Identifier
Identifier
Values
[(1,2)] generating graphics... => 0
[(1,2),(3,4)] generating graphics... => 0
[(1,3),(2,4)] generating graphics... => 1
[(1,4),(2,3)] generating graphics... => 2
[(1,2),(3,4),(5,6)] generating graphics... => 0
[(1,2),(3,5),(4,6)] generating graphics... => 1
[(1,2),(3,6),(4,5)] generating graphics... => 2
[(1,3),(2,4),(5,6)] generating graphics... => 1
[(1,3),(2,5),(4,6)] generating graphics... => 2
[(1,3),(2,6),(4,5)] generating graphics... => 3
[(1,4),(2,3),(5,6)] generating graphics... => 2
[(1,4),(2,5),(3,6)] generating graphics... => 3
[(1,4),(2,6),(3,5)] generating graphics... => 4
[(1,5),(2,3),(4,6)] generating graphics... => 3
[(1,5),(2,4),(3,6)] generating graphics... => 4
[(1,5),(2,6),(3,4)] generating graphics... => 5
[(1,6),(2,3),(4,5)] generating graphics... => 4
[(1,6),(2,4),(3,5)] generating graphics... => 5
[(1,6),(2,5),(3,4)] generating graphics... => 6
[(1,2),(3,4),(8,5),(6,7)] generating graphics... => 2
[(1,2),(3,4),(8,6),(5,7)] generating graphics... => 1
[(1,2),(3,4),(8,7),(5,6)] generating graphics... => 0
[(1,2),(3,5),(8,4),(6,7)] generating graphics... => 3
[(1,2),(3,5),(8,6),(4,7)] generating graphics... => 2
[(1,2),(3,5),(8,7),(4,6)] generating graphics... => 1
[(1,2),(3,6),(8,4),(5,7)] generating graphics... => 4
[(1,2),(3,6),(8,5),(4,7)] generating graphics... => 3
[(1,2),(3,6),(8,7),(4,5)] generating graphics... => 2
[(1,2),(3,7),(8,4),(5,6)] generating graphics... => 5
[(1,2),(3,7),(8,5),(4,6)] generating graphics... => 4
[(1,2),(3,7),(8,6),(4,5)] generating graphics... => 3
[(1,2),(3,8),(4,5),(6,7)] generating graphics... => 4
[(1,2),(3,8),(4,6),(5,7)] generating graphics... => 5
[(1,2),(3,8),(4,7),(5,6)] generating graphics... => 6
[(1,3),(2,4),(8,5),(6,7)] generating graphics... => 3
[(1,3),(2,4),(8,6),(5,7)] generating graphics... => 2
[(1,3),(2,4),(8,7),(5,6)] generating graphics... => 1
[(1,3),(2,5),(8,4),(6,7)] generating graphics... => 4
[(1,3),(2,5),(8,6),(4,7)] generating graphics... => 3
[(1,3),(2,5),(8,7),(4,6)] generating graphics... => 2
[(1,3),(2,6),(8,4),(5,7)] generating graphics... => 5
[(1,3),(2,6),(8,5),(4,7)] generating graphics... => 4
[(1,3),(2,6),(8,7),(4,5)] generating graphics... => 3
[(1,3),(2,7),(8,4),(5,6)] generating graphics... => 6
[(1,3),(2,7),(8,5),(4,6)] generating graphics... => 5
[(1,3),(2,7),(8,6),(4,5)] generating graphics... => 4
[(1,3),(2,8),(4,5),(6,7)] generating graphics... => 5
[(1,3),(2,8),(4,6),(5,7)] generating graphics... => 6
[(1,3),(2,8),(4,7),(5,6)] generating graphics... => 7
[(1,4),(2,3),(8,5),(6,7)] generating graphics... => 4
[(1,4),(2,3),(8,6),(5,7)] generating graphics... => 3
[(1,4),(2,3),(8,7),(5,6)] generating graphics... => 2
[(1,4),(2,5),(8,3),(6,7)] generating graphics... => 5
[(1,4),(2,5),(8,6),(3,7)] generating graphics... => 4
[(1,4),(2,5),(8,7),(3,6)] generating graphics... => 3
[(1,4),(2,6),(8,3),(5,7)] generating graphics... => 6
[(1,4),(2,6),(8,5),(3,7)] generating graphics... => 5
[(1,4),(2,6),(8,7),(3,5)] generating graphics... => 4
[(1,4),(2,7),(8,3),(5,6)] generating graphics... => 7
[(1,4),(2,7),(8,5),(3,6)] generating graphics... => 6
[(1,4),(2,7),(8,6),(3,5)] generating graphics... => 5
[(1,4),(2,8),(3,5),(6,7)] generating graphics... => 6
[(1,4),(2,8),(3,6),(5,7)] generating graphics... => 7
[(1,4),(2,8),(3,7),(5,6)] generating graphics... => 8
[(1,5),(2,3),(8,4),(6,7)] generating graphics... => 5
[(1,5),(2,3),(8,6),(4,7)] generating graphics... => 4
[(1,5),(2,3),(8,7),(4,6)] generating graphics... => 3
[(1,5),(2,4),(8,3),(6,7)] generating graphics... => 6
[(1,5),(2,4),(8,6),(3,7)] generating graphics... => 5
[(1,5),(2,4),(8,7),(3,6)] generating graphics... => 4
[(1,5),(2,6),(8,3),(4,7)] generating graphics... => 7
[(1,5),(2,6),(8,4),(3,7)] generating graphics... => 6
[(1,5),(2,6),(8,7),(3,4)] generating graphics... => 5
[(1,5),(2,7),(8,3),(4,6)] generating graphics... => 8
[(1,5),(2,7),(8,4),(3,6)] generating graphics... => 7
[(1,5),(2,7),(8,6),(3,4)] generating graphics... => 6
[(1,5),(2,8),(3,4),(6,7)] generating graphics... => 7
[(1,5),(2,8),(3,6),(4,7)] generating graphics... => 8
[(1,5),(2,8),(3,7),(4,6)] generating graphics... => 9
[(1,6),(2,3),(8,4),(5,7)] generating graphics... => 6
[(1,6),(2,3),(8,5),(4,7)] generating graphics... => 5
[(1,6),(2,3),(8,7),(4,5)] generating graphics... => 4
[(1,6),(2,4),(8,3),(5,7)] generating graphics... => 7
[(1,6),(2,4),(8,5),(3,7)] generating graphics... => 6
[(1,6),(2,4),(8,7),(3,5)] generating graphics... => 5
[(1,6),(2,5),(8,3),(4,7)] generating graphics... => 8
[(1,6),(2,5),(8,4),(3,7)] generating graphics... => 7
[(1,6),(2,5),(8,7),(3,4)] generating graphics... => 6
[(1,6),(2,7),(8,3),(4,5)] generating graphics... => 9
[(1,6),(2,7),(8,4),(3,5)] generating graphics... => 8
[(1,6),(2,7),(8,5),(3,4)] generating graphics... => 7
[(1,6),(2,8),(3,4),(5,7)] generating graphics... => 8
[(1,6),(2,8),(3,5),(4,7)] generating graphics... => 9
[(1,6),(2,8),(3,7),(4,5)] generating graphics... => 10
[(1,7),(2,3),(8,4),(5,6)] generating graphics... => 7
[(1,7),(2,3),(8,5),(4,6)] generating graphics... => 6
[(1,7),(2,3),(8,6),(4,5)] generating graphics... => 5
[(1,7),(2,4),(8,3),(5,6)] generating graphics... => 8
[(1,7),(2,4),(8,5),(3,6)] generating graphics... => 7
[(1,7),(2,4),(8,6),(3,5)] generating graphics... => 6
[(1,7),(2,5),(8,3),(4,6)] generating graphics... => 9
[(1,7),(2,5),(8,4),(3,6)] generating graphics... => 8
[(1,7),(2,5),(8,6),(3,4)] generating graphics... => 7
[(1,7),(2,6),(8,3),(4,5)] generating graphics... => 10
[(1,7),(2,6),(8,4),(3,5)] generating graphics... => 9
[(1,7),(2,6),(8,5),(3,4)] generating graphics... => 8
[(1,7),(2,8),(3,4),(5,6)] generating graphics... => 9
[(1,7),(2,8),(3,5),(4,6)] generating graphics... => 10
[(1,7),(2,8),(3,6),(4,5)] generating graphics... => 11
[(1,8),(2,3),(4,5),(6,7)] generating graphics... => 6
[(1,8),(2,3),(4,6),(5,7)] generating graphics... => 7
[(1,8),(2,3),(4,7),(5,6)] generating graphics... => 8
[(1,8),(2,4),(3,5),(6,7)] generating graphics... => 7
[(1,8),(2,4),(3,6),(5,7)] generating graphics... => 8
[(1,8),(2,4),(3,7),(5,6)] generating graphics... => 9
[(1,8),(2,5),(3,4),(6,7)] generating graphics... => 8
[(1,8),(2,5),(3,6),(4,7)] generating graphics... => 9
[(1,8),(2,5),(3,7),(4,6)] generating graphics... => 10
[(1,8),(2,6),(3,4),(5,7)] generating graphics... => 9
[(1,8),(2,6),(3,5),(4,7)] generating graphics... => 10
[(1,8),(2,6),(3,7),(4,5)] generating graphics... => 11
[(1,8),(2,7),(3,4),(5,6)] generating graphics... => 10
[(1,8),(2,7),(3,5),(4,6)] generating graphics... => 11
[(1,8),(2,7),(3,6),(4,5)] generating graphics... => 12
click to show generating function       
Description
The number of crossings plus two-nestings of a perfect matching.

This is $C+2N$ where $C$ is the number of crossings (St000042The number of crossings of a perfect matching.) and $N$ is the number of nestings (St000041The number of nestings of a perfect matching.).
The generating series $\sum_{m} q^{\textrm{cn}(m)}$, where the sum is over the perfect matchings of $2n$ and $\textrm{cn}(m)$ is this statistic is $[2n-1]_q[2n-3]_q\cdots [3]_q[1]_q$ where $[m]_q = 1+q+q^2+\cdots + q^{m-1}$ [1, Equation (5,4)].
References
[1] Simion, R., Stanton, D. Octabasic Laguerre polynomials and permutation statistics MathSciNet:1418763
Code
def statistic(x):
    return len(x.crossings()) + 2*len(x.nestings())
Created
Mar 01, 2013 at 03:07 by Alejandro Morales
Updated
May 29, 2015 at 16:21 by Martin Rubey