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
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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
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 $\pi$.