  Let $\kappa$ be a positive real number and $m\in\mathbb{N}\cup\{\infty\}$ be
given. Let $p_{\kappa, m}(n)$ denote the number of partitions of $n$ into the
parts from the Piatestki-Shapiro sequence $(\lfloor
\ell^{\kappa}\rfloor)_{\ell\in \mathbb{N}}$ with at most $m$ times (repetition
allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan
type for $p_{\kappa, m}(n)$, by employing a framework of asymptotics of
partitions established by Roth-Szekeres in 1953, as well as some results on
equidistribution.
