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.

Let G be a group acting on a set X of combinatorial objects, with finite orbits, and consider a statistic ξ : X → C. Propp and Roby defined the triple (X, G, ξ) to be homomesic if for any orbits O 1 , O 2 , the average value of the statistic ξ is the same, that isIn 2013 Propp and Roby conjectured the following instance of homomesy. Let SSYT k (m×n) denote the set of semistandard Young tableaux of shape m × n with entries bounded by k. Let S be any set of boxes in the m × n rectangle fixed under 180 • rotation. For T ∈ SSYT k (m × n), define σ S (T ) to be the sum of the entries of T in the boxes of S. Let P be a cyclic group of order k where P acts on SSYT k (m × n) by promotion. Then (SSYT k (m × n), P , σ S ) is homomesic.We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the Kpromotion of Thomas and Yong, and prove limited results in that direction.

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the 321-avoiding (i.e., 3noncrossing) case. Our approach also provides a more direct proof of a formula of Bóna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilfequivalence of the patterns 321 and 213 which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilfequivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

In Bloom and Saracino (2009) [2] we proved that a natural bijection Γ : S n (321) → S n (132) that Robertson defined by an iterative process in Robertson (2004) [8] preserves the numbers of fixed points and excedances in each σ ∈ S n (321). The proof depended on first showing that Γ (σ −1 ) = (Γ (σ )) −1 for all σ ∈ S n (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 * : S n (213) → S n (321) introduced in Backelin et al. (2007) [1] and studied in BousquetMelou and Steingrimsson (2005) [3].