The $e$-vector of a simplicial complex

TL;DR

This paper introduces the $e$-vector of a simplicial complex, linking it to the exponential Hilbert series of its Stanley-Reisner ring and exploring its relation to classical combinatorial vectors and geometric properties.

Contribution

It defines the $e$-vector, investigates its relationship with $f$- and $h$-vectors, and proves a combinatorial identity for Eulerian manifolds.

Findings

The $e$-vector relates to the exponential Hilbert series coefficients.

The $e$-vector connects to classical $f$- and $h$-vectors.

A combinatorial identity for Eulerian manifolds is established.

Abstract

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.