On Bergeron's Positivity Problem for $q$-Binomial Coefficients

Fabrizio Zanello

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.

Keywords


$q$-Binomial Coefficient; Unimodality; Positivity; KOH Theorem.

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