  The aim of this note is to show that the cycle decomposition of elements of
the symmetric group admits a quite natural formulation in the framework of dual
Coxeter theory, yielding a generalization of it to the family of so-called
parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group
every element is a parabolic quasi-Coxeter element). We show that such an
element admits an analogue of the cycle decomposition. Elements which are not
in this family still admit a generalized cycle decomposition, but it is not
unique in general.
