- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000768: Integer compositions ⟶ ℤ

Values

[1] => [1,0,1,0] => [[1,3],[2,4]] => [3,1] => 0
[2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [2,3,1] => 1
[1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [4,2] => 0
[3] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [3,4,1] => 1
[2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [3,2,1] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [5,3] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [2,2,3,1] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [2,4,2] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [4,2,2] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [2,3,2,1] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [4,3,1] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [3,3,2] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [3,2,2,1] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => [3,3,2,2] => 0
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [[1,3,4,6,7,9],[2,5,8,10,11,12]] => [4,3,2,3] => 0
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [[1,3,5,6,8,9],[2,4,7,10,11,12]] => [3,3,3,3] => 0
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [[1,3,4,7,8,10],[2,5,6,9,11,12]] => [4,4,2,2] => 0

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + q$

Description

The number of peaks in an integer composition.
A peak is an ascent followed by a descent, i.e., a subsequence

$c_{i-1} c_i c_{i+1}$ with

$c_i > \max(c_{i-1}, c_{i+1})$ .

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

peak composition

Description

The composition corresponding to the peak set of a standard tableau.
Let

$T$ be a standard tableau of size

$n$ .
An entry

$i$ of

$T$ is a descent if

$i+1$ is in a lower row (in English notation), otherwise

$i$ is an ascent.
An entry

$2 \leq i \leq n-1$ is a peak, if

$i-1$ is an ascent and

$i$ is a descent.
This map returns the composition

$c_1,\dots,c_k$ of

$n$ such that

$\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the peak set of

$T$ .

Map

to Dyck path

Description

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