- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤ

Values

[1] => [1,0,1,0] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 3
[2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 3
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 5
[2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 8
[] => [] => [1,0] => [[1],[2]] => 0

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$F_{1} = q^{3}$

$F_{2} = q^{3} + q^{5}$

Description

The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation.
The natural comajor index of a tableau of size

$n$ with natural descent set

$D$ is then

$\sum_{d\in D} n-d$ .

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.

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.