Positivity of Warnaar polynomials for general coprime profiles

Establish that, for every coprime profile \(\boldsymbol{c}=(c_0,\ldots,c_{r-1})\) of positive level and every \(m\ge 0\), the polynomial \(Q_{m,\boldsymbol{c}}(q)\) defined by \((q;q)_N C_{\boldsymbol{c},\le N}(q)=\sum_{m=0}^N \begin{bmatrix}N\\m\end{bmatrix}_q Q_{m,\boldsymbol{c}}(q)\) has nonnegative coefficients, thereby proving the remaining positivity assertion in Conjecture 8.4 of Warnaar.

Background

For a coprime profile $\boldsymbol{c}$, the bounded cylindric-partition generating function $C_{\boldsymbol{c},\le N}(q)$ admits an expansion after multiplication by $(q;q)_N$ in Gaussian binomial coefficients, with uniquely determined polynomials $Q_{m,\boldsymbol{c}}(q)$. Warnaar's Conjecture 8.4 asserts that these polynomials have nonnegative coefficients and also specifies their values at $q=1$.

The paper notes that polynomiality and the $q=1$ evaluation were established by unpublished work of Welsh, while the coefficientwise positivity remains unresolved in general. The present paper proves this finer positivity only for the coprime balanced profiles associated with the torus-knot parameters $(a,b)$, so the problem remains open for arbitrary coprime profiles not covered by that balanced family.