The inverse rook problem on Ferrers boards

Rook polynomials have been studied extensively since 1946, principally as a method for enumerating restricted permutations. However, they have also been shown to have many fruitful connections with other areas of mathematics, including graph theory, hypergeometric series, and algebraic geometry. It is known that the rook polynomial of any board can be computed recursively. The naturally arising inverse question – given a polynomial, what board (if any) is associated with it? – remains open. In this paper, we solve the inverse problem completely for the class of Ferrers boards, and show that the increasing Ferrers board constructed from a polynomial is unique.