Charmed roots and the Kroweras complement

TL;DR

This paper establishes a unique, equivariant bijection between noncrossing and nonnesting partitions for the symmetric group, using a new concept called charmed roots and a cyclic action called the Kroweras complement.

Contribution

It introduces charmed roots and constructs a unique equivariant, support-preserving bijection between noncrossing and nonnesting partitions for symmetric groups.

Findings

Constructed an equivariant bijection for symmetric groups.

Defined charmed roots based on Coxeter elements.

Recovers standard bijections in special cases.

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 the Kreweras complement and nonnesting partitions under a Coxeter-theoretically natural cyclic action we call the Kroweras 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 of charmed roots. Charmed roots are determined by the choice of Coxeter element -- in the special case of the linear Coxeter element $(1, 2, \dots, n)$, we recover one of the standard bijections between noncrossing and nonnesting partitions.