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On identities concerning the numbers of crossings and nestings of two edges in matchings. (English) Zbl 1135.05006
Let \(M\) and \(N\) be two matchings on \([2n]\). For an integer \(l\geq 0\), let \({\mathcal T}(M,l)\) be the set of those matchings on \([2n+ 2l]\) which can be obtained from \(M\) by successively adding \(l\) times the first edge, and define similarly \({\mathcal T}(N,l)\). Let \(s,t\in\{\text{cr},\text{ne}\}\), where cr is the statistic of the number of crossings in a matching and ne is the statistic of the number of nestings. Among other more general results, the author shows that if the statistics \(s\) and \(t\) coincide on the set of matchings \({\mathcal T}(M,l)\) and \({\mathcal T}(N,l)\) for \(l= 0\) and \(1\), respectively, then they coincide on these sets for every \(l\geq 0\).

05A19 Combinatorial identities, bijective combinatorics
05C30 Enumeration in graph theory
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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