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Identifier
Values
[1] => [1,0,1,0] => [[1,3],[2,4]] => {{1,3},{2,4}} => 1
[2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => {{1,2,5},{3,4,6}} => 1
[1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => {{1,3,4},{2,5,6}} => 1
[2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => {{1,3,5},{2,4,6}} => 3
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The number of crossings of a set partition.
This is given by the number of i<i′<j<j′ such that i,j are two consecutive entries on one block, and i′,j′ are consecutive entries in another block.
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
rows
Description
The set partition whose blocks are the rows of the tableau.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.