### 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 use Panyushev's map, together with the well-known Kreweras complement, to construct a bijection between nonnesting and noncrossing partitions. Our map is defined uniformly for all root systems, using a recursion in which the map is assumed to be defined already for all parabolic subsystems. Unfortunately, the proof that our map is well defined, and is a bijection, 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. 2013 American Mathematical Society.

Original language | English (US) |
---|---|

Pages (from-to) | 4121-4151 |

Number of pages | 31 |

Journal | Transactions of the American Mathematical Society |

Volume | 365 |

Issue number | 8 |

DOIs | |

State | Published - May 27 2013 |

### Keywords

- Bijective combinatorics
- Coxeter groups
- Cyclic sieving phenomenon
- Noncrossing partitions
- Nonnesting partitions
- Weyl groups

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

## Fingerprint Dive into the research topics of 'A uniform bijection between nonnesting and noncrossing partitions'. Together they form a unique fingerprint.

## Cite this

*Transactions of the American Mathematical Society*,

*365*(8), 4121-4151. https://doi.org/10.1090/S0002-9947-2013-05729-7