Dedication: In memory of Rodica Simion This article is dedicated to the memory of Rodica Simion, one of the greatest enumerators of the 20 th century. Both derangements [8] and restricted permutations [10] were very dear to her heart, and we are sure that she would have appreciated the present surprising connections between these at-first-sight unrelated concepts.Abstract. Define S k n (α) to be the set of permutations of {1, 2,...,n} with exactly k fixed points which avoid the pattern α ∈ S m . Let s k n (α) be the size of S k n (α). We investigate S 0 n (α) for all α ∈ S 3 as well as show that s k n (132) = s k n (213) = s k n (321) and s k n (231) = s k n (312) for all 0 ≤ k ≤ n.

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.

We give a new framework for the construction of homogeneous nilpotent groups and rings which goes a long way toward unifying the two cases, and enables us to extend previous constructions, producing a variety of new examples. In particular we find ingredients for the manufacture of 2No homogeneous nilpotent groups "in nature".

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