Sharp Euler-factor positivity for the BG-rank-refined five-core crank

Abstract

In this paper, we complete the resolution of Stanton's problem OPAC--005 by proving its fixed-BG-rank assertion. We further establish a sharp nine-factor positivity theorem for the joint generating function. Let $K_{n,j}(z)$ enumerate partitions of $5n+4$ with BG-rank $j$ by the integer five-core crank prescribed in OPAC--005, and put $[5]_z=1+z+z^2+z^3+z^4$. We prove that $K_{n,j}(z)/[5]_z$ has nonnegative integer Laurent coefficients for every $n\ge0$ and $j\in \mathbb Z$. The refined equidistribution theorem of Berkovich and Garvan gives divisibility; the new assertion is coefficientwise nonnegativity. Set $Ψ(u,z,q)=\sum_{n\ge0}\sum_{j\in \mathbb Z}q^{n+1}u^jK_{n,j}(z)$ and $D(q)=\prod_{d\ge1}(1-q^{2d})$. For every nonnegative integer $M$, we show that $D(q)^MΨ(u,z,q)/[5]_z$ has nonnegative coefficients if and only if $M\le9$. Thus positivity survives after removing nine of the ten partition factors supplied by the core--quotient decomposition. A coefficient equal to $-4$ at $M=10$ proves sharpness. The key step is a compensation between lattice terms of different core sizes that preserves both the BG-rank and the full integer crank. Theta identities and three local coefficient inequalities provide the required nonnegative contributions, and an explicit allocation ensures that none is used twice. The resulting factorization also yields a nine-color convolution and coefficientwise finite-difference inequalities through order nine.