Durfee Polynomials

Let F(n) be a family of partitions of n and let F(n, d) denote the set of partitions in F(n) with Durfee square of size d. We define the Durfee polynomial of F(n) to be the polynomial PF,n = ∑ |F(n, d)|y, where 0 ≤ d ≤ b √ nc. The work in this paper is motivated by empirical evidence which suggests that for several families F, all roots of the Durfee polynomial are real. Such a result would imply that the corresponding sequence of coefficients {|F(n, d)|} is logconcave and unimodal and that, over all partitions in F(n) for fixed n, the Research supported by National Science Foundation Grant DMS9302505 †Supported in part by National Science Foundation Grants No. DMS 9302505 and DMS 9622772