \pi_i$
This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3].
The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
Matching statistic: St000237
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [2,1] => [2,1] => 1
[1,2] => [1,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[2,1] => [1,1,0,0]
=> [2,3,1] => [3,2,1] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => [3,4,2,1] => 0
[3,1,2] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[3,2,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,1,2,4,3] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,1,5,3,2] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,5,2,1,4] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,2,3,1] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,4,1] => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,4,1] => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,5,3,1] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,5,3,1] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
The following 517 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000296The length of the symmetric border of a binary word. St000488The number of cycles of a permutation of length at most 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. 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 $c_{n−1}$. 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. 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 $g$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000516The number of stretching pairs of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001162The minimum jump of a permutation. St000842The breadth of a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000921The number of internal inversions of a binary word. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001423The number of distinct cubes in a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St000570The Edelman-Greene number of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St001220The width of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. 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. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000478Another weight of a partition according to Alladi. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000799The number of occurrences of the vincular pattern |213 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. 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. St000931The number of occurrences of the pattern UUU in a Dyck path. St000963The 2-shifted major index of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001114The number of odd descents of a permutation. St001130The number of two successive successions in a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001556The number of inversions of the third entry of a permutation. 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. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001731The factorization defect of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 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. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000216The absolute length of a permutation. St000222The number of alignments in the permutation. St000226The convexity of a permutation. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 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. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000461The rix statistic of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. St000538The number of even inversions of a permutation. St000539The number of odd inversions 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. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000565The major index of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. 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. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. 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. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (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. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. 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. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are 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. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000663The number of right floats of a permutation. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000795The mad 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. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000961The shifted major index of a permutation. St000989The number of final rises of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001095The number of non-isomorphic posets with precisely one further covering relation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. 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. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001377The major index minus the number of inversions of a permutation. St001388The number of non-attacking neighbors of a permutation. 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. 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. St001485The modular major index of a binary word. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001718The number of non-empty open intervals in a poset. St001727The number of invisible inversions of a permutation. St001736The total number of cycles in a graph. St001797The number of overfull subgraphs of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001868The number of alignments of type NE of a signed permutation. St001911A descent variant minus the number of inversions. St000047The number of standard immaculate tableaux of a given shape. St000079The number of alternating sign matrices for a given Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000390The number of runs of ones in a binary word. St000402Half the size of the symmetry class of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000526The number of posets with combinatorially isomorphic order polytopes. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000816The number of standard composition tableaux of the composition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000844The size of the largest block in the direct sum decomposition of a permutation. St000862The number of parts of the shifted shape of a permutation. St000910The number of maximal chains of minimal length in a poset. St000919The number of maximal left branches of a binary tree. St000920The logarithmic height of a Dyck path. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001052The length of the exterior of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001246The maximal difference between two consecutive entries of a permutation. St001281The normalized isoperimetric number of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001735The number of permutations with the same set of runs. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000064The number of one-box pattern of a permutation. St000489The number of cycles of a permutation of length at most 3. St000702The number of weak deficiencies of a permutation. 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 $eA$. St001516The number of cyclic bonds of a permutation. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000948The chromatic discriminant of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001057The Grundy value of the game of creating an independent set in a graph. St001691The number of kings in a graph. St000219The number of occurrences of the pattern 231 in a permutation. St000914The sum of the values of the Möbius function of a poset. St000699The toughness times the least common multiple of 1,. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000677The standardized bi-alternating inversion number of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000045The number of linear extensions of a binary tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Möbius function of a poset. St001060The distinguishing index of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000552The number of cut vertices of a graph. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000016The number of attacking pairs of a standard tableau. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000377The dinv defect of an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000671The maximin edge-connectivity for choosing a subgraph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. 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. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. 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. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001172The number of 1-rises at odd height of a Dyck path. 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. St001214The aft of an integer partition. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001721The degree of a binary word. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000929The constant term of the character polynomial of an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St000264The girth of a graph, which is not a tree. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. 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. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000295The length of the border of a binary word. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000995The largest even part of an integer partition. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001248Sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001910The height of the middle non-run of a Dyck path. St000181The number of connected components of the Hasse diagram for the poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000096The number of spanning trees of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000315The number of isolated vertices of a graph. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000145The Dyson rank of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001651The Frankl number of a lattice. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000911The number of maximal antichains of maximal size in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001479The number of bridges of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001826The maximal number of leaves on a vertex of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000338The number of pixed points of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001354The number of series nodes in the modular decomposition of a graph. St001827The number of two-component spanning forests of a graph. St001851The number of Hecke atoms of a signed permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000102The charge of a semistandard tableau. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000623The number of occurrences of the pattern 52341 in a permutation. St000873The aix statistic of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001409The maximal entry of a semistandard tableau. St000928The sum of the coefficients of the character polynomial of an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000286The number of connected components of the complement of a graph. St001637The number of (upper) dissectors of a poset.
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