Chromatic polynomials of partition systems

The theory of the umbral chromatic polynomial of a simplicial complex provides a combinatorial framework for the study of formal group laws over a commutative, torsion-free ring, and our aim in this work is to extend its definition to a class of set systems script P sign, which we label partition systems. When suitably evaluated, our polynomial _χ^φ(script P sign; x) enumerates factorized col-orings, as well as coloring forests of the partition system by type. These colorings are related to the Mullin-Rota concept of reluctant functions, and whenever script P sign is a simplicial complex, they reduce to more familiar notions of coloring. Our three main results demonstrate how several properties of the classical chromatic polynomial _χ(H; x), where H is a simple graph, may be generalized. Firstly, we prove that our polynomial _χ^φ(script P sign; x) retains the property of being expressible as the characteristic polynomial of an appropriate poset, which holds for _φ(H; x) by virtue of Whitney's original definition. Secondly, we provide two generalizations for the formula describ" ing the chromatic polynomial of a disjoint union of graphs; one of these formulas depends explicitly on the context of partition systems and is not available when we restrict attention to graphs or simplicial complexes. Thirdly, we introduce partition systems with group action, thereby providing a combinatorial interpretation of normalized versions of our polynomials. In the case of the symmetric group acting on the trivial partition system, these are the normalized conjugate Bell polynomials, whose interpretation is a vital prerequisite for extending our frame^{script P sign}" work to formal group laws over arbitrary rings of scalars; here, however, we concentrate solely on combinatorial aspects.