We study the exponential Hilbert series (both coarsely- and finely-graded) of the Stanley-Reisner ring of an abstract simplicial complex, $Δ$, and we introduce the $e$-vector of $Δ$, 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 $Δ$ while simultaneously investigating the geometric information that the $e$-vector encodes about $Δ$. We then prove a simple combinatorial identity for the $e$-vector in the case where $Δ$ is an Eulerian manifold.