On the q-log-concavity of Gaussian binomial coefficients (Q749535)

We give a combinatorial proof that \(\left[ \begin{matrix} a\\ k\end{matrix} \right]_ q\left[ \begin{matrix} b\\ \ell \end{matrix} \right]_ q-\left[ \begin{matrix} a\\ k-1\end{matrix} \right]_ q\left[ \begin{matrix} b\\ \ell +1\end{matrix} \right]_ q\) is a polynomial in q with non-negative coefficients for non-negative integers, a, b, k, \(\ell\) with \(a\geq b\) and \(\ell \geq k\). In particular, for \(a=b=n\) and \(\ell =k\), this implies the q-log-concavity of the Gaussian binomial coefficients \(\left[ \begin{matrix} n\\ k\end{matrix} \right]_ q\), which was conjectured by \textit{L. M. Butler} [Proc. Am. Math. Soc. 101, 771-775 (1987; Zbl 0647.20053)].