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

Abstract

F. Bergeron recently asked the intriguing question whether ${\left(\genfrac{}{}{0ex}{}{b+c}{b}\right)}_q-{\left(\genfrac{}{}{0ex}{}{a+d}{d}\right)}_q$ 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.

Final version. To appear in the Electronic J. Combinatorics