Pattern avoidance and the fundamental bijection

The fundamental bijection is a bijection θ: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 π ∈ Sₙ that avoids a pattern σ ∈ S₃, whose image θ(π) also avoids σ. We additionally consider what happens under repeated iterations of θ; in particular, we enumerate permutations π ∈ Sₙ that have the property that π and its first k iterations under θ all avoid a pattern σ. Finally, we consider permutations with the property that π=θ²(π) that avoid a given pattern σ, and end the paper with some directions for future study.