  Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq
\mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of
$n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$.
Under hypotheses on the roots of $f-f(0)$, we use the Hardy--Littlewood circle
method, a polylogarithm identity, and the Matsumoto--Weng zeta function to
derive asymptotic formulae for $p_{\mathcal{A}_f}(n)$ as $n$ tends to infinity.
This generalises asymptotic formulae for the number of partitions into perfect
$d$th powers, established by Vaughan for $d=2$, and Gafni for the case $d \geq
2$, in 2015 and 2016 respectively.
