Pattern avoidance for set partitions \`a la Klazar

TL;DR

This paper studies pattern avoidance in set partitions introduced by Klazar, focusing on Wilf-equivalence classifications, inequalities between avoidance sets, and enumerations for partitions of small size.

Contribution

It classifies Wilf-equivalence for certain set partitions, establishes inequalities between avoidance set sizes, and enumerates partitions avoiding all patterns of size four.

Findings

All Wilf-equivalences for partitions with two blocks including a singleton are determined.

For partitions of size k, the avoidance set with a single block is larger than those with two blocks for all n > k.

Enumerations of avoidance sets are provided for all partitions of size four.

Abstract

In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| <|\Pi_n(\tau_2)|$ for all $n>k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$.