On the enumeration of permutations avoiding chains of patterns

TL;DR

This paper provides explicit formulas for counting permutations that avoid specific chains of patterns, advancing the understanding of chain avoidance and confirming two prior conjectures.

Contribution

It introduces explicit enumeration formulas for permutations avoiding chains of patterns, confirming two conjectures by Archer and Geary.

Findings

Explicit formulas for chain-avoiding permutations

Confirmation of two Archer and Geary conjectures

Advancement in pattern avoidance enumeration

Abstract

In 2019, B\'ona and Smith introduced the notion of strong pattern avoidance, saying that a permutation $\pi$ strongly avoids a pattern $\sigma$ if $\pi$ and $\pi^2$ both avoid $\sigma$. Recently, Archer and Geary generalized the idea of strong pattern avoidance to chain avoidance, in which a permutation $\pi$ avoids a chain of patterns $(\tau^{(1)}:\tau^{(2)}:\cdots:\tau^{(k)})$ if the $i$-th power of the permutation avoids the pattern $\tau^{(i)}$ for $1\leq i\leq k$. In this paper, we give explicit formulae for the number of sets of permutations avoiding certain chains of patterns. Our results give affirmative answers to two conjectures proposed by Archer and Geary.