The generalized cluster complex: refined enumeration of faces and related parking spaces

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. As consequences of our enumeration of faces, we completely explain some observations by Fomin and Reading about f-vectors, we obtain identities that essentially refine the transformation between f-vectors and h-vectors, and introduce new kinds of parking spaces that we call cluster parking spaces