- FindStat

Identifier

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤ

Values

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

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$F_{1} = q$

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

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

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

Description

The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux

$T_1$ and

$T_2$ is obtained by adding the largest entry of

$T_1$ to each entry of

$T_2$ , and then appending the rows of the result to

$T_1$ , see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.

Map

inverse zeta map

Description

The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.

Map

reflect parallelogram polyomino

Description

Reflect the corresponding parallelogram polyomino, such that the first column becomes the first row.
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.

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.