Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000354: Permutations ⟶ ℤ
Values
[[1,2]] => [1,2] => [1,2] => [1,2] => 0
[[1],[2]] => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [1,3,2] => [3,1,2] => 1
[[1,2],[3]] => [3,1,2] => [1,2,3] => [1,2,3] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]] => [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] => [3,1,2,4] => 1
[[1,2,4],[3]] => [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]] => [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[[1,2],[3,4]] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,3,4,5]] => [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] => [3,1,2,4,5] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,2,4,5,3] => [4,1,2,3,5] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [1,3,5,2,4] => [3,5,1,2,4] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => [1,5,2,3,4] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,3,4,2,5] => [1,3,2,4,5] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,2,4,3,5] => [1,4,2,3,5] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => [4,5,1,2,3] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => [3,5,1,2,4] => 2
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,2,5,3,4] => [1,5,2,3,4] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,3,4,2,5] => [1,3,2,4,5] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,2,4,3,5] => [1,4,2,3,5] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,4,2,5,3] => [4,1,5,2,3] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => [3,1,5,2,4] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]] => [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] => [3,1,2,4,5,6] => 1
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,2,4,5,6,3] => [4,1,2,3,5,6] => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,3,5,6,4] => [5,1,2,3,4,6] => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,5,6] => [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] => [3,5,1,2,4,6] => 2
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => 1
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [1,3,4,6,2,5] => [3,6,1,2,4,5] => 2
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,2,4,6,3,5] => [4,6,1,2,3,5] => 2
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 1
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => [1,3,2,4,5,6] => 1
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,2,4,5,3,6] => [1,4,2,3,5,6] => 1
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => [1,5,2,3,4,6] => 1
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,2,3,4,5,6] => [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] => [4,5,1,2,3,6] => 1
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,3,5,6,2,4] => [3,5,1,2,4,6] => 2
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => 1
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [1,3,4,6,2,5] => [3,6,1,2,4,5] => 2
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,2,4,6,3,5] => [4,6,1,2,3,5] => 2
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,3,4,5,2,6] => [1,3,2,4,5,6] => 1
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => [1,4,2,3,5,6] => 1
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => [1,5,2,3,4,6] => 1
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,3,4,5,6] => [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] => [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] => [1,2,5,3,4,6] => 1
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => [1,3,4,2,5,6] => 1
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => [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] => [6,4,5,1,2,3] => 2
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,3,6,2,5,4] => [6,3,5,1,2,4] => 3
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => 2
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [1,3,6,2,4,5] => [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] => [1,2,6,3,4,5] => 1
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [1,4,5,2,6,3] => [4,1,5,2,3,6] => 1
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [1,3,5,2,6,4] => [3,1,5,2,4,6] => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,2,5,3,6,4] => [5,1,6,2,3,4] => 1
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [1,3,4,2,6,5] => [3,1,6,2,4,5] => 2
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,2,4,3,6,5] => [4,1,6,2,3,5] => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,3,5,2,4,6] => [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] => [1,2,5,3,4,6] => 1
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,3,4,2,5,6] => [1,3,4,2,5,6] => 1
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,2,3,4,5,6] => [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] => [1,5,6,2,3,4] => 1
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,4,6,2,3,5] => [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] => [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] => [1,2,6,3,4,5] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => 1
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,3,5,2,4,6] => [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] => [1,2,5,3,4,6] => 1
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,3,4,2,5,6] => [1,3,4,2,5,6] => 1
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,4,5,6] => [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] => [1,4,2,5,3,6] => 1
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [1,3,2,5,4,6] => [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] => [1,5,2,3,4,6] => 1
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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.
Map
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
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