Define a hypergraph to be a pair $H = (V,E)$ where $V$ is a set of vertices and $E$ is any set of subsets of $V$ called edges. Thus if every edge $U \in E$ has only two elements, then the hypergraph $H$ is equivalent to an ordinary simple graph. We say that a coloring $\chi: V \rightarrow \mathbb{N}$ of a hypergraph $H$ is proper for each $U \in E$ there are $u_1,u_2 \in U$ with $\chi(u_1) \neq \chi(u_2)$. It might seem more natural to require that $\chi(u_1)\neq \chi(u_2)$ for every pair of distinct elements $u_1,u_2 \in U$, but then hypergraph colorings would be equivalent to graph colorings as we could replace every edge $U\in E$ with a clique.
Given a hypergraph $G = (V,E)$, define the chromatic symmetric function of $G$ to be $$X_G = \sum_{\chi} x^\chi$$ where the sum is over proper colorings $\chi$ of $G$, and $x^\chi = \prod_{v \in V} x_v$ with $x_1, x_2, \ldots$ being indeterminates.
I believe Stanley introduced the chromatic symmetric function of a hypergraph in the last section of this paper (pdf). He computes $X_H$ for hypergraphs of the form $H = (V,E)$ where $E = \{S \subseteq V: |S| = k\}$. But he writes:
There is an extensive theory of hypergraph coloring, but little of this theory is enumerative.
This was in 1995. Is this statement still true? What are some nice examples of work counting the number of proper colorings of hypergraphs? In particular, are there other nice classes of hypergraphs $H$ for which we can compute $X_H$? Are there any conjectures as to when $X_H$ is Schur-positive? I am interested in cases where $H$ is not equivalent to an ordinary graph.