A uniform bijection between nonnesting and noncrossing partitions

Drew Armstrong, Christian Stump, Hugh Thomas

Research output: Contribution to journalArticle

22 Scopus citations

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 - 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

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