Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000795: 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] => [1,3,2] => 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] => [1,4,3,2] => 2
[[1,2,4],[3]] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 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] => 2
[[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] => [1,5,4,3,2] => 3
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 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] => [1,5,3,2,4] => 3
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 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] => [1,5,2,4,3] => 4
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => [1,5,3,2,4] => 3
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => 2
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 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] => [1,5,4,2,3] => 3
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 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] => [1,5,2,3,4] => 3
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 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] => [1,6,5,4,3,2] => 4
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => 3
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,4,6,5] => [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] => [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] => [1,6,3,2,5,4] => 5
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => 4
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [1,3,4,6,2,5] => [1,6,4,3,2,5] => 4
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => 3
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 2
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => [1,5,4,3,2,6] => 3
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 2
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => [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] => [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] => [1,6,2,5,4,3] => 6
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => 5
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => 4
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [1,3,4,6,2,5] => [1,6,4,3,2,5] => 4
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => 3
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 2
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,3,4,5,2,6] => [1,5,4,3,2,6] => 3
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => 2
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => [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] => [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] => [1,5,3,2,4,6] => 3
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => 2
[[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] => [1,5,6,2,4,3] => 6
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,3,6,2,5,4] => [1,5,6,3,2,4] => 5
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,2,6,3,5,4] => [1,2,5,6,3,4] => 4
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => 4
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 3
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [1,4,5,2,6,3] => [1,6,5,2,4,3] => 5
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [1,3,5,2,6,4] => [1,6,5,3,2,4] => 4
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,2,5,3,6,4] => [1,2,6,5,3,4] => 3
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 3
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,2,4,3,6,5] => [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] => [1,2,3,4,6,5] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,3,5,2,4,6] => [1,5,3,2,4,6] => 3
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => 2
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => 2
[[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] => [1,6,2,5,3,4] => 6
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => 5
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => 4
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 3
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,2,3,6] => [1,5,2,4,3,6] => 4
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,3,5,2,4,6] => [1,5,3,2,4,6] => 3
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => 2
[[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] => [1,5,4,2,3,6] => 3
[[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] => [1,2,3,5,4,6] => 1
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Description
The mad of a permutation.
According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
This is the inverse of the map $\Phi$ in [1, sec.3].
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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