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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}} => 1
[1,1,0,0] => [[1,2],[3,4]] => {{1,2},{3,4}} => {{1,2},{3,4}} => 1
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => {{1,3,5},{2,4,6}} => {{1,4,5},{2,3,6}} => 2
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => {{1,3,4},{2,5,6}} => {{1,5,6},{2,3,4}} => 1
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => {{1,2,5},{3,4,6}} => {{1,2,4,5},{3,6}} => 2
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => {{1,2,4},{3,5,6}} => {{1,2,5,6},{3,4}} => 1
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => 1
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => {{1,2,3,5},{4,6,7,8}} => {{1,2,3,6,7,8},{4,5}} => 1
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Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
Map
inverse Wachs-White-rho
Description
The inverse of a transformation of set partitions due to Wachs and White.
Return the set partition of {1,...,n} corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection ρ−1.
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with (i,j) to the cell in the i-th column from the right and the j-th row from the top.
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.