- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤ

Values

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

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$F_{1} = 2\ q^{2}$

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

Description

The orbit size of a standard tableau under promotion.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.

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.