### 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 - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

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

**A uniform bijection between nonnesting and noncrossing partitions.** / Armstrong, Drew; Stump, Christian; Thomas, Hugh.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 365, no. 8, pp. 4121-4151. https://doi.org/10.1090/S0002-9947-2013-05729-7

}

TY - JOUR

T1 - A uniform bijection between nonnesting and noncrossing partitions

AU - Armstrong, Drew

AU - Stump, Christian

AU - Thomas, Hugh

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Bijective combinatorics

KW - Coxeter groups

KW - Cyclic sieving phenomenon

KW - Noncrossing partitions

KW - Nonnesting partitions

KW - Weyl groups

UR - http://www.scopus.com/inward/record.url?scp=84877974174&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877974174&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2013-05729-7

DO - 10.1090/S0002-9947-2013-05729-7

M3 - Article

VL - 365

SP - 4121

EP - 4151

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -