A uniform bijection between nonnesting and noncrossing partitions

Understand

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.