Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000354: Permutations ⟶ ℤ
Values
[[1,2]] => [1,2] => [1,2] => 0
[[1],[2]] => [2,1] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]] => [3,1,2] => [1,2,3] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]] => [2,1,3,4] => [1,3,4,2] => 1
[[1,2,4],[3]] => [3,1,2,4] => [1,2,4,3] => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,2,3,4] => 0
[[1,3],[2,4]] => [2,4,1,3] => [1,3,2,4] => 1
[[1,2],[3,4]] => [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [1,4,2,3] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,2,4] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,3,4] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,2,4,5,3] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,3,5,4] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [1,3,5,2,4] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,3,4,2,5] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,2,4,3,5] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => 2
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,2,5,3,4] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,3,4,2,5] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,2,4,3,5] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,4,2,5,3] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,3,5,4] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,3,2,4,5] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,5,2,3,4] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,4,2,3,5] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,3,2,4,5] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 1
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,2,4,5,6,3] => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,3,5,6,4] => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,4,6,5] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [1,3,5,6,2,4] => 2
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => 1
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [1,3,4,6,2,5] => 2
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,2,4,6,3,5] => 2
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,3,6,4,5] => 1
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => 1
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,2,4,5,3,6] => 1
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => 1
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,2,3,4,5,6] => 0
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,4,5,6,2,3] => 1
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,3,5,6,2,4] => 2
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,2,5,6,3,4] => 1
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [1,3,4,6,2,5] => 2
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,2,4,6,3,5] => 2
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,3,6,4,5] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,3,4,5,2,6] => 1
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => 1
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => 1
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,3,4,5,6] => 0
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [1,3,5,2,4,6] => 2
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,2,5,3,4,6] => 1
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => 1
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,2,4,3,5,6] => 1
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [1,4,6,2,5,3] => 2
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,3,6,2,5,4] => 3
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,2,6,3,5,4] => 2
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [1,3,6,2,4,5] => 2
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,2,6,3,4,5] => 1
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [1,4,5,2,6,3] => 1
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [1,3,5,2,6,4] => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,2,5,3,6,4] => 1
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [1,3,4,2,6,5] => 2
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,2,4,3,6,5] => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,2,3,4,6,5] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,3,5,2,4,6] => 2
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,2,5,3,4,6] => 1
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,3,4,2,5,6] => 1
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,2,4,3,5,6] => 1
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,2,3,4,5,6] => 0
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,5,6,2,3,4] => 1
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,4,6,2,3,5] => 2
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [1,3,6,2,4,5] => 2
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,2,6,3,4,5] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,2,3,6] => 1
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,3,5,2,4,6] => 2
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,2,5,3,4,6] => 1
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,3,4,2,5,6] => 1
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,4,3,5,6] => 1
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,4,5,6] => 0
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [1,4,2,5,3,6] => 1
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [1,3,2,5,4,6] => 2
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [1,2,3,5,4,6] => 1
>>> Load all 331 entries. <<<
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Description
The number of recoils of a permutation.
A recoil, or inverse descent of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
A recoil, or inverse descent of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
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