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Mp00252: Permutations restrictionPermutations
St000325: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 2
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 2
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 2
[3,1,4,2] => [3,1,2] => 2
[3,2,1,4] => [3,2,1] => 3
[3,2,4,1] => [3,2,1] => 3
[3,4,1,2] => [3,1,2] => 2
[3,4,2,1] => [3,2,1] => 3
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 2
[4,3,2,1] => [3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => 3
Description
The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.
Mp00252: Permutations restrictionPermutations
St000470: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 2
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 2
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 2
[3,1,4,2] => [3,1,2] => 2
[3,2,1,4] => [3,2,1] => 3
[3,2,4,1] => [3,2,1] => 3
[3,4,1,2] => [3,1,2] => 2
[3,4,2,1] => [3,2,1] => 3
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 2
[4,3,2,1] => [3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => 3
Description
The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.
Mp00252: Permutations restrictionPermutations
St000021: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 0 = 1 - 1
[2,1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => 1 = 2 - 1
[2,3,1] => [2,1] => 1 = 2 - 1
[3,1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => 2 = 3 - 1
[3,2,4,1] => [3,2,1] => 2 = 3 - 1
[3,4,1,2] => [3,1,2] => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => 2 = 3 - 1
[4,1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => 2 = 3 - 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Mp00252: Permutations restrictionPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
St000155: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [2,1] => 1 = 2 - 1
[2,3,1] => [2,1] => [2,1] => 1 = 2 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [2,3,1] => 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [2,3,1] => 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [2,3,1] => 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [2,3,1] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Mp00252: Permutations restrictionPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [[1]]
=> 0 = 1 - 1
[2,1] => [1] => [[1]]
=> 0 = 1 - 1
[1,2,3] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [[1],[2]]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [[1],[2]]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [[1],[2]]
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Mp00252: Permutations restrictionPermutations
Mp00326: Permutations weak order rowmotionPermutations
St000245: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [2,1] => 0 = 1 - 1
[1,3,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 1 = 2 - 1
[2,3,1] => [2,1] => [1,2] => 1 = 2 - 1
[3,1,2] => [1,2] => [2,1] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [3,4,2,1] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [3,4,2,1] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [3,4,2,1] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [3,2,4,1] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [3,2,4,1] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [3,2,4,1] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [2,4,3,1] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [2,4,3,1] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [2,3,4,1] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [2,3,4,1] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [2,4,3,1] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [2,3,4,1] => 2 = 3 - 1
Description
The number of ascents of a permutation.
Mp00252: Permutations restrictionPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
St000662: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [2,1] => 1 = 2 - 1
[2,3,1] => [2,1] => [2,1] => 1 = 2 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [2,3,4,1] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [2,3,1,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [2,3,1,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [2,3,1,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [2,1,4,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [2,1,4,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [3,4,2,1] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [3,4,2,1] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [2,1,4,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [3,4,2,1] => 2 = 3 - 1
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Mp00252: Permutations restrictionPermutations
Mp00086: Permutations first fundamental transformationPermutations
St000703: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [2,1] => 1 = 2 - 1
[2,3,1] => [2,1] => [2,1] => 1 = 2 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [2,3,1] => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [2,3,1] => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [3,1,2] => 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [3,1,2] => 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [2,3,1] => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [3,1,2] => 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [2,3,1] => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [3,1,2] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2 = 3 - 1
Description
The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].
Mp00252: Permutations restrictionPermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1]
=> 1
[2,1] => [1] => [1] => [1]
=> 1
[1,2,3] => [1,2] => [2] => [2]
=> 1
[1,3,2] => [1,2] => [2] => [2]
=> 1
[2,1,3] => [2,1] => [1,1] => [1,1]
=> 2
[2,3,1] => [2,1] => [1,1] => [1,1]
=> 2
[3,1,2] => [1,2] => [2] => [2]
=> 1
[3,2,1] => [2,1] => [1,1] => [1,1]
=> 2
[1,2,3,4] => [1,2,3] => [3] => [3]
=> 1
[1,2,4,3] => [1,2,3] => [3] => [3]
=> 1
[1,3,2,4] => [1,3,2] => [2,1] => [2,1]
=> 2
[1,3,4,2] => [1,3,2] => [2,1] => [2,1]
=> 2
[1,4,2,3] => [1,2,3] => [3] => [3]
=> 1
[1,4,3,2] => [1,3,2] => [2,1] => [2,1]
=> 2
[2,1,3,4] => [2,1,3] => [1,2] => [2,1]
=> 2
[2,1,4,3] => [2,1,3] => [1,2] => [2,1]
=> 2
[2,3,1,4] => [2,3,1] => [2,1] => [2,1]
=> 2
[2,3,4,1] => [2,3,1] => [2,1] => [2,1]
=> 2
[2,4,1,3] => [2,1,3] => [1,2] => [2,1]
=> 2
[2,4,3,1] => [2,3,1] => [2,1] => [2,1]
=> 2
[3,1,2,4] => [3,1,2] => [1,2] => [2,1]
=> 2
[3,1,4,2] => [3,1,2] => [1,2] => [2,1]
=> 2
[3,2,1,4] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[3,4,1,2] => [3,1,2] => [1,2] => [2,1]
=> 2
[3,4,2,1] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[4,1,2,3] => [1,2,3] => [3] => [3]
=> 1
[4,1,3,2] => [1,3,2] => [2,1] => [2,1]
=> 2
[4,2,1,3] => [2,1,3] => [1,2] => [2,1]
=> 2
[4,2,3,1] => [2,3,1] => [2,1] => [2,1]
=> 2
[4,3,1,2] => [3,1,2] => [1,2] => [2,1]
=> 2
[4,3,2,1] => [3,2,1] => [1,1,1] => [1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => [3,1]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => [3,1]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [4] => [4]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => [3,1]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => [3,1]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => [3,1]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => [3,1]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000015
Mp00252: Permutations restrictionPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> [1,0]
=> 1
[2,1] => [1] => [.,.]
=> [1,0]
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[2,3,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 3
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 3
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 3
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3
Description
The number of peaks of a Dyck path.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000288The number of ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000168The number of internal nodes of an ordered tree. St000272The treewidth of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001277The degeneracy of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001935The number of ascents in a parking function.