On Identities Concerning the Numbers of Crossings and Nestings of Two Edges in Matchings

Let M, N be two matchings on [2n] (possibly M = N) and for an integer l ≥ 0 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 ∈ {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.