A central limit theorem for integer partitions into small powers

Abstract

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