Identifier
Values
[1,0] => [[1],[2]] => [[1,2]] => [2] => 1
[1,0,1,0] => [[1,3],[2,4]] => [[1,2,4],[3]] => [3,1] => 1
[1,1,0,0] => [[1,2],[3,4]] => [[1,2,3,4]] => [4] => 1
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [[1,2,4,6],[3,5]] => [3,2,1] => 1
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [[1,2,4,5,6],[3]] => [3,3] => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [[1,2,3,4,6],[5]] => [5,1] => 1
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [[1,2,3,5,6],[4]] => [4,2] => 1
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [[1,2,3,4,5,6]] => [6] => 1
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [[1,2,4,6,8],[3,5,7]] => [3,2,2,1] => 1
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [[1,2,4,6,7,8],[3,5]] => [3,2,3] => 2
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [[1,2,4,5,6,8],[3,7]] => [3,4,1] => 2
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [[1,2,4,5,7,8],[3,6]] => [3,3,2] => 2
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [[1,2,4,5,6,7,8],[3]] => [3,5] => 2
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [[1,2,3,4,6,8],[5,7]] => [5,2,1] => 1
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [[1,2,3,4,7,8],[5,6]] => [5,3] => 1
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [[1,2,3,5,6,8],[4,7]] => [4,3,1] => 1
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [[1,2,3,5,7,8],[4,6]] => [4,2,2] => 1
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [[1,2,3,5,6,7,8],[4]] => [4,4] => 2
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [[1,2,3,4,5,6,8],[7]] => [7,1] => 1
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [[1,2,3,4,5,7,8],[6]] => [6,2] => 1
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => [[1,2,3,4,6,7,8],[5]] => [5,3] => 1
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [[1,2,3,4,5,6,7,8]] => [8] => 1
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Description
The number of weak records in an integer composition.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
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
catabolism
Description
Remove the first row of the standard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry $k$ into tableau $T$ is as follows:
If $k$ is larger than all entries in the first column, place $k$ at the bottom of the first column and the procedure is finished. Otherwise, place $k$ in the first column, replacing the smallest entry, $y$, greater than $k$. Now insert $y$ into the second column using the same procedure: if $y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place $y$ in the second column, replacing, or 'bumping', the smallest entry, $z$, larger than $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).
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 $T$.