Faces of Generalized Cluster Complexes and Noncrossing Partitions

Let Φ be an finite root system with corresponding reflection group W and let m be a nonnegative integer. We consider the generalized cluster complex ∆(Φ) defined by S. Fomin and N. Reading and the poset NC(m)(W ) of m-divisible noncrossing partitions defined by D. Armstrong. We give a characterization of the faces of ∆(Φ) in terms of NC(m)(W ), generalizing that of T. Brady and C. Watt given in the case m = 1. Making use of this, we give a case free proof of a conjecture of F. Chapoton and D. Armstrong, which relates a certain refined face count of ∆(Φ) with the Möbius function of NC(m)(W ).