Identifier
Values
[(1,2)] => [2,1] => [1,2] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [3,2,1,4] => [4,3,2,1] => 0
[(1,3),(2,4)] => [3,4,1,2] => [4,1,2,3] => [1,3,4,2] => 0
[(1,4),(2,3)] => [4,3,2,1] => [1,4,3,2] => [2,1,3,4] => 0
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [3,2,5,4,1,6] => [4,3,6,5,2,1] => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [4,5,2,3,1,6] => [5,6,3,4,2,1] => 1
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [5,4,3,2,1,6] => [6,5,4,3,2,1] => 0
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [6,4,3,1,2,5] => [1,6,5,3,4,2] => 1
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,4,3,6,5,2] => [2,5,4,1,3,6] => 1
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,5,6,3,4,2] => [2,6,1,5,3,4] => 1
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [6,5,1,3,2,4] => [1,2,4,6,5,3] => 0
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [5,6,1,2,3,4] => [6,1,3,4,5,2] => 1
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [4,6,2,1,3,5] => [5,1,4,3,6,2] => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [3,2,6,1,4,5] => [4,3,1,5,6,2] => 1
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [3,2,1,6,5,4] => [4,3,2,1,5,6] => 0
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [4,1,2,6,5,3] => [5,2,3,1,4,6] => 1
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [5,1,6,2,4,3] => [6,2,1,4,3,5] => 1
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [6,1,5,4,2,3] => [1,3,2,4,6,5] => 0
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,6,5,4,3,2] => [2,1,3,4,5,6] => 0
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => [4,3,6,5,8,7,2,1] => 1
[(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => [8,5,4,7,6,3,2,1] => 1
[(1,6),(2,4),(3,5),(7,8)] => [6,4,5,2,3,1,8,7] => [7,5,6,3,4,2,1,8] => [8,6,7,4,5,3,2,1] => 1
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [5,6,7,2,3,4,1,8] => [6,7,8,3,4,5,2,1] => 2
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [4,6,2,7,3,5,1,8] => [5,7,3,8,4,6,2,1] => 1
[(1,5),(2,6),(3,4),(7,8)] => [5,6,4,3,1,2,8,7] => [6,7,5,4,2,3,1,8] => [7,8,6,5,3,4,2,1] => 2
[(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => 0
[(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,7,8] => 0
[(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [3,2,1,8,7,6,5,4] => [4,3,2,1,5,6,7,8] => 0
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [1,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8] => 0
[(1,8),(2,7),(3,6),(4,5),(9,10)] => [8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,1,10] => [10,9,8,7,6,5,4,3,2,1] => 0
[(1,6),(2,5),(3,4),(7,10),(8,9)] => [6,5,4,3,2,1,10,9,8,7] => [7,6,5,4,3,2,1,10,9,8] => [8,7,6,5,4,3,2,1,9,10] => 0
[(1,4),(2,3),(5,10),(6,9),(7,8)] => [4,3,2,1,10,9,8,7,6,5] => [5,4,3,2,1,10,9,8,7,6] => [6,5,4,3,2,1,7,8,9,10] => 0
[(1,2),(3,10),(4,9),(5,8),(6,7)] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,10,9,8,7,6,5,4] => [4,3,2,1,5,6,7,8,9,10] => 0
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [1,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10] => 0
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)] => [10,9,8,7,6,5,4,3,2,1,12,11] => [11,10,9,8,7,6,5,4,3,2,1,12] => [12,11,10,9,8,7,6,5,4,3,2,1] => 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
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.