On Bergeron's positivity problem for $q$-binomial coefficients

TL;DR

This paper investigates Bergeron's positivity problem for differences of q-binomial coefficients, conjectures unimodality, and proves the conjecture for specific parameter ranges using Zeilberger's KOH theorem.

Contribution

It introduces a conjecture on the unimodality of certain q-binomial difference polynomials and proves it for cases where a ≤ 3 using a novel application of Zeilberger's KOH theorem.

Findings

Conjecture that the polynomial difference is unimodal.

Proof of the conjecture for a ≤ 3 and any b,c ≥ 4.

Application of Zeilberger's KOH theorem to this problem.

Abstract

F. Bergeron recently asked the intriguing question whether $\binom{b+c}{b}_q -\binom{a+d}{d}_q$ has nonnegative coefficients as a polynomial in $q$, whenever $a,b,c,d$ are positive integers, $a$ is the smallest, and $ad=bc$. We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for $a\le 3$ and any $b,c\ge 4$. The main ingredient will be a novel (and rather technical) application of Zeilberger's KOH theorem.