Powers of permutations that avoid chains of patterns

Abstract

In a recent paper, Bona and Smith define the notion of \textit{strong avoidance}, in which a permutation and its square both avoid a given pattern. In this paper, we generalize this idea to what we call \textit{chain avoidance}. We say that a permutation avoids a chain of patterns $(τ_1 : τ_2: \cdots : τ_k)$ if the $i$-th power of the permutation avoids the pattern $τ_i$. We enumerate the set of permutations $π$ which avoid the chain $(213, 312 : τ)$, i.e.,~unimodal permutations whose square avoids $τ$, for $τ\in §_3$ and use this to find a lower bound on the number of permutations that avoid the chain $(312: τ)$ for $τ\in §_3$. We finish the paper by discussing permutations that avoid longer chains.