Let $H$ be a hypergraph and let $P_H$ denote it's chromatic polynomial. I am interested in the best results interpreting $P_H(-1)$. I am interested both in the general case (which I think is hard) as well as any special cases where good descriptions are known. One well known special case is for a graph $G$ we have $P_G(-1)$ is the number of acyclic orientations by a classic result of Stanley [S]. I am happy with references or arguments typed below for any $P_H(-1)$. More details as well as references and examples of what I would like to know follows...

See [ZD] for precise definitions and a contrast of hypergraph behavior compared to chromatic polynomials of graphs. Furthermore, see [ZD, Corollary 1] for a special case where a good description is known.

Now for the reason I came to this question. I put it at the end since it contains self-promotion (but hopefully not view as shameless self-promotion). The best result I personally know for describing $P_H(-1)$ in general is in [BB] and [BBM]. Both of these papers take a Hopf algebra/monoid perspective where $P_H(-1)$ shows up in the antipode computation. Hence, it is not so easy to find these looking just for hypergraph theoretic terms (so maybe I am missing other resources in different terminology?). We have $P_H(-1)$ as a signed sum over a version of acyclic orientations of the hypergraph. The acyclic orienations and antipode originally come from [BB], but [BBM] has less Hopf overhead. See [BBM, Section 2.1] and [BBM, Theorem 2.10]. A polyhedral Euler characteristic version is added by [BBM, Corollary 2.20]. Usually there is lots of cancellation (and everything can cancel resulting in $P_H(-1)=0$). Maybe we can simplify to a cancellation free formula for some special cases like in [ZD, Corollary 1] or [BBM, Proposition 2.25] for hyperforests?

References

[ZD] Zhang, R.; Dong, F. Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs. European J. Combin. 64 (2017), 138–151.

[S] Stanley, R. P. Acyclic orientations of graphs. Discrete Math. 5 (1973), 171–178.

[BB] Benedetti, C.; Bergeron, N. The antipode of linearized Hopf monoids. Algebr. Comb. 2 (2019), no. 5, 903–935.

[BBM] Benedetti, C.; Bergeron, N.; Machacek, J. Hypergraphic polytopes: combinatorial properties and antipode. J. Comb. 10 (2019), no. 3, 515–544.