Inversion Sets and Quotient Root Systems

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.
Comment: Preliminary Version