Symmetric quantum binomial coefficient (${\left[\begin{array}{c}n\\ k\end{array}\right]}_q=q^{k(n-k)}{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_{q^{-2}}$)

The quotient $[n{]}_q[n-1{]}_q\cdots [n-k+1{]}_q/([k{]}_q\cdots [1{]}_q)$ is the displayed shifted Gaussian binomial coefficient. It is the formal character of ${\Lambda }^k{L}_{n-1}$, where $L_{n-1}$ is the $n$-dimensional irreducible sl2 Lie algebra representation. Indeed its exterior-power weights are sums of $k$ distinct terms of $n-1,n-3,\dots ,1-n$. Parity unimodality of an sl2 character proves unimodality after removing the intervening zero coefficients; equivalently, the ordinary Gaussian polynomial has unimodal coefficients.