  An element $w$ of the Weyl group is called rationally smooth if the
corresponding Schubert variety is rationally smooth. This happens exactly when
the lower interval $[id,w]$ in the Bruhat order is palindromic. For each
element $w$ of the Weyl group, we construct a certain hyperplane arrangement.
After analyzing the palindromic intervals inside the maximal quotients, we use
this result to show that the generating function for regions of the arrangement
coincides with the Poincar\'e polynomial of the corresponding Schubert variety
if and only if the Schubert variety is rationally smooth.
