Pattern avoidance in ordered set partitions and words

TL;DR

This paper explores the enumeration of ordered set partitions avoiding permutation patterns, establishing identities with pattern-avoiding words, determining growth rates, and analyzing specific pattern cases.

Contribution

It introduces an explicit identity linking pattern-avoiding partitions to words, enabling transfer of results and detailed analysis of pattern avoidance.

Findings

Derived asymptotic growth rates for pattern-avoiding partitions.

Confirmed a conjecture on sequence variation for pattern-avoiding partitions.

Analyzed the number of partitions avoiding length-3 patterns.

Abstract

We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let $\op_{n,k}(p)$ be the number of ordered set partitions of $\{1,2,\ldots,n\}$ into $k$ blocks that avoid a permutation pattern $p$. We establish an explicit identity between the number $\op_{n,k}(p)$ and the numbers of words avoiding the inverse of $p$. This identity allows us to easily translate results on pattern-avoiding words obtained in earlier works into equivalent results on pattern-avoiding ordered set partitions. In particular, \emph{(a)} we determine the asymptotic growth rate of the sequence $(\op_{n,k}(p))_{n\geq 1}$ for every positive $k$ and every permutation pattern $p$, \emph{(b)} we partially confirm a conjecture of Godbole et al. concerning the variation of the sequences $(\op_{n,k}p))_{1\leq k\leq n}$, \emph{(c)} we undertake a detailed study of the number of ordered set partitions avoiding a pattern of length 3.