On bijections for pattern-avoiding permutations

By considering bijections from the set of Dyck paths of length 2n onto each of Sn(321) and Sn(132), Elizalde and Pak in [S. Elizalde, I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A 105 (2004) 207–219] gave a bijection that preserves the number of fixed points and the number of excedances in each σ∈Sn(321). We show that a direct bijection Γ:Sn(321)→Sn(132) introduced by Robertson in [A. Robertson, Restricted permutations from Catalan to Fine and back, Sém. Lothar. Combin. 50 (2004) B50g] also preserves the number of fixed points and the number of excedances in each σ. We also show that a bijection ϕ∗:Sn(213)→Sn(321) studied in [J. Backelin, J. West, G. Xin, Wilf-equivalence for singleton classes, Adv. in Appl. Math. 38 (2007) 133–148] and [M. Bousquet-Melou, E. Steingrimsson, Decreasing subsequences in permutations and Wilf equivalence for involutions, J. Algebraic Combin. 22 (2005) 383–409] preserves these same statistics, and we show that an analogous bijection from Sn(132) onto Sn(213) does the same.