Charmed roots and the Kroweras complement

. Although both noncrossing partitions and nonnesting partitions are uniformly enu-merated 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 ele- ment (1 , 2 ,...,n ), we recover one of the standard bijections between noncrossing and nonnesting partitions.

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