Abstract
We give the first type-independent proof of the Kreweras-style formulas for
the enumeration of noncrossing partitions in a real reflection group W, with respect to
parabolic type. This answers a central open question in Coxeter-Catalan combinatorics,
originally asked by Athanasiadis-Reiner in 2003, special cases of which have been open
even longer. Our proof also covers the m-Fuss version of the problem, as well as similar
Loday-style formulas for the refined-by-type enumeration of faces of the m-cluster
complex of W. It proceeds by developing a family of combinatorial recursions that
completely determine the enumeration and proving their algebraic counterparts.