Identifier
Values
[(1,2)] => [(1,2)] => [2,1] => 0
[(1,2),(3,4)] => [(1,2),(3,4)] => [2,1,4,3] => 0
[(1,3),(2,4)] => [(1,4),(2,3)] => [3,4,2,1] => 1
[(1,4),(2,3)] => [(1,3),(2,4)] => [3,4,1,2] => 2
[(1,2),(3,4),(5,6)] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => 1
[(1,4),(2,3),(5,6)] => [(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => 2
[(1,5),(2,3),(4,6)] => [(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => 3
[(1,6),(2,3),(4,5)] => [(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => 4
[(1,6),(2,4),(3,5)] => [(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => 5
[(1,5),(2,4),(3,6)] => [(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => 4
[(1,4),(2,5),(3,6)] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => 3
[(1,3),(2,5),(4,6)] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => 2
[(1,2),(3,5),(4,6)] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => 1
[(1,2),(3,6),(4,5)] => [(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => 2
[(1,3),(2,6),(4,5)] => [(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => 3
[(1,4),(2,6),(3,5)] => [(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => 4
[(1,5),(2,6),(3,4)] => [(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => 5
[(1,6),(2,5),(3,4)] => [(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => 6
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Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
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
Kasraoui-Zeng
Description
The Kasraoui-Zeng involution for perfect matchings.
This yields the perfect matching with the number of nestings and crossings exchanged.