Powers of permutations that avoid chains of patterns

In a recent paper, B & oacute;na and Smith define the notion of 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 chain avoidance. We say that a permutation avoids a chain of patterns (tau 1 : tau 2 : <middle dot> <middle dot> <middle dot> : tau k) if the i-th power of the permutation avoids the pattern tau i. We enumerate the set of permutations pi which avoid the chain (213, 312 : tau), i.e., unimodal permutations whose square avoids tau, for tau e S3 and use this to find a lower bound on the number of permutations that avoid the chain (312 : tau) for tau e S3. We finish the paper by discussing permutations that avoid longer chains.