  We study the exponential Hilbert series (both coarsely- and finely-graded) of
the Stanley-Reisner ring of an abstract simplicial complex, $\Delta$, and we
introduce the $e$-vector of $\Delta$, which relates to the coefficients of the
exponential Hilbert series. We explore the relationship of the $e$-vector with
the classical $f$-vector and $h$-vector of $\Delta$ while simultaneously
investigating the geometric information that the $e$-vector encodes about
$\Delta$. We then prove a simple combinatorial identity for the $e$-vector in
the case where $\Delta$ is an Eulerian manifold.
