CROSSINGS AND NESTINGS OF TWO EDGES IN SET PARTITIONS

Let pi and lambda be two set partitions with the same number of blocks. Assume pi is a partition of [n]. For any integer l, m >= 0, let T (pi, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to p, and T (pi, l, m) the subset of T (pi, l) consisting of those partitions with exactly m blocks. Similarly define T (lambda, l) and T (lambda, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T(pi, l) and T (lambda, l) for l = 0, 1, then it coincides on T (pi, l, m) and T (lambda, l, m) for all l, m >= 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings.