Crossings and Nestings of Two Edges in Set Partitions
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Let and be two set partitions with the same number of blocks. Assume is a partition of [n]. For any integer l, m 0, let T(, l) be the set of partitions of [n + l]whose restrictions to the last n elements are isomorphic to , and T(, l, m) the subset of T(, l) consisting of those partitions with exactly m blocks. Similarly define T(, l) and T(, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T(, l) and T(, l)for l = 0, 1, then it coincides on T(,l, m)and T(, l, m) for all l, m 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. 2009 Society for Industrial and Applied Mathematics.
