- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤ

Values

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

$F_{1} = q$

$F_{2} = 2\ q$

Description

The number of connected components of a skew partition.

Map

Cori-Le Borgne involution

Description

The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite

$\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$ , where

$\zeta$ is Mp00030zeta map.

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of

$\gamma(D)$ .

Map

to Dyck path

Description

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