Durfee polynomials

Abstract

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)yd, where 0 ≤ d ≤ ⌊√n⌋. 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 log-concave and unimodal and that, over all partitions in F(n) for fixed n, the average size of the Durfee square, aF(n), and the most likely size of the Durfee square, m F(n), differ by less than 1. In this paper, we prove results in support of the conjecture that for the family of ordinary partitions, P(n), the Durfee polynomial has all roots real. Specifically, we find an asymptotic formula for |P(n,d)|, deriving in the process a simple upper bound on the number of partitions of n with at most k parts which generalizes the upper bound of Erdös for |P(n)|. We show that as n tends to infinity, the sequence {|P(n;d)|}, 1 ≤ d ≤ √n, is asymptotically normal, unimodal, and log concave; in addition, formulas are found for aP(n) and m P(n) which differ asymptotically by at most 1. Experimental evidence also suggests that for several families F(n) which satisfy a recurrence of a certain form, mF(n) grows as c√n, for an appropriate constant c = cF. We prove this under an assumption about the asymptotic form of |F(n,d)| and show how to produce, from recurrences for the |F(n,d)|, analytical expressions for the constants cF which agree numerically with the observed values.