On bijections for pattern-avoiding permutations

By considering bijections from the set of Dyck paths of length 2n onto each of S n ( 321 ) and S n ( 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 Θ : S n ( 321 ) → S n ( 132 ) that preserves the number of fixed points and the number of excedances in each σ ∈ S n ( 321 ) . We show that a direct bijection Γ : S n ( 321 ) → S n ( 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 ∗ : S n ( 213 ) → S n ( 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 S n ( 132 ) onto S n ( 213 ) does the same.