  We compute the asymptotic behaviour of the number of partitions of large
vectors $(n_1,n_2)$ of $\mathbb{Z}_+^2$ in the critical regime $n_1 \asymp
\sqrt{n_2}$ and in the subcritical regime $n_1 = o(\sqrt{n_2})$. This work
completes the results established in the fifties by Auluck, Nanda, and Wright.
