  We consider the number of partitions of $n$ whose Young diagrams fit inside
an $m \times \ell$ rectangle; equivalently, we study the coefficients of the
$q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics
throughout the regime $\ell = \Theta (m)$ and $n = \Theta (m^2)$. Previously,
sharp asymptotics were derived by Tak\'acs only in the regime where $|n - \ell
m /2| = O(\sqrt{\ell m (\ell + m)})$ using a local central limit theorem. Our
approach is to solve a related large deviation problem: we describe the tilted
measure that produces configurations whose bounding rectangle has the given
aspect ratio and is filled to the given proportion. Our results are
sufficiently sharp to yield the first asymptotic estimates on the consecutive
differences of these numbers when $n$ is increased by one and $m, \ell$ remain
the same, hence significantly refining Sylvester's unimodality theorem.
