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 $l, m \geq 0$, 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 $l, m \geq 0$. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings.