  The study of the well-known partition function $p(n)$ counting the number of
solutions to $n = a_{1} + \dots + a_{\ell}$ with integers $1 \leq a_{1} \leq
\dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we
study a variant, namely partitions of integers into \begin{equation*}
  n=\lfloor a_1^\alpha\rfloor + \cdots + \lfloor a_\ell^\alpha\rfloor
\end{equation*} with $1\leq a_1 < \cdots < a_\ell$ and some fixed $0 < \alpha <
1$. In particular, we prove a central limit theorem for the number of summands
in such partitions, using the saddle point method.
