Set maps, umbral calculus, and the chromatic polynomial

Abstract

Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying p S ( x + y ) = ∑ T ⊎ U = S p T ( x ) p U ( y ) . Using umbral calculus, we give a formula for the expansion of such a set map in terms of any polynomial sequence of binomial type. This leads to some new expansions of the chromatic polynomial. We also describe a set map generalization of Abel polynomials.