  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.
