Pattern avoidance in compositions and powers of permutations

A permutation $\pi$ is said to avoid a chain $(\sigma :\tau )$ of patterns if $\pi$ avoids $\sigma$ and ${\pi }^2$ avoids $\tau \mathrm{.}$ In this paper, we define a notion of pattern avoidance for compositions of positive integers and use that idea to enumerate permutations of length $n$ that avoid the chain $(312,321:\sigma )$ for any pattern $\sigma \in {\bigcup }_{m\ge 1}{S}_m$. We also enumerate those permutations that avoid the chain $(312,4321:\sigma )$ for any $\sigma \in S_3\mathrm{.}$