- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001960: Permutations ⟶ ℤ

Values

[[1,2]] => [1,2] => [2,1] => [1,2] => 0
[[1],[2]] => [2,1] => [1,2] => [2,1] => 0
[[1,2,3]] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [2,3,1] => [3,1,2] => 0
[[1,2],[3]] => [3,1,2] => [1,3,2] => [2,1,3] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => [2,3,1] => 0
[[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[[1,3,4],[2]] => [2,1,3,4] => [3,4,2,1] => [4,1,2,3] => 0
[[1,2,4],[3]] => [3,1,2,4] => [2,4,3,1] => [3,1,2,4] => 0
[[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => [2,1,3,4] => 0
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => [3,2,1,4] => 1
[[1,4],[2],[3]] => [3,2,1,4] => [2,3,4,1] => [3,4,1,2] => 0
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,4,2] => [2,4,1,3] => 0
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => [2,3,1,4] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[[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]] => [2,1,3,4,5] => [4,5,3,2,1] => [5,1,2,3,4] => 0
[[1,2,4,5],[3]] => [3,1,2,4,5] => [3,5,4,2,1] => [4,1,2,3,5] => 0
[[1,2,3,5],[4]] => [4,1,2,3,5] => [2,5,4,3,1] => [3,1,2,4,5] => 0
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [4,2,5,3,1] => [5,3,1,2,4] => 1
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,2,5,4,1] => [4,3,1,2,5] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [4,1,5,3,2] => [5,2,1,3,4] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [3,1,5,4,2] => [4,2,1,3,5] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,5,4,3] => [3,2,1,4,5] => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,4,5,2,1] => [4,5,1,2,3] => 0
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,5,3,1] => [3,5,1,2,4] => 0
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [2,3,5,4,1] => [3,4,1,2,5] => 0
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,4,5,3,2] => [2,5,1,3,4] => 0
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,3,5,4,2] => [2,4,1,3,5] => 0
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,4,1,5,2] => [4,5,2,1,3] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => [3,5,2,1,4] => 1
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,3,1,5,4] => [3,4,2,1,5] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,4,2,5,3] => [2,5,3,1,4] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,2,5,4] => [2,4,3,1,5] => 1
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,3,4,5,2] => [2,4,5,1,3] => 0
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,2,4,5,3] => [2,3,5,1,4] => 0
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => [2,3,4,1,5] => 0
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 0

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$F_{2} = 2$

$F_{3} = 4$

$F_{4} = 8 + 2\ q$

$F_{5} = 16 + 10\ q$

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

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

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

Lehmer code rotation

Description

Sends a permutation

$\pi$ to the unique permutation

$\tau$ (of the same length) such that every entry in the Lehmer code of

$\tau$ is cyclically one larger than the Lehmer code of

$\pi$ .