Identifier
Values
[1] => [1,0,1,0] => [[1,3],[2,4]] => [3,1] => 0
[2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [2,3,1] => 1
[1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [4,2] => 0
[3] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [3,4,1] => 1
[2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [3,2,1] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [5,3] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [2,2,3,1] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [2,4,2] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [4,2,2] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [2,3,2,1] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [4,3,1] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [3,3,2] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [3,2,2,1] => 0
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Description
The number of ascents in an integer composition.
A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
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
peak composition
Description
The composition corresponding to the peak 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 peak, if $i-1$ is an ascent and $i$ is a descent.
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 peak set of $T$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.