A central limit theorem for integer partitions into small powers

Abstract

The study of the well-known partition function p(n) counting the number of solutions to \(n = a_{1} + \dots + a_{\ell }\) with integers \(1 \le a_{1} \le \dots \le a_{\ell }\) has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into

$$\begin{aligned} n=\left\lfloor a_1^\alpha \right\rfloor +\cdots +\left\lfloor a_\ell ^\alpha \right\rfloor \end{aligned}$$

with \(1\le a_1< \cdots < a_\ell \) 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.