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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 $$l\geq 0$$.

##### MSC:
 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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