  Solomon showed that the Poincar\'e polynomial of a Coxeter group $W$
satisfies a product decomposition depending on the exponents of $W$. This
polynomial coincides with the rank-generating function of the poset of regions
of the underlying Coxeter arrangement. In this note we determine all instances
when the analogous factorization property of the rank-generating function of
the poset of regions holds for a restriction of a Coxeter arrangement. It turns
out that this is always the case with the exception of some instances in type
$E_8$.
