Remarks on umbral evaluations of chromatic polynomials (Q1318843)

The author has studied the polynomial \(\chi (u) = a_ 1u^ 1+ a_ 2u^ 2 + \cdots + a_ nu^ n\) which generalizes the concept of chromatic polynomial of graphs. It is shown that a suitable umbral evaluation \(a_ 1u_ 1+ a_ 2u_ 2 + \cdots + a_ nu_ n\) of the polynomial gives the solution of a class of coloring enumeration problems depending on a group action on the set of colors. These results are obtained using the Pólya-de Bruijn theory. Applications include proofs of several formulas for \(B_ n\), the number of partitions of an \(n\)-set.