Identifier
Values
[1,0] => [[1],[2]] => {{1},{2}} => {{1},{2}} => 0
[1,0,1,0] => [[1,3],[2,4]] => {{1,3},{2,4}} => {{1,4},{2,3}} => 0
[1,1,0,0] => [[1,2],[3,4]] => {{1,2},{3,4}} => {{1,2,4},{3}} => 0
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => {{1,3,5},{2,4,6}} => {{1,4,6},{2,3,5}} => 1
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => {{1,3,4},{2,5,6}} => {{1,5},{2,3,4,6}} => 1
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => {{1,2,5},{3,4,6}} => {{1,2,4},{3,5,6}} => 1
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => {{1,2,4},{3,5,6}} => {{1,2},{3,4,5,6}} => 0
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => {{1,2,3,5,6},{4}} => 0
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => {{1,3,4,5},{2,6,7,8}} => {{1,6},{2,3,4,5,7,8}} => 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}} => {{1,2,3,4,6,7,8},{5}} => 0
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Description
The number of internal points of a set partition.
An element $e$ is internal, if there are $f < e < g$ such that the blocks of $f$ and $g$ have larger minimal element than the block of $e$. See Section 5.5 of [1]
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
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
Yip
Description
A transformation of set partitions due to Yip.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Yip's bijection $\psi$.