Counting chains in the noncrossing partition lattice via the W-Laplacian

We give an elementary, case-free, Coxeter-theoretic derivation of the formula hnn!/|W| for the number of maximal chains in the noncrossing partition lattice NC(W) of a real reflection group W. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the W-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.