Pattern avoidance in compositions and powers of permutations

Abstract

A permutation $\ddot{I}€$ is said to avoid a chain $(\ddot{I}ƒ:\ddot{I}„)$ of patterns if $\ddot{I}€$ avoids $\ddot{I}ƒ$ and $\ddot{I}{€}^2$ avoids $\ddot{I}„.$ 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 $(312,321:\ddot{I}ƒ)$ for any pattern $\ddot{I}ƒ\in {\bigcup }_{m\ge 1}{S}_m$. We also enumerate those permutations that avoid the chain $(312,4321:\ddot{I}ƒ)$ for any $\ddot{I}ƒ\in S_3.$