- FindStat

Identifier

St001997: Perfect matchings ⟶ ℤ

Values

=>
                            Cc0012;cc-rep
                            
                            
                            

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

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Description

The number of edges in common with the identity matching.
Can be seen as an analogue of counting the number of fixed points of a permutation St000022The number of fixed points of a permutation.. After fixing one perfect matching as the identity matching, we can count the number of edges that a perfect matching has in common with the identity matching. Usually, one chooses the matching $12, 34, 56, \ldots$ as the identity matching.
A deranged matching is a matching that has no edges in common with the identity matching. Corresponds to counting derangements in the symmetric group. The problem of counting deranged matchings was posed in [1] and solved in [2]. For more information on deranged matchings, see [3].

References

[1] Brawner, J. N. Dinner, Dancing, and Tennis, Anyone? MathSciNet:1573426
[2] Margolius, B. H. Avoiding your spouse at a bridge party MathSciNet:2075960
[3] Johnston, D., Kayll, P. M., Palmer, C. Deranged matchings: proofs and conjectures arXiv:2209.11319

Code

#Sage code, sample usage:
#M = PerfectMatching([(1, 3), (2, 4)])
#statistic(M)
#Assuming 12,34,56,... as identity matching
def statistic(x):
    n = len(x)
    identity = PerfectMatching([(2*i - 1, 2*i) for i in range(1, n + 1)])
    return len(set(x).intersection(identity))

Created

Sep 20, 2026 at 17:26 by Lukas Klawuhn

Updated

Sep 20, 2026 at 17:26 by Lukas Klawuhn