The $e$-vector of a simplicial complex

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