Identifier
Values
[(1,2)] => [2,1] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 2
[(1,3),(2,4)] => [3,4,1,2] => [4,3,2,1] => 4
[(1,4),(2,3)] => [3,4,2,1] => [3,4,2,1] => 4
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,3,2,1,6,5] => 5
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [3,4,2,1,6,5] => 5
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [3,6,4,5,1,2] => 7
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [5,4,2,6,3,1] => 7
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [4,6,5,3,2,1] => 9
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [6,4,5,3,2,1] => 9
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [6,5,4,3,2,1] => 9
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [6,3,4,5,1,2] => 7
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,6,5,4,3] => 5
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,5,6,4,3] => 5
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [5,3,2,6,4,1] => 7
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [5,6,4,3,2,1] => 9
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [4,6,5,3,2,1] => 9
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [4,5,6,3,2,1] => 9
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of visible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.
Map
catalanization
Description
The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
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.