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Your data matches 179 different statistics following compositions of up to 3 maps.
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Matching statistic: St000534
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[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] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 2
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 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: St000648
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[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] => 1
[3,2,1] => 1
[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] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 2
[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] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
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 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[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] => 1
[3,2,1] => [1,3,2] => 1
[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] => 1
[1,4,3,2] => [1,2,4,3] => 1
[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] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => 0
[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] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[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] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0
[1,4,5,2,3] => [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: St000237
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[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] => 1
[3,2,1] => [1,3,2] => 1
[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] => 1
[1,4,3,2] => [1,2,4,3] => 1
[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] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => 0
[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] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[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] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [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: St000441
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[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] => 1
[3,2,1] => [1,3,2] => [3,1,2] => 1
[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] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1
[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] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 0
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 0
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 2
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 0
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 2
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 0
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0
[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] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[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] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,4,5,2,3] => [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''.
Matching statistic: St001640
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 0
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 2
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 0
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 2
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 0
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 0
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 0
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 0
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 1
[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] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 0
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St001810
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001810: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001810: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => 2
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => 2
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => 0
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => 0
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => 0
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 0
[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] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => 1
Description
The number of fixed points of a permutation smaller than its largest moved point.
Matching statistic: St000731
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 2
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St001216
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001216: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001216: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,4,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,2,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,2,5,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,2,5,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
Description
The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module.
Matching statistic: St000007
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1 = 0 + 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1 = 0 + 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1 = 0 + 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [1,5,2,3,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [1,5,2,4,3] => 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [1,5,4,2,3] => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [1,5,4,3,2] => 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1 = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [1,4,3,2,5] => 1 = 0 + 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].
The following 169 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000248The number of anti-singletons of a set partition. St000502The number of successions of a set partitions. St000247The number of singleton blocks of a set partition. St000504The cardinality of the first block of a set partition. St001948The number of augmented double ascents of a permutation. St001644The dimension of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000150The floored half-sum of the multiplicities of a partition. St000377The dinv defect of an integer partition. St000480The number of lower covers of a partition in dominance order. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001274The number of indecomposable injective modules with projective dimension equal to two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001484The number of singletons of an 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. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001730The number of times the path corresponding to a binary word crosses the base line. St000506The number of standard desarrangement tableaux of shape equal to the given partition. 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. 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. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000658The number of rises of length 2 of a Dyck path. St000934The 2-degree of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000137The Grundy value of an integer partition. 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. St001175The size of a partition minus the hook length of the base cell. 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. St001525The number of symmetric hooks on the diagonal of a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000260The radius of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. 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. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001877Number of indecomposable injective modules with projective dimension 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001596The number of two-by-two squares inside a skew partition. St001964The interval resolution global dimension of a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St001280The number of parts of an integer partition that are at least two. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000699The toughness times the least common multiple of 1,. St000455The second largest eigenvalue of a graph if it is integral. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000936The number of even values of the symmetric group character corresponding to the partition. St000454The largest eigenvalue of a graph if it is integral. 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. St000944The 3-degree of an integer partition. 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. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001570The minimal number of edges to add to make a graph Hamiltonian. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001651The Frankl number of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000894The trace of an alternating sign matrix. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001568The smallest positive integer that does not appear twice in the partition. 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. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001820The size of the image of the pop stack sorting operator. St001561The value of the elementary symmetric function evaluated at 1. 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 $eA$. St000850The number of 1/2-balanced pairs in a poset. St001399The distinguishing number of a poset. St000633The size of the automorphism group of a poset. St001624The breadth of a lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000451The length of the longest pattern of the form k 1 2. St000842The breadth 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. St001330The hat guessing number of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001937The size of the center of a parking function. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001060The distinguishing index of a graph.
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