Identifier
Values
[1] => [1,0,1,0] => [[1,3],[2,4]] => [2,2] => 0
[2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,3] => 0
[1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,3,1] => -1
[2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,2,2] => 0
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The variation of a composition.
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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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 $T$.