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 $n=a_1+\cdots +a_{\ell }$ with integers $1\le a_1\le \cdots \le a_{\ell }$ has a long history in combinatorics. In this paper, we study a variant, namely partitions of integers into \begin{equation*} n=\lfloor a_1^α\rfloor + \cdots + \lfloor a_\ell^α\rfloor \end{equation*} 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.

17 pages

Cited by in corpus (2)

• Boltzmann Distribution on "Short" Integer Partitions with Power Parts: Limit Laws and Sampling
• A Local Limit Theorem for Integer Partitions into Small Powers