A uniform bijection between nonnesting and noncrossing partitions

TL;DR

This paper establishes a uniform bijection between nonnesting and noncrossing partitions by identifying Panyushev's map with the Kreweras complement, involving new combinatorics and confirming several conjectures.

Contribution

It provides the first uniform bijection between nonnesting and noncrossing partitions, linking Panyushev's map to the Kreweras complement, with case-by-case verification and new combinatorial insights.

Findings

Proved the bijection between nonnesting and noncrossing partitions.

Confirmed conjectural properties of Panyushev's map.

Established two cyclic sieving phenomena.

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.