A Central Limit Theorem for Integer Partitions into Small Powers

The study of the well-known partition function $p(n)$ counting the number of solutions to $n=a_1+\cdots +a_{\mathrm{\ell }}$ 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 $1\le a_1<\cdots <a_{\mathrm{\ell }}$ 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.