Set maps, umbral calculus, and the chromatic polynomialAlgebra & Discrete Mathematics
|Gus Wiseman, UC Davis
|Mon, Apr 25 2005, 3:10PM
Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying a certain binomial type identity. Using elementary umbral calculus, we give a formula for the expansion of such a set map in terms of any associated sequence (or Sheffer sequence). Applications to graph coloring are numerous, including several new expansions of the chromatic polynomial. By extending associated sequences to symmetric function bases, we also obtain corresponding expansions of the chromatic symmetric function.