Identifier
Values
[(1,2)] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => 2
[(1,3),(2,4)] => [3,4,1,2] => 0
[(1,4),(2,3)] => [3,4,2,1] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 6
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => 4
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => 5
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => 3
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => 4
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => 2
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => 0
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => 2
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => 4
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => 5
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => 3
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => 2
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => 3
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Description
The number of cyclic alignments of a permutation.
The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, j, \pi(j), \pi(i)$ are cyclically ordered and all distinct, see Section 5 of [1]
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.