Acyclic orientations of graphs

Let GG be a finite graph with pp vertices and χχ its chromatic polynomial. A combinatorial interpretation is given to the positive integer (-1)pχ(-λ)(-1)pχ(-λ), where λλ is a positive integer, in terms of acyclic orientations of GG. In particular, (-1)pχ(-1)(-1)pχ(-1) is the number of acyclic orientations of GG. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in the sense of Doubilet–Rota–Stanley, is constructed which yields the generating functions which occur in the above context.