AMS subject classifications. 05A18, 05A15 Let π and λ be two set partitions with the same number of blocks. Assume π is a partition of [n]. For any integer l, m ≥ 0, let T (π, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to π, and T (π, l, m) the subset of T (π, l) consisting of those partitions with exactly m blocks. Similarly define T (λ, l) and T (λ, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (π, l) and T (λ, l) for l = 0, 1, then it coincides on T (π, l, m) and T (λ, l, m) for all l, m ≥ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. 1 Introduction and Statement of Main Result In a recent paper [5], Klazar studied distributions of the numbers of crossings and nestings of two edges in (perfect) matchings. All matchings form an infinite tree T rooted at the empty matching ∅, in which the children of a matching M are the matchings obtained from M by adding to M in all possible ways a new first edge. Given two matchings M and N on [2n], Klazar decided whe
