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Abstract Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under theKreweras complementand nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call theKroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition ofcharmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions.
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Noncrossing partitions of an annulus (Extended abstract)
The noncrossing partition poset associated to a Coxeter group $$W$$ and Coxeter element $$c$$ is the interval $$[1,c]_T$$ in the absolute order on $$W$$. We construct a new model of noncrossing partititions for $$W$$ of classical affine type, using planar diagrams. The model in type $$\afftype{A}$$ consists of noncrossing partitions of an annulus. In type~$$\afftype{C}$$, the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete $$[1,c]_T$$ to a lattice by factoring the translations in $$[1,c]_T$$, but the combinatorics of the planar diagrams leads us to make different choices about how to factor.
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- Award ID(s):
- 2054489
- PAR ID:
- 10530463
- Publisher / Repository:
- The Séminaire Lotharingien de Combinatoire
- Date Published:
- ISSN:
- 1286-4889
- Format(s):
- Medium: X
- Location:
- https://www.mat.univie.ac.at/~slc/
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
