  We give an elementary, case-free, Coxeter-theoretic derivation of the formula
$h^nn!/|W|$ 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.
