Partitions into Piatetski-Shapiro sequences

Let $kappa$ be a positive real number and $minmathbb{N}cup{infty}$ be given. Let $p_{kappa, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(lfloor ell^{kappa}rfloor)_{ellin mathbb{N}}$ 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.