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
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
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
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 $T$.