under whose guidance the original draft [2] was written. I thank my wife Heather for her thorough reading of the manuscript and for helpful mathematical discussions. Thanks to the anonymous referee for many valuable suggestions. And thanks to Hugh Thomas who made several contributions to my understanding; in particular, he suggested the proof of Theorem 3.7.2.I have benefitted from valuable conversations with many mathematicians; their suggestions have improved this memoir in countless ways. Thanks to: (in alphabetical order)

Abstract. In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we identify Panyushev's map with the Kreweras complement on the set of noncrossing partitions, and hence construct the first uniform bijection between nonnesting and noncrossing partitions. Unfortunately, the proof that our construction is well-defined is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner.

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ABSTRACT. An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects.In particular, we prove that 2n-and (2mn + 1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type Cn, generalizing a result of Fishel-Vazirani for type A. We also introduce a major index statistic on simultaneous n-and (n + 1)-core partitions and on self-conjugate simultaneous 2n-and (2n + 1)-core partitions that yield q-analogues of the Coxeter-Catalan numbers of type A and type C.We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q, t-Catalan numbers.