A uniform bijection between nonnesting and noncrossing partitions

Drew Armstrong, Christian Stump, Hugh Thomas

Research output: Contribution to journalArticle

15 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)4121-4151
Number of pages31
JournalTransactions of the American Mathematical Society
Volume365
Issue number8
DOIs
StatePublished - 2013

Fingerprint

Noncrossing Partitions
Bijection
Antichain
Weyl Group
Root System
Combinatorics
Poset
Recursion
Well-defined
Subsystem
Finite Group
Complement
Partition
Roots

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 365, No. 8, 2013, p. 4121-4151.

Research output: Contribution to journalArticle

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