Pattern avoidance for set partitions à la Klazar

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 $τ$ is a partition of $[k]$ and $Π_n(τ)$ denotes the set of all partitions of $[n]$ that avoid $τ$, we establish inequalities between $|Π_n(τ_1)|$ and $|Π_n(τ_2)|$ for several choices of $τ_1$ and $τ_2$, and we prove that if $τ_2$ is the partition of $[k]$ with only one block, then $|Π_n(τ_1)| <|Π_n(τ_2)|$ for all $n>k$ and all partitions $τ_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $τ_1$ of $[k]$. Finally, we enumerate $Π_n(τ)$ for all partitions $τ$ of $[4]$.