Identifier
Values
[(1,2)] => [2,1] => [2,1] => [1,2] => 0
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[(1,3),(2,4)] => [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 0
[(1,4),(2,3)] => [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 0
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,1,3,2,6,5] => [5,6,2,3,1,4] => 0
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,2,3,1,6,5] => [5,6,1,3,2,4] => 1
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [6,2,3,1,5,4] => [4,5,1,3,2,6] => 1
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 2
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,2,5,3,4,1] => [1,4,3,5,2,6] => 2
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [6,2,5,1,4,3] => [3,4,1,5,2,6] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6,1,5,2,4,3] => [3,4,2,5,1,6] => 0
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [6,1,3,2,5,4] => [4,5,2,3,1,6] => 0
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,3,5,4] => [4,5,3,6,1,2] => 0
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,4,5,3] => [3,5,4,6,1,2] => 1
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [6,4,5,1,3,2] => [2,3,1,5,4,6] => 1
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [6,1,5,3,4,2] => [2,4,3,5,1,6] => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,3,5,1,4,2] => [2,4,1,5,3,6] => 2
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [6,3,5,2,4,1] => [1,4,2,5,3,6] => 3
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => 0
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [6,4,5,2,3,1,8,7] => [7,8,1,3,2,5,4,6] => 2
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8,6,7,4,5,2,3,1] => [1,3,2,5,4,7,6,8] => 3
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [8,6,7,3,5,2,4,1] => [1,4,2,5,3,7,6,8] => 4
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [2,1,8,6,7,4,5,3] => [3,5,4,7,6,8,1,2] => 2
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8,5,7,3,6,2,4,1] => [1,4,2,6,3,7,5,8] => 5
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [2,1,8,4,7,3,6,5] => [5,6,3,7,4,8,1,2] => 1
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [4,2,3,1,8,6,7,5] => [5,7,6,8,1,3,2,4] => 2
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [8,5,7,4,6,2,3,1] => [1,3,2,6,4,7,5,8] => 4
[(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [8,4,7,3,6,2,5,1] => [1,5,2,6,3,7,4,8] => 6
[(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => 0
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Description
The number of occurrences of the pattern 13-2.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern $13\!\!-\!\!2$.
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
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.