  For $k\geqslant 1$, denote by $p_k(n)$ the number of partitions of an integer
$n$ into $k$-th powers. In this note, we apply the saddle-point method to
provide a new proof for the well-known asymptotic expansion of $p_k(n)$. This
approach turns out to significantly simplify those of Wright (1934), Vaughan
(2015) and Gafni (2016).
