  Let $\Phi$ be an 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 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 $\Delta^m(\Phi)$ 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
$\Delta^m(\Phi)$ with the M\"obius function of $NC_{(m)}(W)$.
