  It is well-known that the Eulerian polynomials, which count permutations in
$S_n$ by their number of descents, give the $h$-polynomial/$h$-vector of the
simple polytopes known as permutohedra, the convex hull of the $S_n$-orbit for
a generic weight in the weight lattice of $S_n$. Therefore the Eulerian
polynomials give the Betti numbers for certain smooth toric varieties
associated with the permutohedra.
  In this paper we derive recurrences for the $h$-vectors of a family of
polytopes generalizing this. The simple polytopes we consider arise as the
orbit of a non-generic weight, namely a weight fixed by only the simple
reflections $J=\{s_{n},s_{n-1},s_{n-2} \cdots,s_{n-k+2},s_{n-k+1}\}$ for some
$k$ with respect to the $A_n$ root lattice. Furthermore, they give rise to
certain rationally smooth toric varieties $X(J)$ that come naturally from the
theory of algebraic monoids. Using effectively the theory of reductive
algebraic monoids and the combinatorics of simple polytopes, we obtain a
recurrence formula for the Poincar\'e polynomial of $X(J)$ in terms of the
Eulerian polynomials.
