Some Thoughts on Rook Polynomials on Square Chessboards

Abstract

A rook polynomial is a polynomial whose x k coefficient is the number of ways k rooks can be placed on the squares of an arbitrarily shaped chessboard so that no rooks share the same rows or columns. The k rooks are called non-taking. Rook polynomials pattern combinatorial situations, especially those involving restricted permutations. The conventional square board used in the game, chess, is but one configuration.