Identifier
Values
[(1,2)] => [2,1] => [[.,.],.] => [1,2] => 0
[(1,2),(3,4)] => [2,1,4,3] => [[.,.],[[.,.],.]] => [1,3,4,2] => 0
[(1,3),(2,4)] => [3,4,1,2] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 0
[(1,4),(2,3)] => [3,4,2,1] => [[[.,.],.],[.,.]] => [1,2,4,3] => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 1
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 1
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 0
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 0
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => 0
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => 0
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 0
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 0
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 0
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 0
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 0
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => 0
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => 0
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 0
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => [1,3,5,7,8,6,4,2] => 1
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]] => [2,1,4,3,8,7,6,5] => 0
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]] => [2,1,4,3,8,7,6,5] => 0
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]] => [2,1,4,3,8,7,6,5] => 0
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] => [4,3,2,1,8,7,6,5] => 0
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
Description
The genus of a permutation.
The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation
$$ n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ), $$
where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.
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
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.