  We provide a recursive description of all decompositions of the positive
roots $R^+$ of a quotient root system $R$ into disjoint unions of inversion
sets. Our description is type-independent and generalizes the analogous result
for type $\mathbb A$ root systems in [USRA]. The main tool is the notion of an
inflation of a subset of a quotient root system. This new notion allows us to
treat all root systems (and their quotients) uniformly. We also obtain some
numerical results about the number of special decompositions. The new sequences
we obtain may be considered as extensions of Catalan numbers.
