Identifier
Values
[(1,2)] => [2,1] => {{1,2}} => {{1,2}} => 0
[(1,2),(3,4)] => [2,1,4,3] => {{1,2},{3,4}} => {{1,2,4},{3}} => 0
[(1,3),(2,4)] => [3,4,1,2] => {{1,3},{2,4}} => {{1,4},{2,3}} => 0
[(1,4),(2,3)] => [3,4,2,1] => {{1,3},{2,4}} => {{1,4},{2,3}} => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}} => {{1,2,4},{3,6},{5}} => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => {{1,3},{2,4},{5,6}} => {{1,4},{2,3,6},{5}} => 1
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => {{1,3},{2,4},{5,6}} => {{1,4},{2,3,6},{5}} => 1
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => {{1,3},{2,5},{4,6}} => {{1,6},{2,3,5},{4}} => 0
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => {{1,3},{2,5},{4,6}} => {{1,6},{2,3,5},{4}} => 0
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 0
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 0
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 0
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => {{1,3},{2,5},{4,6}} => {{1,6},{2,3,5},{4}} => 0
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => {{1,2},{3,5},{4,6}} => {{1,2,6},{3,5},{4}} => 0
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => {{1,2},{3,5},{4,6}} => {{1,2,6},{3,5},{4}} => 0
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => {{1,3},{2,5},{4,6}} => {{1,6},{2,3,5},{4}} => 0
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 0
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 0
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => {{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => 0
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,5),(2,6),(3,8),(4,7)] => [5,6,7,8,1,2,4,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,6),(2,5),(3,8),(4,7)] => [5,6,7,8,2,1,4,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,7),(2,5),(3,8),(4,6)] => [5,6,7,8,2,4,1,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,8),(2,5),(3,7),(4,6)] => [5,6,7,8,2,4,3,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,8),(2,6),(3,7),(4,5)] => [5,6,7,8,4,2,3,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,7),(2,6),(3,8),(4,5)] => [5,6,7,8,4,2,1,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,6),(2,7),(3,8),(4,5)] => [5,6,7,8,4,1,2,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,5),(2,7),(3,8),(4,6)] => [5,6,7,8,1,4,2,3] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,5),(2,8),(3,7),(4,6)] => [5,6,7,8,1,4,3,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,6),(2,8),(3,7),(4,5)] => [5,6,7,8,4,1,3,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,7),(2,8),(3,6),(4,5)] => [5,6,7,8,4,3,1,2] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => {{1,5},{2,6},{3,7},{4,8}} => {{1,8},{2,7},{3,6},{4,5}} => 0
[(1,6),(2,7),(3,8),(4,9),(5,10)] => [6,7,8,9,10,1,2,3,4,5] => {{1,6},{2,7},{3,8},{4,9},{5,10}} => {{1,10},{2,9},{3,8},{4,7},{5,6}} => 0
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => {{1,6},{2,7},{3,8},{4,9},{5,10}} => {{1,10},{2,9},{3,8},{4,7},{5,6}} => 0
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [7,8,9,10,11,12,6,5,4,3,2,1] => {{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}} => {{1,12},{2,11},{3,10},{4,9},{5,8},{6,7}} => 0
[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)] => [7,8,9,10,11,12,1,2,3,4,5,6] => {{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}} => {{1,12},{2,11},{3,10},{4,9},{5,8},{6,7}} => 0
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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
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
Map
inverse Yip
Description
The inverse of 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 $\psi^{-1}$.