Rook polynomials in three and higher dimensions

Abstract

The rook polynomial of a board counts the number of ways of placing nonattacking rooks on the board. In this paper, we describe how the properties of the two-dimensional rook polynomials generalize to the rook polynomials of “boards” in three and higher dimensions. We also define families of three-dimensional boards which generalize the two-dimensional triangle boards and the boards representing the problème des rencontres. The rook coefficients of these three-dimensional boards are shown to be related to famous number sequences such as the central factorial numbers, the number of Latin rectangles and the Genocchi numbers.