Factorisations of a Coxeter element
and discriminant of a reflection group

When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a very rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (only proved case-by-case for now) expresses the number of multichains of a given length in NCP_W as a generalized Fuss-Catalan number, depending on the invariant degrees of W. We explain how to comprehend geometrically some specifications of this formula. We use an interpretation of the chains of NCP_W as fibers of a "Lyashko-Looijenga covering", constructed from the discriminant hypersurface of W. We deduce new enumeration formulas for certain factorisations of a Coxeter element of W.