Zeilberger's KOH theorem and the strict unimodality of q-binomial coefficients

Abstract

A recent nice result due to I. Pak and G. Panova is the strict unimodality of the $q$-binomial coefficients $\binom{a+b}{b}_q$ (see \cite{PP} and also \cite{PP2} for a slightly revised version of their theorem). Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of $\binom{a+b}{b}_q$ can get arbitrarily large, when we assume that $b$ is fixed and $a$ is large enough.