Set maps, umbral calculus, and the chromatic polynomial

In order to motivate our results we will begin by describing two classical expansions of the chromatic polynomial. Let G be a simple graph with finite vertex set V . Let χG(x) be the number of proper colorings of G with x colors (assignments of colors 1, 2, . . . , x to the vertices of G so that no two adjacent vertices have the same color). If e is an edge of G, let G\e be G with e removed, and let G/e be G with e contracted to a single vertex. Then χG(x) satisfies the deletion-contraction property,