Abstract

A rook polynomial counts the number of placements of non-attacking rooks on a board. In this paper we describe generalizations of the definition and properties of rook polynomials to "boards" in three and higher dimensions. We also defefine generalizations of special two dimensional boards to three dimensions, including the triangle board and the board representing the probleme des rencontres. The number of rook placements on these three dimensional families of rook boards are shown to be related to famous number sequences, such as central factorial numbers, the number of Latin rectangles and the Genocchi numbers.