Binary operations on pattern-avoiding cycles

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 $n\geq 2.$