A q-rious positivity

Abstract

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of $\qbinom{n}{m}$. In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of $q$-factorials that happen to be polynomials.