Identifier
-
Mp00108:
Permutations
—cycle type⟶
Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001960: Permutations ⟶ ℤ
Values
[1,2] => [1,1] => [[1],[2]] => [2,1] => 0
[2,1] => [2] => [[1,2]] => [1,2] => 0
[1,2,3] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[1,3,2] => [2,1] => [[1,2],[3]] => [3,1,2] => 0
[2,1,3] => [2,1] => [[1,2],[3]] => [3,1,2] => 0
[2,3,1] => [3] => [[1,2,3]] => [1,2,3] => 0
[3,1,2] => [3] => [[1,2,3]] => [1,2,3] => 0
[3,2,1] => [2,1] => [[1,2],[3]] => [3,1,2] => 0
[1,2,3,4] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 2
[1,2,4,3] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 1
[1,3,2,4] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 1
[1,3,4,2] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[1,4,2,3] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[1,4,3,2] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 1
[2,1,3,4] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 1
[2,1,4,3] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[2,3,1,4] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[2,3,4,1] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[2,4,1,3] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[2,4,3,1] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[3,1,2,4] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[3,1,4,2] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[3,2,1,4] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 1
[3,2,4,1] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[3,4,1,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[3,4,2,1] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[4,1,2,3] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[4,1,3,2] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[4,2,1,3] => [3,1] => [[1,2,3],[4]] => [4,1,2,3] => 0
[4,2,3,1] => [2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => 1
[4,3,1,2] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[4,3,2,1] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,2,3,4,5] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 3
[1,2,3,5,4] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[1,2,4,3,5] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[1,2,4,5,3] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,2,5,3,4] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[1,3,2,4,5] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[1,3,2,5,4] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[1,3,4,2,5] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,3,4,5,2] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[1,3,5,2,4] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[1,3,5,4,2] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,4,2,3,5] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,4,2,5,3] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[1,4,3,2,5] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[1,4,3,5,2] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,4,5,2,3] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[1,4,5,3,2] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[1,5,2,3,4] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[1,5,2,4,3] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,5,3,2,4] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[1,5,3,4,2] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[1,5,4,2,3] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[1,5,4,3,2] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[2,1,3,4,5] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[2,1,3,5,4] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[2,1,4,3,5] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[2,1,4,5,3] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[2,1,5,3,4] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[2,1,5,4,3] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[2,3,1,4,5] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[2,3,1,5,4] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[2,3,4,1,5] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[2,3,4,5,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,3,5,1,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[2,4,1,3,5] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[2,4,1,5,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,4,3,1,5] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[2,4,3,5,1] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[2,4,5,1,3] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[2,4,5,3,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,5,1,3,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,5,1,4,3] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[2,5,3,1,4] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[2,5,3,4,1] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[2,5,4,1,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,5,4,3,1] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[3,1,2,4,5] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[3,1,2,5,4] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[3,1,4,2,5] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[3,1,4,5,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,1,5,2,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[3,2,1,4,5] => [2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => 2
[3,2,1,5,4] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[3,2,4,1,5] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[3,2,4,5,1] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[3,2,5,1,4] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[3,2,5,4,1] => [3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => 1
[3,4,1,2,5] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[3,4,1,5,2] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[3,4,2,1,5] => [4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => 0
[3,4,2,5,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,4,5,1,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[3,5,1,2,4] => [3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => 0
[3,5,1,4,2] => [2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => 1
[3,5,2,1,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
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Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
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
cycle type
Description
The cycle type of a permutation as a partition.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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