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Your data matches 234 different statistics following compositions of up to 3 maps.
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Matching statistic: St000649
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000649: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
[1,4,5,3,2] => 0
Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St000650
St000650: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The number of 3-rises of a permutation.
A 3-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+3 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]], for 2-rises see [[St000534]].
Matching statistic: St000648
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => 0
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000214
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => 0
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => 0
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 1
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000215
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => 0
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => 0
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 1
Description
The number of adjacencies of a permutation, zero appended.
An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended.
$(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000237
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000241
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,5,3,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 1
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000534
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,2,3] => [3,2,1] => 0
[2,1,3] => [1,2,3] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 0
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 0
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 0
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 0
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => 0
Description
The number of 2-rises of a permutation.
A 2-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+2 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]].
Matching statistic: St000022
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [3,4,2,1] => 0
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [3,1,2,4] => 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [3,4,2,1] => 0
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [3,1,2,4] => 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [3,4,2,1] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [3,4,2,1] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => [3,1,4,2] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,2,3] => [3,4,1,2] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => [3,1,4,2] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,2,3] => [3,4,1,2] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,2,3] => [3,4,1,2] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,2,3] => [3,4,1,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => [2,3,5,1,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => [2,3,5,1,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,5,3,4] => [2,3,5,1,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,5,3,4] => [2,3,5,1,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1
Description
The number of fixed points of a permutation.
Matching statistic: St000441
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 0
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 0
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => [1,4,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => [1,4,2,3] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
The following 224 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000502The number of successions of a set partitions. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001810The number of fixed points of a permutation smaller than its largest moved point. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000661The number of rises of length 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000944The 3-degree of an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001092The number of distinct even parts of a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001964The interval resolution global dimension of a poset. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000934The 2-degree of an integer partition. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000260The radius of a connected graph. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001877Number of indecomposable injective modules with projective dimension 2. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St000699The toughness times the least common multiple of 1,. St001498The normalised height of a Nakayama algebra with magnitude 1. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000454The largest eigenvalue of a graph if it is integral. 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. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001330The hat guessing number of a graph. St001651The Frankl number of a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001520The number of strict 3-descents. St001948The number of augmented double ascents 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. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001862The number of crossings of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000068The number of minimal elements in a poset. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001864The number of excedances of a signed permutation. St000731The number of double exceedences of a permutation. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001866The nesting alignments of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001867The number of alignments of type EN of a signed permutation. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. 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 $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001768The number of reduced words of a signed permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000679The pruning number of an ordered tree. St001624The breadth of a lattice. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 4. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph. St001271The competition number of a graph.
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