A q-rious positivity

TL;DR

This paper explores a conjecture that extends the known non-negativity of $q$-binomial coefficients to a broader class of polynomials formed from ratios of $q$-factorials, motivated by arithmetic considerations.

Contribution

It proposes a new conjecture generalizing the non-negativity property of $q$-binomial coefficients to products of ratios of $q$-factorials, with potential combinatorial implications.

Findings

Conjecture extends non-negativity to new polynomial classes

Provides arithmetic motivation for the generalization

Suggests possible combinatorial interpretations

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.