- FindStat

Identifier

St000043: Perfect matchings ⟶ ℤ

Values

=>
                            Cc0012;cc-rep
                            
                            
                            

                        [(1,2)]=>0
[(1,2),(3,4)]=>0
[(1,3),(2,4)]=>1
[(1,4),(2,3)]=>2
[(1,2),(3,4),(5,6)]=>0
[(1,3),(2,4),(5,6)]=>1
[(1,4),(2,3),(5,6)]=>2
[(1,5),(2,3),(4,6)]=>3
[(1,6),(2,3),(4,5)]=>4
[(1,6),(2,4),(3,5)]=>5
[(1,5),(2,4),(3,6)]=>4
[(1,4),(2,5),(3,6)]=>3
[(1,3),(2,5),(4,6)]=>2
[(1,2),(3,5),(4,6)]=>1
[(1,2),(3,6),(4,5)]=>2
[(1,3),(2,6),(4,5)]=>3
[(1,4),(2,6),(3,5)]=>4
[(1,5),(2,6),(3,4)]=>5
[(1,6),(2,5),(3,4)]=>6
[(1,8),(2,3),(4,5),(6,7)]=>6
[(1,8),(2,4),(3,5),(6,7)]=>7
[(1,8),(2,5),(3,4),(6,7)]=>8
[(1,8),(2,6),(3,4),(5,7)]=>9
[(1,2),(3,8),(4,5),(6,7)]=>4
[(1,3),(2,8),(4,5),(6,7)]=>5
[(1,4),(2,8),(3,5),(6,7)]=>6
[(1,5),(2,8),(3,4),(6,7)]=>7
[(1,6),(2,8),(3,4),(5,7)]=>8
[(1,7),(2,8),(3,4),(5,6)]=>9
[(1,8),(2,7),(3,4),(5,6)]=>10
[(1,8),(2,7),(3,5),(4,6)]=>11
[(1,7),(2,8),(3,5),(4,6)]=>10
[(1,6),(2,8),(3,5),(4,7)]=>9
[(1,5),(2,8),(3,6),(4,7)]=>8
[(1,4),(2,8),(3,6),(5,7)]=>7
[(1,3),(2,8),(4,6),(5,7)]=>6
[(1,2),(3,8),(4,6),(5,7)]=>5
[(1,8),(2,6),(3,5),(4,7)]=>10
[(1,8),(2,5),(3,6),(4,7)]=>9
[(1,8),(2,4),(3,6),(5,7)]=>8
[(1,8),(2,3),(4,6),(5,7)]=>7
[(1,8),(2,3),(4,7),(5,6)]=>8
[(1,8),(2,4),(3,7),(5,6)]=>9
[(1,8),(2,5),(3,7),(4,6)]=>10
[(1,8),(2,6),(3,7),(4,5)]=>11
[(1,2),(3,8),(4,7),(5,6)]=>6
[(1,3),(2,8),(4,7),(5,6)]=>7
[(1,4),(2,8),(3,7),(5,6)]=>8
[(1,5),(2,8),(3,7),(4,6)]=>9
[(1,6),(2,8),(3,7),(4,5)]=>10
[(1,7),(2,8),(3,6),(4,5)]=>11
[(1,8),(2,7),(3,6),(4,5)]=>12
[(1,2),(3,4),(8,5),(6,7)]=>2
[(1,2),(3,4),(8,6),(5,7)]=>1
[(1,2),(3,4),(8,7),(5,6)]=>0
[(1,2),(3,5),(8,4),(6,7)]=>3
[(1,2),(3,5),(8,6),(4,7)]=>2
[(1,2),(3,5),(8,7),(4,6)]=>1
[(1,2),(3,6),(8,4),(5,7)]=>4
[(1,2),(3,6),(8,5),(4,7)]=>3
[(1,2),(3,6),(8,7),(4,5)]=>2
[(1,2),(3,7),(8,4),(5,6)]=>5
[(1,2),(3,7),(8,5),(4,6)]=>4
[(1,2),(3,7),(8,6),(4,5)]=>3
[(1,3),(2,4),(8,5),(6,7)]=>3
[(1,3),(2,4),(8,6),(5,7)]=>2
[(1,3),(2,4),(8,7),(5,6)]=>1
[(1,3),(2,5),(8,4),(6,7)]=>4
[(1,3),(2,5),(8,6),(4,7)]=>3
[(1,3),(2,5),(8,7),(4,6)]=>2
[(1,3),(2,6),(8,4),(5,7)]=>5
[(1,3),(2,6),(8,5),(4,7)]=>4
[(1,3),(2,6),(8,7),(4,5)]=>3
[(1,3),(2,7),(8,4),(5,6)]=>6
[(1,3),(2,7),(8,5),(4,6)]=>5
[(1,3),(2,7),(8,6),(4,5)]=>4
[(1,4),(2,3),(8,5),(6,7)]=>4
[(1,4),(2,3),(8,6),(5,7)]=>3
[(1,4),(2,3),(8,7),(5,6)]=>2
[(1,4),(2,5),(8,3),(6,7)]=>5
[(1,4),(2,5),(8,6),(3,7)]=>4
[(1,4),(2,5),(8,7),(3,6)]=>3
[(1,4),(2,6),(8,3),(5,7)]=>6
[(1,4),(2,6),(8,5),(3,7)]=>5
[(1,4),(2,6),(8,7),(3,5)]=>4
[(1,4),(2,7),(8,3),(5,6)]=>7
[(1,4),(2,7),(8,5),(3,6)]=>6
[(1,4),(2,7),(8,6),(3,5)]=>5
[(1,5),(2,3),(8,4),(6,7)]=>5
[(1,5),(2,3),(8,6),(4,7)]=>4
[(1,5),(2,3),(8,7),(4,6)]=>3
[(1,5),(2,4),(8,3),(6,7)]=>6
[(1,5),(2,4),(8,6),(3,7)]=>5
[(1,5),(2,4),(8,7),(3,6)]=>4
[(1,5),(2,6),(8,3),(4,7)]=>7
[(1,5),(2,6),(8,4),(3,7)]=>6
[(1,5),(2,6),(8,7),(3,4)]=>5
[(1,5),(2,7),(8,3),(4,6)]=>8
[(1,5),(2,7),(8,4),(3,6)]=>7
[(1,5),(2,7),(8,6),(3,4)]=>6
[(1,6),(2,3),(8,4),(5,7)]=>6
[(1,6),(2,3),(8,5),(4,7)]=>5
[(1,6),(2,3),(8,7),(4,5)]=>4
[(1,6),(2,4),(8,3),(5,7)]=>7
[(1,6),(2,4),(8,5),(3,7)]=>6
[(1,6),(2,4),(8,7),(3,5)]=>5
[(1,6),(2,5),(8,3),(4,7)]=>8
[(1,6),(2,5),(8,4),(3,7)]=>7
[(1,6),(2,5),(8,7),(3,4)]=>6
[(1,6),(2,7),(8,3),(4,5)]=>9
[(1,6),(2,7),(8,4),(3,5)]=>8
[(1,6),(2,7),(8,5),(3,4)]=>7
[(1,7),(2,3),(8,4),(5,6)]=>7
[(1,7),(2,3),(8,5),(4,6)]=>6
[(1,7),(2,3),(8,6),(4,5)]=>5
[(1,7),(2,4),(8,3),(5,6)]=>8
[(1,7),(2,4),(8,5),(3,6)]=>7
[(1,7),(2,4),(8,6),(3,5)]=>6
[(1,7),(2,5),(8,3),(4,6)]=>9
[(1,7),(2,5),(8,4),(3,6)]=>8
[(1,7),(2,5),(8,6),(3,4)]=>7
[(1,7),(2,6),(8,3),(4,5)]=>10
[(1,7),(2,6),(8,4),(3,5)]=>9
[(1,7),(2,6),(8,5),(3,4)]=>8
                    

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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