Set partition patterns and statistics

TL;DR

This paper studies the distribution of key statistics on set partitions that avoid certain patterns, extending known enumeration results to multivariate distributions and exploring new combinatorial properties.

Contribution

It introduces the first detailed analysis of the distributions of fundamental statistics on pattern-avoiding set partitions, linking combinatorial patterns with statistical distributions.

Findings

Distribution formulas for statistics over avoidance classes

Multivariate analogues of avoidance class sizes

Identification of open problems in the field

Abstract

A set partition $\sigma$ of $[n]=\{1,\dots,n\}$ contains another set partition $\pi$ if restricting $\sigma$ to some $S\subseteq[n]$ and then standardizing the result gives $\pi$. Otherwise we say $\sigma$ avoids $\pi$. For all sets of patterns consisting of partitions of $[3]$, the sizes of the avoidance classes were determined by Sagan and by Goyt. Set partitions are in bijection with restricted growth functions (RGFs) for which Wachs and White defined four fundamental statistics. We consider the distributions of these statistics over various avoidance classes, thus obtaining multivariate analogues of the previously cited cardinality results. This is the first in-depth study of such distributions. We end with a list of open problems.