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The generalized cluster complex was introduced by Fomin and Reading, as anatural extension of the Fomin-Zelevinsky cluster complex coming from finitetype cluster algebras. In this work, to each face of this complex we associatea parabolic conjugacy class of the underlying finite Coxeter group. We showthat the refined enumeration of faces (respectively, positive faces) accordingto this data gives an explicit formula in terms of the correspondingcharacteristic polynomial (equivalently, in terms of Orlik-Solomon exponents).This characteristic polynomial originally comes from the theory of hyperplanearrangements, but it is conveniently defined via the parabolic Burnside ring.This makes a connection with the theory of parking spaces: our resultseventually rely on some enumeration of chains of noncrossing partitions thatwere obtained in this context. The precise relations between the formulascounting faces and the one counting chains of noncrossing partitions arecombinatorial reciprocities, generalizing the one between Narayana and Kirkmannumbers.

The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces