Crossings and Nestings of Two Edges in Set Partitions

TL;DR

This paper proves that if the crossing and nesting statistics are equal for certain set partitions, then they are equal for all related partitions, extending previous results on matchings.

Contribution

It establishes a general equivalence for crossing and nesting statistics across extended classes of set partitions, broadening Klazar's earlier work on matchings.

Findings

Crossing and nesting statistics coincide for base partitions imply they coincide for all extensions.

Results extend Klazar's distribution results from matchings to general set partitions.

Provides a unifying framework for understanding crossings and nestings 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 $l, m \geq 0$, 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.