Identifier
Values
[1] => [1,0,1,0] => [[1,3],[2,4]] => [2,2] => 1
[2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,3] => 1
[1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,3,1] => 2
[3] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [4,4] => 1
[2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,2,2] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [2,4,2] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => [5,5] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [3,2,3] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [3,4,1] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [2,3,2,1] => 2
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[1,2,3,4,5,11],[6,7,8,9,10,12]] => [6,6] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [3,3,2] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [2,3,3] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [2,2,3,1] => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,2,2,2] => 1
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => [3,2,2,3] => 1
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [[1,2,3,5,7,11],[4,6,8,9,10,12]] => [4,2,2,4] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => [3,3,2,2] => 1
[4,2,2,1] => [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
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [[1,2,3,6,8,11],[4,5,7,9,10,12]] => [4,3,2,3] => 1
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [[1,2,4,5,8,11],[3,6,7,9,10,12]] => [3,3,3,3] => 1
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [[1,2,4,6,7,11],[3,5,8,9,10,12]] => [3,2,3,4] => 2
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[1,2,3,7,9,11],[4,5,6,8,10,12]] => [4,4,2,2] => 1
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[1,3,5,6,7,11],[2,4,8,9,10,12]] => [2,2,4,4] => 2
[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[1,2,5,7,8,11],[3,4,6,9,10,12]] => [3,3,3,3] => 1
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [2,2,2,2,2,2] => 1
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Description
The number of strong records in an integer composition.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record.
Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
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.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.