Binary operations on pattern-avoiding cycles

TL;DR

This paper introduces a partial groupoid structure on cyclic pattern-avoiding permutations to establish recursive bounds on their counts, revealing growth rates and confirming a conjecture for most patterns.

Contribution

It defines a new algebraic structure on cyclic pattern-avoiding permutations and uses it to derive growth bounds and prove a conjecture for pattern avoidance counts.

Findings

Growth rate at least 3 for certain patterns

Growth rate at least 2.6 for others

Confirmed a conjecture relating counts of pattern-avoiding permutations

Abstract

Suppose $c_n(\sigma)$ denotes the number of cyclic permutations in $\mathcal{S}_n$ that avoid a pattern $\sigma$. In this paper, we define partial groupoid structures on cyclic pattern-avoiding permutations that allow us to build larger cyclic pattern-avoiding permutations from smaller ones. We use this structure to find recursive lower bounds on $c_n(\sigma)$. These bounds imply that $c_n(\sigma)$ has a growth rate of at least 3 for $\sigma\in\{231,312,321\}$ and a growth rate of at least 2.6 for $\sigma\in\{123,132,213\}$. In the process, we prove (and sometimes improve) a conjecture of B\'{o}na and Cory that $c_n(\sigma)\geq 2 c_{n-1}(\sigma)$ for all $\sigma\in\mathcal{S}_3\setminus\{123\}$ and $n\geq 2.$