Crossings and Nestings of Two Edges in Set Partitions

Abstract

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

19 pages, 2 figures

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