Rook theory. V rook polynomials, Möbius inversion and the umbral calculus

Abstract

In this paper we provide the first general expressions for the rook and factorial polynomials associated with a general board. We use Möbius inversion over the lattice of partitions of a set to provide explicit formulas. Alternate formulas are given by the methods of the umbral calculus. We also state a general conjecture concerning the roots of rook polynomials and prove the conjecture for a special class of boards.