- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St001960: Permutations ⟶ ℤ

Values

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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = 2$

$F_{3} = 2 + 2\ q$

$F_{4} = 2 + 6\ q + 2\ q^{2}$

$F_{5} = 4 + 12\ q + 8\ q^{2} + 2\ q^{3}$

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

toric promotion

Description

Toric promotion of a permutation.
Let

$\sigma\in\mathfrak S_n$ be a permutation and let

$\tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases}$
The toric promotion operator is the product

$\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$ .
This is the special case of toric promotion on graphs for the path graph. Its order is

$n-1$ .

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

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.