Faces of Generalized Cluster Complexes and Noncrossing Partitions

Let $\Phi$ be a finite root system with corresponding reflection group W, and let m be a nonnegative integer. We consider the generalized cluster complex $\Delta^m(\Phi)$ defined by Fomin and Reading and the poset $NC_{(m)}(W)$ of m-divisible noncrossing partitions defined by Armstrong. We give a characterization of the faces of $\Delta^m(\Phi)$ in terms of $NC_{(m)}(W)$, generalizing that of Brady and Watt given in the case $m=1$. Making use of this, we give a case free proof of a conjecture of Chapoton and Armstrong, which relates a certain refined face count of $\Delta^m(\Phi)$ with the Möbius function of $NC_{(m)}(W)$.