  We show that an element $w$ of a finite Weyl group $W$ is rationally smooth
if and only if the hyperplane arrangement $I$ associated to the inversion set
of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the
coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval
$[e,w]$, where $e$ is the identity in $W$. As part of the proof, we describe
exactly when a rationally smooth element in a finite Weyl group has a chain
Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain
Billey-Postnikov decompositions are connected with certain modular coatoms of
$I$.
