  We determine the asymptotics of the number of independent sets of size
$\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for
any fixed $\beta \in [0,1]$ as $d \to \infty$, extending a result of Galvin for
$\beta \in [1-1/\sqrt{2},1]$. Moreover, we prove a multivariate local central
limit theorem for structural features of independent sets in $Q_d$ drawn
according to the hard core model at any fixed fugacity $\lambda>0$. In proving
these results we develop several general tools for performing combinatorial
enumeration using polymer models and the cluster expansion from statistical
physics along with local central limit theorems.
