  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.
