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

TL;DR

This paper proves that if two matchings have identical crossing and nesting statistics after specific edge additions, then these statistics remain identical for all such extensions, revealing a fundamental symmetry in their combinatorial properties.

Contribution

It establishes a general identity showing the invariance of crossing and nesting statistics under certain edge addition operations in matchings.

Findings

Statistics coincide for initial matchings and their extensions.

Identities hold for joint crossing and nesting statistics.

Results apply to weighted versions with abelian group elements.

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.