  Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We
study the rationality of the Poincar\'e series of the set of representatives of
minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends
mostly on the rationality of the Poincar\'e series of the normalizers of finite
parabolic subgroups of $W$. For affine Weyl groups, we prove that all these
series are rational and we give some explicit examples.
