Identifier
Values
[[1,2]] => [[1],[2]] => [2,1] => [1,2] => 0
[[1],[2]] => [[1,2]] => [1,2] => [2,1] => 0
[[1,2,3]] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[[1,3],[2]] => [[1,2],[3]] => [3,1,2] => [1,3,2] => 1
[[1,2],[3]] => [[1,3],[2]] => [2,1,3] => [2,3,1] => 0
[[1],[2],[3]] => [[1,2,3]] => [1,2,3] => [3,2,1] => 1
[[1,2,3,4]] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[[1,3,4],[2]] => [[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => 1
[[1,2,4],[3]] => [[1,3],[2],[4]] => [4,2,1,3] => [1,3,4,2] => 1
[[1,2,3],[4]] => [[1,4],[2],[3]] => [3,2,1,4] => [2,3,4,1] => 0
[[1,3],[2,4]] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 1
[[1,2],[3,4]] => [[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => 1
[[1,4],[2],[3]] => [[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => 2
[[1,3],[2],[4]] => [[1,2,4],[3]] => [3,1,2,4] => [2,4,3,1] => 1
[[1,2],[3],[4]] => [[1,3,4],[2]] => [2,1,3,4] => [3,4,2,1] => 1
[[1],[2],[3],[4]] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 2
[[1,2,3,4,5]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => 1
[[1,2,4,5],[3]] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,2,4,5,3] => 1
[[1,2,3,5],[4]] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,3,4,5,2] => 1
[[1,2,3,4],[5]] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[[1,3,5],[2,4]] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,2,5,4] => 2
[[1,2,5],[3,4]] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[[1,3,4],[2,5]] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,3,1,5,4] => 1
[[1,2,4],[3,5]] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => 1
[[1,2,3],[4,5]] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,4,1,5,2] => 1
[[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => 2
[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,3,5,4,2] => 2
[[1,2,5],[3],[4]] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,4,5,3,2] => 2
[[1,3,4],[2],[5]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [2,3,5,4,1] => 1
[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,5,3,1] => 1
[[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,4,5,2,1] => 1
[[1,4],[2,5],[3]] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,5,4,3] => 2
[[1,3],[2,5],[4]] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[[1,2],[3,5],[4]] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [4,1,5,3,2] => 2
[[1,3],[2,4],[5]] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,2,5,4,1] => 2
[[1,2],[3,4],[5]] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [4,2,5,3,1] => 2
[[1,5],[2],[3],[4]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => 3
[[1,4],[2],[3],[5]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [2,5,4,3,1] => 2
[[1,3],[2],[4],[5]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [3,5,4,2,1] => 2
[[1,2],[3],[4],[5]] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [4,5,3,2,1] => 2
[[1],[2],[3],[4],[5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 3
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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].
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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
conjugate
Description
Sends a standard tableau to its conjugate tableau.