= Symmetric quantum binomial coefficient
{title2=$\left[\begin{smallmatrix}n\\k\end{smallmatrix}\right]_q=q^{k(n-k)}\binom nk_{q^{-2}}$}
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 $\Lambda^kL_{n-1}$, 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 $n-1,n-3,\ldots,1-n$. <Parity unimodality of an sl2 character> proves unimodality after removing the intervening zero coefficients; equivalently, the ordinary Gaussian polynomial has unimodal coefficients.
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