Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler

We investigate Conjecture 7.4 of Guo and Krattenthaler \cite{GK}, which predicts that for every positive integer $n$, the polynomial $$P_n(q)=\frac{(1-q{)}^2}{(1-q^{10n-1})(1-q^{15n-1})}\text{qbin}30n5n$$ has non-negative coefficients, except for $q^1$ and $q^{125n^2-25n+3}$, which equal $-1$. We study this problem through three main perspectives: First, by linking the boundary coefficients to the second differences of the partition function, we settle the first and last $25n+1$ coefficients via a novel partition injection. Second, by introducing a ``defect’’ generalization of Andrews’ classical lemma, we reduce the conjecture to an almost-unimodality property and employ a uniform saddle-point analysis to prove it analytically for all large $n\ge 2\cdot {10}^{29}$. Finally, we connect these quotients to graph binomial coefficients, providing a natural orbit-counting interpretation via Burnside’s lemma on directed cycles and paths.