Another look at bijections for pattern-avoiding permutations

In Bloom and Saracino (2009) [2], we proved that a natural bijection Γ:Sn(321)→Sn(132) that Robertson defined by an iterative process in Robertson (2004) [8], preserves the numbers of fixed points and excedances in each σ∈Sn(321). The proof depended on first showing that Γ(σ−1)=(Γ(σ))−1 for all σ∈Sn(321). Here we give a noniterative definition of Γ that frees the result about fixed points and excedances from its dependence on the result about inverses, while also greatly simplifying and elucidating the result about inverses. We also establish a simple connection between Γ and an analogous bijection ϕ∗:Sn(213)→Sn(321) introduced in Backelin et al. (2007) [1], and studied in Bousquet-Melou and Steingrimsson (2005) [3].