Identifier
Identifier
Values
[(1,2)] generating graphics... => 2
[(1,2),(3,4)] generating graphics... => 3
[(1,3),(2,4)] generating graphics... => 1
[(1,4),(2,3)] generating graphics... => 3
[(1,2),(3,4),(5,6)] generating graphics... => 4
[(1,2),(3,5),(4,6)] generating graphics... => 2
[(1,2),(3,6),(4,5)] generating graphics... => 4
[(1,3),(2,4),(5,6)] generating graphics... => 2
[(1,3),(2,5),(4,6)] generating graphics... => 2
[(1,3),(2,6),(4,5)] generating graphics... => 2
[(1,4),(2,3),(5,6)] generating graphics... => 4
[(1,4),(2,5),(3,6)] generating graphics... => 2
[(1,4),(2,6),(3,5)] generating graphics... => 2
[(1,5),(2,3),(4,6)] generating graphics... => 2
[(1,5),(2,4),(3,6)] generating graphics... => 2
[(1,5),(2,6),(3,4)] generating graphics... => 2
[(1,6),(2,3),(4,5)] generating graphics... => 4
[(1,6),(2,4),(3,5)] generating graphics... => 2
[(1,6),(2,5),(3,4)] generating graphics... => 4
[(1,2),(3,4),(5,8),(6,7)] generating graphics... => 5
[(1,2),(3,4),(5,7),(6,8)] generating graphics... => 3
[(1,2),(3,4),(5,6),(7,8)] generating graphics... => 5
[(1,2),(3,5),(4,8),(6,7)] generating graphics... => 3
[(1,2),(3,5),(4,7),(6,8)] generating graphics... => 3
[(1,2),(3,5),(4,6),(7,8)] generating graphics... => 3
[(1,2),(3,6),(4,8),(5,7)] generating graphics... => 3
[(1,2),(3,6),(4,7),(5,8)] generating graphics... => 3
[(1,2),(3,6),(4,5),(7,8)] generating graphics... => 5
[(1,2),(3,7),(4,8),(5,6)] generating graphics... => 3
[(1,2),(3,7),(4,6),(5,8)] generating graphics... => 3
[(1,2),(3,7),(4,5),(6,8)] generating graphics... => 3
[(1,2),(3,8),(4,5),(6,7)] generating graphics... => 5
[(1,2),(3,8),(4,6),(5,7)] generating graphics... => 3
[(1,2),(3,8),(4,7),(5,6)] generating graphics... => 5
[(1,3),(2,4),(5,8),(6,7)] generating graphics... => 3
[(1,3),(2,4),(5,7),(6,8)] generating graphics... => 1
[(1,3),(2,4),(5,6),(7,8)] generating graphics... => 3
[(1,3),(2,5),(4,8),(6,7)] generating graphics... => 3
[(1,3),(2,5),(4,7),(6,8)] generating graphics... => 1
[(1,3),(2,5),(4,6),(7,8)] generating graphics... => 3
[(1,3),(2,6),(4,8),(5,7)] generating graphics... => 3
[(1,3),(2,6),(4,7),(5,8)] generating graphics... => 1
[(1,3),(2,6),(4,5),(7,8)] generating graphics... => 3
[(1,3),(2,7),(4,8),(5,6)] generating graphics... => 3
[(1,3),(2,7),(4,6),(5,8)] generating graphics... => 1
[(1,3),(2,7),(4,5),(6,8)] generating graphics... => 3
[(1,3),(2,8),(4,5),(6,7)] generating graphics... => 3
[(1,3),(2,8),(4,6),(5,7)] generating graphics... => 1
[(1,3),(2,8),(4,7),(5,6)] generating graphics... => 3
[(1,4),(2,3),(5,8),(6,7)] generating graphics... => 5
[(1,4),(2,3),(5,7),(6,8)] generating graphics... => 3
[(1,4),(2,3),(5,6),(7,8)] generating graphics... => 5
[(1,4),(2,5),(3,8),(6,7)] generating graphics... => 3
[(1,4),(2,5),(3,7),(6,8)] generating graphics... => 1
[(1,4),(2,5),(3,6),(7,8)] generating graphics... => 3
[(1,4),(2,6),(3,8),(5,7)] generating graphics... => 1
[(1,4),(2,6),(3,7),(5,8)] generating graphics... => 3
[(1,4),(2,6),(3,5),(7,8)] generating graphics... => 3
[(1,4),(2,7),(3,8),(5,6)] generating graphics... => 3
[(1,4),(2,7),(3,6),(5,8)] generating graphics... => 3
[(1,4),(2,7),(3,5),(6,8)] generating graphics... => 1
[(1,4),(2,8),(3,5),(6,7)] generating graphics... => 3
[(1,4),(2,8),(3,6),(5,7)] generating graphics... => 1
[(1,4),(2,8),(3,7),(5,6)] generating graphics... => 3
[(1,5),(2,3),(4,8),(6,7)] generating graphics... => 3
[(1,5),(2,3),(4,7),(6,8)] generating graphics... => 3
[(1,5),(2,3),(4,6),(7,8)] generating graphics... => 3
[(1,5),(2,4),(3,8),(6,7)] generating graphics... => 3
[(1,5),(2,4),(3,7),(6,8)] generating graphics... => 3
[(1,5),(2,4),(3,6),(7,8)] generating graphics... => 3
[(1,5),(2,6),(3,8),(4,7)] generating graphics... => 3
[(1,5),(2,6),(3,7),(4,8)] generating graphics... => 1
[(1,5),(2,6),(3,4),(7,8)] generating graphics... => 3
[(1,5),(2,7),(3,8),(4,6)] generating graphics... => 1
[(1,5),(2,7),(3,6),(4,8)] generating graphics... => 3
[(1,5),(2,7),(3,4),(6,8)] generating graphics... => 3
[(1,5),(2,8),(3,4),(6,7)] generating graphics... => 3
[(1,5),(2,8),(3,6),(4,7)] generating graphics... => 1
[(1,5),(2,8),(3,7),(4,6)] generating graphics... => 3
[(1,6),(2,3),(4,8),(5,7)] generating graphics... => 3
[(1,6),(2,3),(4,7),(5,8)] generating graphics... => 3
[(1,6),(2,3),(4,5),(7,8)] generating graphics... => 5
[(1,6),(2,4),(3,8),(5,7)] generating graphics... => 1
[(1,6),(2,4),(3,7),(5,8)] generating graphics... => 1
[(1,6),(2,4),(3,5),(7,8)] generating graphics... => 3
[(1,6),(2,5),(3,8),(4,7)] generating graphics... => 3
[(1,6),(2,5),(3,7),(4,8)] generating graphics... => 3
[(1,6),(2,5),(3,4),(7,8)] generating graphics... => 5
[(1,6),(2,7),(3,8),(4,5)] generating graphics... => 3
[(1,6),(2,7),(3,5),(4,8)] generating graphics... => 1
[(1,6),(2,7),(3,4),(5,8)] generating graphics... => 3
[(1,6),(2,8),(3,4),(5,7)] generating graphics... => 3
[(1,6),(2,8),(3,5),(4,7)] generating graphics... => 1
[(1,6),(2,8),(3,7),(4,5)] generating graphics... => 3
[(1,7),(2,3),(4,8),(5,6)] generating graphics... => 3
[(1,7),(2,3),(4,6),(5,8)] generating graphics... => 3
[(1,7),(2,3),(4,5),(6,8)] generating graphics... => 3
[(1,7),(2,4),(3,8),(5,6)] generating graphics... => 3
[(1,7),(2,4),(3,6),(5,8)] generating graphics... => 1
[(1,7),(2,4),(3,5),(6,8)] generating graphics... => 1
[(1,7),(2,5),(3,8),(4,6)] generating graphics... => 1
[(1,7),(2,5),(3,6),(4,8)] generating graphics... => 1
[(1,7),(2,5),(3,4),(6,8)] generating graphics... => 3
[(1,7),(2,6),(3,8),(4,5)] generating graphics... => 3
[(1,7),(2,6),(3,5),(4,8)] generating graphics... => 3
[(1,7),(2,6),(3,4),(5,8)] generating graphics... => 3
[(1,7),(2,8),(3,4),(5,6)] generating graphics... => 3
[(1,7),(2,8),(3,5),(4,6)] generating graphics... => 1
[(1,7),(2,8),(3,6),(4,5)] generating graphics... => 3
[(1,8),(2,3),(4,5),(6,7)] generating graphics... => 5
[(1,8),(2,3),(4,6),(5,7)] generating graphics... => 3
[(1,8),(2,3),(4,7),(5,6)] generating graphics... => 5
[(1,8),(2,4),(3,5),(6,7)] generating graphics... => 3
[(1,8),(2,4),(3,6),(5,7)] generating graphics... => 3
[(1,8),(2,4),(3,7),(5,6)] generating graphics... => 3
[(1,8),(2,5),(3,4),(6,7)] generating graphics... => 5
[(1,8),(2,5),(3,6),(4,7)] generating graphics... => 3
[(1,8),(2,5),(3,7),(4,6)] generating graphics... => 3
[(1,8),(2,6),(3,4),(5,7)] generating graphics... => 3
[(1,8),(2,6),(3,5),(4,7)] generating graphics... => 3
[(1,8),(2,6),(3,7),(4,5)] generating graphics... => 3
[(1,8),(2,7),(3,4),(5,6)] generating graphics... => 5
[(1,8),(2,7),(3,5),(4,6)] generating graphics... => 3
[(1,8),(2,7),(3,6),(4,5)] generating graphics... => 5
click to show generating function       
Description
The number of vertices of the unicellular map given by a perfect matching.
If the perfect matching of $2n$ elements is viewed as a fixed point-free involution $\epsilon$ This statistic is counting the number of cycles of the permutation $\gamma \circ \epsilon$ where $\gamma$ is the long cycle $(1,2,3,\ldots,2n)$.
Example The perfect matching $[(1,3),(2,4)]$ corresponds to the permutation in $S_4$ with disjoint cycle decomposition $(1,3)(2,4)$. Then the permutation $(1,2,3,4)\circ (1,3)(2,4) = (1,4,3,2)$ has only one cycle.
Let $\epsilon_v(n)$ is the number of matchings of $2n$ such that yield $v$ cycles in the process described above. Harer and Zagier [1] gave the following expression for the generating series of the numbers $\epsilon_v(n)$.
$$ \sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k. $$
References
[1] Harer, J., Zagier, D. The Euler characteristic of the moduli space of curves MathSciNet:0848681
[2] Lando, S. K., Zvonkin, A. K. Graphs on surfaces and their applications MathSciNet:2036721
Code
def statistic(pm):
    p = Permutation(pm.to_permutation())*Permutation(list(range(2,pm.size()+1)) + [1,])
    return len(p.cycle_type())

Created
Mar 01, 2013 at 03:31 by Alejandro Morales
Updated
Nov 12, 2017 at 17:34 by Martin Rubey