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St000534: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 0 = 1 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 0 = 1 - 1
[1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 0 = 1 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => 0 = 1 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => 2 = 3 - 1
[1,2,4,5,3] => 1 = 2 - 1
[1,2,5,3,4] => 0 = 1 - 1
[1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => 2 = 3 - 1
[1,3,2,5,4] => 1 = 2 - 1
[1,3,4,2,5] => 1 = 2 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 3 = 4 - 1
[1,3,5,4,2] => 2 = 3 - 1
[1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => 0 = 1 - 1
[1,4,3,2,5] => 0 = 1 - 1
[1,4,3,5,2] => 1 = 2 - 1
[1,4,5,2,3] => 0 = 1 - 1
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]].
St000648: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 1 = 2 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,3,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => 1 = 2 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => 1 = 2 - 1
[1,4,3,2,5] => 1 = 2 - 1
[1,4,3,5,2] => 1 = 2 - 1
[1,4,5,2,3] => 2 = 3 - 1
Description
The number of 2-excedences of a permutation. This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Mp00090: Permutations cycle-as-one-line notationPermutations
St000214: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[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
[3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => 2 = 3 - 1
[4,2,3,1] => [1,4,2,3] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => 0 = 1 - 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
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 2 - 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
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 1 = 2 - 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].
Mp00090: Permutations cycle-as-one-line notationPermutations
St000237: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[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
[3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 2 = 3 - 1
[3,2,1,4] => [1,3,2,4] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 2 = 3 - 1
[3,4,1,2] => [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => 0 = 1 - 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
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 2 - 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
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 1 = 2 - 1
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00069: Permutations complementPermutations
St000441: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 2 = 3 - 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 1 = 2 - 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''.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
St001640: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 2 = 3 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 2 = 3 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 0 = 1 - 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St001810: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [3,2,1] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,2,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => 2 = 3 - 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => 2 = 3 - 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => 1 = 2 - 1
Description
The number of fixed points of a permutation smaller than its largest moved point.
Matching statistic: St000007
Mp00223: Permutations runsortPermutations
Mp00069: Permutations complementPermutations
Mp00149: Permutations Lehmer code rotationPermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [1,2] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,3,2] => 2
[2,1,3] => [1,3,2] => [3,1,2] => [1,3,2] => 2
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 3
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,2,5,3,4] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,2,5,4,3] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [1,5,2,3,4] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [1,5,2,4,3] => 3
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [1,5,4,2,3] => 3
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [1,5,4,3,2] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [1,4,3,2,5] => 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].
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[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]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[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]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[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]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[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]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[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]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[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]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[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]
=> [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]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[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]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[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]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[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]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[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]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[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]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[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]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[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]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[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]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000731
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck 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 - 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 = 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
[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 = 1 - 1
[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 = 1 - 1
[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 = 1 - 1
[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 = 1 - 1
[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 - 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 = 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 - 1
[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 = 3 - 1
[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 = 1 - 1
[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 = 1 - 1
[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 = 1 - 1
[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 = 1 - 1
[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 = 1 - 1
[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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 = 2 - 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 = 2 - 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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 1
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St000248The number of anti-singletons of a set partition. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St000247The number of singleton blocks 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000306The bounce count of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001142The projective dimension of the socle of the regular module as a bimodule 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 $eA$. 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 $eA$. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001571The Cartan determinant of the integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001732The number of peaks visible from the left. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000708The product of the parts of an integer 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. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001432The order dimension of the partition. St000260The radius of a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001128The exponens consonantiae of a partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. 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. St000456The monochromatic index of a connected graph. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001964The interval resolution global dimension of a poset. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000707The product of the factorials of the parts. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000706The product of the factorials of the multiplicities of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000454The largest eigenvalue of a graph if it is integral. St000681The Grundy value of Chomp on Ferrers diagrams. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000894The trace of an alternating sign matrix. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000455The second largest eigenvalue of a graph if it is integral. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001820The size of the image of the pop stack sorting operator. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. 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. St001330The hat guessing number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001889The size of the connectivity set of a signed permutation. St001937The size of the center of a parking function. St001868The number of alignments of type NE of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. 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. St001570The minimal number of edges to add to make a graph Hamiltonian.