Pattern avoidance and the fundamental bijection

The fundamental bijection is a bijection: Sₙₙ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations Sₙ that avoids a pattern S₃, whose image () also avoids. We additionally consider what happens under repeated iterations of ; in particular, we enumerate permutations Sₙ that have the property that and its first k iterations under all avoid a pattern. Finally, we consider permutations with the property that =² () that avoid a given pattern, and end the paper with some directions for future study.