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On identities concerning the numbers of crossings and nestings of two edges in matchings

Martin Klazar
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Abstract

Let M,N be two matchings on [2n]={1, 2, ..., 2n} (possibly M=N) and for a nonnegative integer l let T(M,l) be the set of those matchings on [2n+2l] which can be obtained from M by successively adding l times in all ways the first edge, and similarly for T(N,l). Let s,t in {cr,ne} where cr is the statistic of the number of crossings (in a matching) and ne is the statistic of the number of nestings (possibly s=t). We prove that if the statistics s and t coincide on the sets of matchings T(M,l) and T(N,l) for l=0,1, they must coincide on these sets for every l >= 0; similar identities hold for the joint statistic of cr and ne. These results are instances of a general identity in which crossings and nestings are weighted by elements from an abelian group.

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Martin Klazar. 2005-03-01. On identities concerning the numbers of crossings and nestings of two edges in matchings. https://arxiv.org/abs/math/0503012

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