  The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of
integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are
often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart
$\delta$-vector), which is the vector of coefficients of $L_P$ with respect to
a certain binomial basis and which coincides with the $h$-vector of a regular
unimodular triangulation of $P$ (if one exists). One important result by
Stanley about $h^*$-vectors of polytopes is that their entries are always
non-negative. However, recent combinatorial applications of Ehrhart theory give
rise to polytopal complexes with $h^*$-vectors that have negative entries.
  In this article we introduce the Ehrhart $f^*$-vector of polytopes or, more
generally, of polytopal complexes $K$. These are again coefficient vectors of
$L_K$ with respect to a certain binomial basis of the space of polynomials and
they have the property that the $f^*$-vector of a unimodular simplicial complex
coincides with its $f$-vector. The main result of this article is a counting
interpretation for the $f^*$-coefficients which implies that $f^*$-coefficients
of integral polytopal complexes are always non-negative integers. This holds
even if the polytopal complex does not have a unimodular triangulation and if
its $h^*$-vector does have negative entries. Our main technical tool is a new
partition of the set of lattice points in a simplicial cone into discrete
cones. Further results include a complete characterization of Ehrhart
polynomials of integral partial polytopal complexes and a non-negativity
theorem for the $f^*$-vectors of rational polytopal complexes.
