  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.
