- FindStat

Identifier

Mp00163: Signed permutations —permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤ

Values

[1] => [1] => [1,0] => [[1],[2]] => 1
[-1] => [1] => [1,0] => [[1],[2]] => 1
[1,2] => [1,2] => [1,0,1,0] => [[1,3],[2,4]] => 2
[1,-2] => [1,2] => [1,0,1,0] => [[1,3],[2,4]] => 2
[-1,2] => [1,2] => [1,0,1,0] => [[1,3],[2,4]] => 2
[-1,-2] => [1,2] => [1,0,1,0] => [[1,3],[2,4]] => 2
[2,1] => [2,1] => [1,1,0,0] => [[1,2],[3,4]] => 2
[2,-1] => [2,1] => [1,1,0,0] => [[1,2],[3,4]] => 2
[-2,1] => [2,1] => [1,1,0,0] => [[1,2],[3,4]] => 2
[-2,-1] => [2,1] => [1,1,0,0] => [[1,2],[3,4]] => 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[1,2,-3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[1,-2,3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[1,-2,-3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[-1,2,3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[-1,2,-3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[-1,-2,3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[-1,-2,-3] => [1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[1,3,-2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[1,-3,2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[1,-3,-2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[-1,3,2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[-1,3,-2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[-1,-3,2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[-1,-3,-2] => [1,3,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 3
[2,1,3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[2,1,-3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[2,-1,3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[2,-1,-3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[-2,1,3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[-2,1,-3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[-2,-1,3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[-2,-1,-3] => [2,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[2,3,1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[2,3,-1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[2,-3,1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[2,-3,-1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[-2,3,1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[-2,3,-1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[-2,-3,1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[-2,-3,-1] => [2,3,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[3,1,-2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[3,-1,2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[3,-1,-2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,1,2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,1,-2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,-1,2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,-1,-2] => [3,1,2] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[3,2,-1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[3,-2,1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[3,-2,-1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,2,1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,2,-1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,-2,1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[-3,-2,-1] => [3,2,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3

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$F_{1} = 2\ q$

$F_{2} = 8\ q^{2}$

$F_{3} = 16\ q^{2} + 32\ q^{3}$

Description

The orbit size of a standard tableau under promotion.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

permutation

Description

The permutation obtained by forgetting the colours.

Map

to two-row standard tableau

Description

Return a standard tableau of shape

$(n,n)$ where

$n$ is the semilength of the Dyck path.
Given a Dyck path

$D$ , its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.