Proof of a nonnegativity conjecture of Prellberg--Stanton

Abstract

In 1994, Friedman, Joichi and Stanton conjectured that coefficients of the power-series $q+(1-q)\prod_{i=n}^{2n-1}\frac{1}{1-q^{i}}$ are nonnegative for $n\geq3$ and $n$ being odd. Let $\qbinom nm$ denote the Gaussian polynomial. In 2003, Prellberg and Stanton conjectured that coefficients of the power-series $(1-q)\qbinom nm\prod_{i=0}^{m-1}\frac{1}{1-q^{i+m}}$ are nonnegative for $0<m<n$, except when $n$ is even and $m\in\{2,n-2\}$. The latter conjecture implies the former. In this paper, we prove a stronger result concerning coefficientwise nonnegativity of another power series, which implies the conjecture of Prellberg and Stanton. The proof uses a Möbius correction from Gaussian polynomials, weight-preserving injections on partitions, explicit linearly independent vectors in the kernel of a derivation operator and the KOH identity.