searching the database
Your data matches 1011 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000007
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[2,1,3,4,5] => [1,2,3,4,5] => 1
[2,1,3,5,4] => [1,2,3,4,5] => 1
[2,1,4,3,5] => [1,2,3,4,5] => 1
[2,1,4,5,3] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [1,2,3,4,5] => 1
[2,3,1,5,4] => [1,2,3,4,5] => 1
[2,3,4,1,5] => [1,2,3,4,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000470
(load all 49 compositions to match this statistic)
(load all 49 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[2,1,3,4,5] => [1,2,3,4,5] => 1
[2,1,3,5,4] => [1,2,3,4,5] => 1
[2,1,4,3,5] => [1,2,3,4,5] => 1
[2,1,4,5,3] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [1,2,3,4,5] => 1
[2,3,1,5,4] => [1,2,3,4,5] => 1
[2,3,4,1,5] => [1,2,3,4,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000619
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[2,1,3,4,5] => [1,2,3,4,5] => 1
[2,1,3,5,4] => [1,2,3,4,5] => 1
[2,1,4,3,5] => [1,2,3,4,5] => 1
[2,1,4,5,3] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [1,2,3,4,5] => 1
[2,3,1,5,4] => [1,2,3,4,5] => 1
[2,3,4,1,5] => [1,2,3,4,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000669
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000669: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000669: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[2,1,3,4,5] => [1,2,3,4,5] => 1
[2,1,3,5,4] => [1,2,3,4,5] => 1
[2,1,4,3,5] => [1,2,3,4,5] => 1
[2,1,4,5,3] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [1,2,3,4,5] => 1
[2,3,1,5,4] => [1,2,3,4,5] => 1
[2,3,4,1,5] => [1,2,3,4,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
Description
The number of permutations obtained by switching ascents or descents of size 2.
For a permutation $\pi$, this statistic is the size of its equivalence class of the equivalence relation generated by the interchange of any two adjacent elements $\pi_i$ and $\pi_{i+1}$ such that $|\pi_i-\pi_{i+1}|=2$.
Matching statistic: St000862
(load all 87 compositions to match this statistic)
(load all 87 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[2,1,3,4,5] => [1,2,3,4,5] => 1
[2,1,3,5,4] => [1,2,3,4,5] => 1
[2,1,4,3,5] => [1,2,3,4,5] => 1
[2,1,4,5,3] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [1,2,3,4,5] => 1
[2,3,1,5,4] => [1,2,3,4,5] => 1
[2,3,4,1,5] => [1,2,3,4,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St000920
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001735
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001735: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001735: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[2,1,3,4,5] => [1,2,3,4,5] => 1
[2,1,3,5,4] => [1,2,3,4,5] => 1
[2,1,4,3,5] => [1,2,3,4,5] => 1
[2,1,4,5,3] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [1,2,3,4,5] => 1
[2,3,1,5,4] => [1,2,3,4,5] => 1
[2,3,4,1,5] => [1,2,3,4,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
Description
The number of permutations with the same set of runs.
For example, the set of runs of $4132$ is $\{(13), (2), (4)\}$. The only other permutation with this set of runs is $4213$, so the statistic equals $2$ for these two permutations.
Matching statistic: St001741
(load all 86 compositions to match this statistic)
(load all 86 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[4,1,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[2,1,3,4,5] => [1,2,3,4,5] => 1
[2,1,3,5,4] => [1,2,3,4,5] => 1
[2,1,4,3,5] => [1,2,3,4,5] => 1
[2,1,4,5,3] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [1,2,3,4,5] => 1
[2,3,1,5,4] => [1,2,3,4,5] => 1
[2,3,4,1,5] => [1,2,3,4,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
Description
The largest integer such that all patterns of this size are contained in the permutation.
Matching statistic: St000028
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,1,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 0 = 1 - 1
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St000035
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,1,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 0 = 1 - 1
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
The following 1001 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000354The number of recoils of a permutation. St000446The disorder of a permutation. St000539The number of odd inversions of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000829The Ulam distance of a permutation to the identity permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. 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. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001928The number of non-overlapping descents in a permutation. St000010The length of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000058The order of a permutation. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. 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. St000374The number of exclusive right-to-left minima of a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000402Half the size of the symmetry class of a permutation. St000451The length of the longest pattern of the form k 1 2. St000485The length of the longest cycle of a permutation. St000527The width of the poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000568The hook number of a binary tree. St000630The length of the shortest palindromic decomposition of a binary word. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000668The least common multiple of the parts of the partition. St000670The reversal length of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000886The number of permutations with the same antidiagonal sums. St000933The number of multipartitions of sizes given by an integer partition. St000935The number of ordered refinements of an integer partition. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001029The size of the core of a graph. St001062The maximal size of a block of a set partition. St001081The number of minimal length factorizations of a permutation into star transpositions. St001116The game chromatic number of a graph. St001128The exponens consonantiae of a partition. St001246The maximal difference between two consecutive entries of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001597The Frobenius rank 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. St001670The connected partition number of a graph. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St001883The mutual visibility number of a graph. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001963The tree-depth of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000169The cocharge of a standard tableau. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000204The number of internal nodes of a binary tree. St000211The rank of the set partition. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000272The treewidth of a graph. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000336The leg major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000356The number of occurrences of the pattern 13-2. St000362The size of a minimal vertex cover of a graph. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000387The matching number of a graph. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000486The number of cycles of length at least 3 of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000534The number of 2-rises of a permutation. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000538The number of even 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. St000546The number of global descents of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000565The major index of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000628The balance of a binary word. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000730The maximal arc length of a set partition. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000779The tier of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000833The comajor index of a permutation. St000834The number of right outer peaks of a permutation. St000836The number of descents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000891The number of distinct diagonal sums of a permutation matrix. St000919The number of maximal left branches of a binary tree. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with 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. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001114The number of odd descents of a permutation. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001393The induced matching number of a graph. St001394The genus of a permutation. St001403The number of vertical separators in a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001697The shifted natural comajor index of a standard Young tableau. St001743The discrepancy of a graph. St001746The coalition number of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001842The major index of a set partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001931The weak major index of an integer composition regarded as a word. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000001The number of reduced words for a permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000068The number of minimal elements in a poset. St000086The number of subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000124The cardinality of the preimage of the Simion-Schmidt map. St000167The number of leaves of an ordered tree. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000259The diameter of a connected graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000306The bounce count of a Dyck path. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000343The number of spanning subgraphs of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000397The Strahler number of a rooted tree. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000482The (zero)-forcing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000507The number of ascents of a standard tableau. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000532The total number of rook placements on a Ferrers board. St000544The cop number of a graph. St000553The number of blocks of a graph. St000626The minimal period of a binary word. St000675The number of centered multitunnels of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000679The pruning number of an ordered tree. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000701The protection number of a binary tree. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000795The mad of a permutation. St000808The number of up steps of the associated bargraph. 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. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000831The number of indices that are either descents or recoils. St000839The largest opener of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000903The number of different parts of an integer composition. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St000925The number of topologically connected components of a set partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000971The smallest closer of a set partition. St000972The composition number of a graph. St000982The length of the longest constant subword. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001052The length of the exterior of a permutation. St001057The Grundy value of the game of creating an independent set in a graph. St001058The breadth of the ordered tree. St001061The number of indices that are both descents and recoils of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001096The size of the overlap set of a permutation. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama 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:
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001286The annihilation number of a graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001352The number of internal nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001363The Euler characteristic of a graph according to Knill. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001366The maximal multiplicity of a degree of a vertex of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001400The total number of Littlewood-Richardson tableaux of given shape. St001427The number of descents of a signed permutation. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001468The smallest fixpoint of a permutation. St001471The magnitude of a Dyck path. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001486The number of corners of the ribbon associated with an integer composition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001512The minimum rank of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001642The Prague dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001734The lettericity of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001884The number of borders of a binary word. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000009The charge of a standard tableau. St000012The area of a Dyck path. St000016The number of attacking pairs of a standard tableau. St000021The number of descents of a permutation. St000022The number of fixed points of a permutation. St000053The number of valleys of the Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000065The number of entries equal to -1 in an alternating sign matrix. St000081The number of edges of a graph. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000143The largest repeated part of a partition. St000148The number of odd parts 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. St000154The sum of the descent bottoms of a permutation. St000160The multiplicity of the smallest part of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000171The degree of the graph. St000183The side length of the Durfee square of an integer partition. St000203The number of external nodes of a binary tree. St000215The number of adjacencies of a permutation, zero appended. St000218The number of occurrences of the pattern 213 in a permutation. St000228The size of a partition. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000247The number of singleton blocks of a set 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. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000330The (standard) major index of a standard tableau. St000347The inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000361The second Zagreb index of a graph. St000367The number of simsun double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000386The number of factors DDU in a Dyck path. St000391The sum of the positions of the ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. 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. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000490The intertwining number of a set partition. St000491The number of inversions of a set partition. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000552The number of cut vertices of a graph. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal 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. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) 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. 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. St000640The rank of the largest boolean interval in a poset. St000648The number of 2-excedences of a permutation. St000653The last descent of a permutation. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000693The modular (standard) major index of a standard tableau. St000710The number of big deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000731The number of double exceedences of a permutation. 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. St000778The metric dimension of a graph. St000784The maximum of the length and the largest part of the integer partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000792The Grundy value for the game of ruler on a binary word. St000794The mak of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000871The number of very big ascents of a permutation. St000877The depth of the binary word interpreted as a path. St000897The number of different multiplicities of parts of an integer partition. St000918The 2-limited packing number of a graph. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer partition. St000944The 3-degree of an integer partition. St000947The major index east count of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000981The length of the longest zigzag subpath. St000984The number of boxes below precisely one peak. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St000992The alternating sum of the parts of an integer partition. St001011Number of simple modules of projective dimension 2 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001153The number of blocks with even minimum in a set partition. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. 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. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001375The pancake length of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001459The number of zero columns in the nullspace of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001479The number of bridges of a graph. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001521Half the total irregularity of a graph. St001552The number of inversions between excedances and fixed points of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001649The length of a longest trail in a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001671Haglund's hag of a permutation. St001673The degree of asymmetry of an integer composition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001721The degree of a binary word. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001742The difference of the maximal and the minimal degree in a graph. 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. St001783The number of odd automorphisms of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001869The maximum cut size of a graph. St001874Lusztig's a-function for the symmetric group. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001949The rigidity index of a graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St000472The sum of the ascent bottoms of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000422The energy of a graph, if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000741The Colin de Verdière graph invariant. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000219The number of occurrences of the pattern 231 in a permutation. St000358The number of occurrences of the pattern 31-2. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000056The decomposition (or block) number of a permutation. St000092The number of outer peaks of a permutation. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000638The number of up-down runs of a permutation. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000039The number of crossings of a permutation. St000133The "bounce" of a permutation. St000216The absolute length of a permutation. St000222The number of alignments in the permutation. St000242The number of indices that are not cyclical small weak excedances. St000304The load of a permutation. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000848The balance constant multiplied with the number of linear extensions of a poset. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001273The projective dimension of the first term in an injective coresolution of the regular module. St001388The number of non-attacking neighbors of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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). St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000004The major index of a permutation. St000015The number of peaks of a Dyck path. St000020The rank of the permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000060The greater neighbor of the maximum. St000061The number of nodes on the left branch of a binary tree. St000079The number of alternating sign matrices for a given Dyck path. St000082The number of elements smaller than a binary tree in Tamari order. St000087The number of induced subgraphs. St000109The number of elements less than or equal to the given element in Bruhat order. St000156The Denert index of a permutation. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000224The sorting index of a permutation. St000239The number of small weak excedances. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000479The Ramsey number of a graph. St000636The hull number of a graph. St000694The number of affine bounded permutations that project to a given permutation. St000717The number of ordinal summands of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000906The length of the shortest maximal chain in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000917The open packing number of a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000959The number of strong Bruhat factorizations of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. 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. 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. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001497The position of the largest weak excedence of a permutation. St001530The depth of a Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001624The breadth of a lattice. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001778The largest greatest common divisor of an element and its image in a permutation. St001802The number of endomorphisms of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001959The product of the heights of the peaks of a Dyck path. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000051The size of the left subtree of a binary tree. St000055The inversion sum of a permutation. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000161The sum of the sizes of the right subtrees of a binary tree. St000168The number of internal nodes of an ordered tree. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000338The number of pixed points of a permutation. St000341The non-inversion sum of a permutation. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000461The rix statistic of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000671The maximin edge-connectivity for choosing a subgraph. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000961The shifted major index of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001119The length of a shortest maximal path in a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001480The number of simple summands of the module J^2/J^3. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. 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. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001731The factorization defect of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001911A descent variant minus the number of inversions. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001569The maximal modular displacement of a permutation. St001964The interval resolution global dimension of a poset. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001948The number of augmented double ascents of a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001487The number of inner corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001555The order of a signed permutation. St000117The number of centered tunnels of a Dyck path. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000824The sum of the number of descents and the number of recoils of a permutation. St000872The number of very big descents of a permutation. St000963The 2-shifted major index of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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 $eA$. St001429The number of negative entries in a signed permutation. St001470The cyclic holeyness of a permutation. St001488The number of corners of a skew partition. St001513The number of nested exceedences of 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. St001537The number of cyclic crossings of a permutation. St001557The number of inversions of the second entry of a permutation. St001668The number of points of the poset minus the width of the poset. St001684The reduced word complexity of a permutation. St001769The reflection length of a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001892The flag excedance statistic of a signed permutation. St001895The oddness of a signed permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in 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. 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 $eA$. 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. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001889The size of the connectivity set of a signed permutation. St001937The size of the center of a parking function. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001209The pmaj statistic of a parking function. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001822The number of alignments of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001618The cardinality of the Frattini sublattice of a lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!