A Central Limit Theorem for Integer Partitions into Small Powers

TL;DR

This paper proves a central limit theorem for the number of summands in integer partitions into small powers, extending classical partition results to a new variant involving fractional powers.

Contribution

It introduces a novel analysis of partitions into fractional powers and establishes a central limit theorem for the number of summands in this setting.

Findings

Central limit theorem for partitions into fractional powers

Asymptotic distribution of the number of summands

Application of saddle point method to this partition variant

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 $1 \leq a_{1} \leq \dots \leq a_{\ell}$ 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.