Factorization of the Cover Polynomial

Chung and Grahamʹs cover polynomial is a generalization of the factorial rook polynomial in which the second variable keeps track of cycles. We factor the cover polynomial completely for Ferrers boards with either increasing or decreasing column heights. For column-permuted Ferrers boards, we find a sufficient condition for partial factorization. We apply this result to several special cases, including column-permuted “staircase boards,” getting a partial factorization in terms of the column permutation, as well as a sufficient condition for complete factorization. We conclude with some conjectures.