Crossings and Nestings of Two Edges in Set Partitions

Abstract

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

19 pages, 2 figures