# Counting avoiding improper 3-colorings

Given a graph $$G=(V,E)$$, what I call an improper $$3$$-coloring of $$G$$ is simply a function $$c:V \to \{1,2,3\}$$. I say that $$c$$ is $$1$$-$$2$$-avoiding when there do not exist two adjacent nodes $$u,v$$ with $$c(u)=1$$ and $$c(v)=2$$. I am interested in the problem that takes as input a graph and counts the number of $$1$$-$$2$$-avoiding improper $$3$$-colorings of $$G$$. Or equivalently, this is the number of pairs of disjoint node subsets $$(S,S')$$ that are not adjacent (in the sense that $$E \cap (S\times S') = \emptyset$$).

Question: has this problem already been studied? It is obviously in the complexity class #P, but is it also #P-hard?