Pattern Avoidance in Set Partitions

TL;DR

This paper investigates pattern avoidance in set partitions, providing exact formulas for avoiding specific patterns, characterizing partitions avoiding 3-element set patterns, and introducing a new pattern notion based on restricted growth functions.

Contribution

It extends pattern avoidance theory to set partitions, deriving formulas, characterizations, and a new pattern concept based on restricted growth functions.

Findings

Exact formulas for pattern-avoiding set partitions

Characterization of partitions avoiding 3-element patterns

Sequences are P-recursive for these avoiding partitions

Abstract

The study of patterns in permutations in a very active area of current research. Klazar defined and studied an analogous notion of pattern for set partitions. We continue this work, finding exact formulas for the number of set partitions which avoid certain specific patterns. In particular, we enumerate and characterize those partitions avoiding any partition of a 3-element set. This allows us to conclude that the corresponding sequences are P-recursive. Finally, we define a second notion of pattern in a set partition, based on its restricted growth function. Related results are obtained for this new definition.