Let $$ \binom{n}{k}_{\!q} = \frac{(1-q^n) \cdots (1-q^{n-k+1})}{(1-q) \cdots (1-q^k)} $$ be the Gaussian polynomials. For example, $$ \binom{6}{3}_{\!q} = 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9. $$ It seems that the coefficients in $\binom n k _q$ are always weakly increasing up to $q^{\lfloor d/2\rfloor}$, where $d$ is the degree of the polynomial. How would one prove this?
If $p_{k\ell}(n)$ is the number of partitions of $n$ that fit into an $k\times\ell$-rectangle, I know that $\sum_np_{k\ell}(n)q^n=\binom{k+\ell}{\ell}_q$ (generating functions), so it seems reasonable to try to find an injection from the set of partitions counted by $p_{k\ell}(n)$ to the set of partitions counted by $p_{k\ell}(n+1)$, where $n\leq\lfloor k\ell/2\rfloor$, but I cannot find such an injection.
