- FindStat

Identifier

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000757: Integer compositions ⟶ ℤ

Values

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

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$F_{1} = q$

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

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

$F_{4} = q + 8\ q^{2} + 4\ q^{3} + q^{4}$

Description

The length of the longest weakly inreasing subsequence of parts of an integer composition.

Map

valley composition

Description

The composition corresponding to the valley 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 valley if

$i-1$ is a descent and

$i$ is an ascent.
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 valley set of

$T$ .

Map

bounce path

Description

Sends a Dyck path

$D$ of length

$2n$ to its bounce path.
This path is formed by starting at the endpoint

$(n,n)$ of

$D$ and travelling west until encountering the first vertical step of

$D$ , then south until hitting the diagonal, then west again to hit

$D$ , etc. until the point

$(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.

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.