Pattern Avoidance in Set Partitions

TL;DR

This paper studies pattern avoidance in set partitions, characterizing and enumerating classes based on size four patterns, and analyzing the distribution of specific statistics across these classes.

Contribution

It provides a detailed characterization and enumeration of avoidance classes for size four patterns and explores the distribution and equidistribution of statistics within these classes.

Findings

Characterized avoidance classes for size four patterns

Enumerated the avoidance classes

Analyzed distribution of Wachs and White statistics

Abstract

A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of interest to this work. We first characterize the restricted growth functions of several avoidance classes based on partitions of size four, enumerate these avoidance classes, and consider the distribution of the Wachs and White statistics across these avoidance classes. We also investigate the equidistribution of statistics between avoidance classes based on multiple patterns.