Identifier
Values
[1,0] => [[1],[2]] => [2] => [2] => 2
[1,0,1,0] => [[1,3],[2,4]] => [2,2] => [2,2] => 2
[1,1,0,0] => [[1,2],[3,4]] => [3,1] => [3,1] => 3
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,2,2] => [2,2,2] => 2
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,3,1] => [2,3,1] => 3
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,3] => [3,3] => 3
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [3,2,1] => [3,2,1] => 3
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [4,2] => [4,2] => 4
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,2,2,2] => [2,2,2,2] => 2
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [2,2,3,1] => [2,2,3,1] => 3
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [2,3,3] => [2,3,3] => 3
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [2,3,2,1] => [3,2,2,1] => 3
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [2,4,2] => [4,2,2] => 4
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [3,3,2] => [3,3,2] => 3
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [3,4,1] => [3,4,1] => 4
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [3,2,3] => [3,2,3] => 3
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [3,2,2,1] => [2,3,2,1] => 3
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [3,3,2] => [3,3,2] => 3
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [4,4] => [4,4] => 4
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [4,3,1] => [4,3,1] => 4
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => [4,2,2] => [2,4,2] => 4
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [5,3] => [5,3] => 5
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => [2,2,3,3] => [2,2,3,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => [2,3,3,2] => [3,3,2,2] => 3
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => [2,3,2,3] => [3,2,2,3] => 3
[1,0,1,1,0,1,1,0,0,0] => [[1,3,4,6,7],[2,5,8,9,10]] => [2,3,3,2] => [3,3,2,2] => 3
[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => [5,5] => [5,5] => 5
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Description
The largest part of an integer composition.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00314Foata bijection.
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
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.