A uniform bijection between nonnesting and noncrossing partitions
By Drew Armstrong, Christian Stump, Hugh Thomas. 2011
91 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/7c60fa911b50e277a0d0dbaa7e6b46b39fb0404f
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
In Mathematical Physics
Citations in the corpus
- Rowmotion in slow motion (2017, methodology): “t bijections to natural combinatorial objects, and Stanley completely characterized the orbit structure of rowmotion on the product of two chains combinatorially [Sta09]. Armstrong, Stump, and Thomas [AST13] then went on to resolve Panyushev’s conjectures using noncrossing partitions, while Striker ”