  The generalized cluster complex was introduced by Fomin and Reading, as a
natural extension of the Fomin-Zelevinsky cluster complex coming from finite
type cluster algebras. In this work, to each face of this complex we associate
a parabolic conjugacy class of the underlying finite Coxeter group. We show
that the refined enumeration of faces (respectively, positive faces) according
to this data gives an explicit formula in terms of the corresponding
characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents).
This characteristic polynomial originally comes from the theory of hyperplane
arrangements, but it is conveniently defined via the parabolic Burnside ring.
This makes a connection with the theory of parking spaces: our results
eventually rely on some enumeration of chains of noncrossing partitions that
were obtained in this context. The precise relations between the formulas
counting faces and the one counting chains of noncrossing partitions are
combinatorial reciprocities, generalizing the one between Narayana and Kirkman
numbers.
