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 $1 \leq a_{1} \leq \dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we study a variant, namely partitions of integers into \begin{equation*} n=\lfloor a_1^α\rfloor + \cdots + \lfloor a_\ell^α\rfloor \end{equation*} with $1\leq a_1 < \cdots < a_\ell$ and some fixed $0 < α< 1$. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle point method.

Theorems & Definitions (13)

• Lemma 1
• proof
• Theorem 2
• Theorem 3: Converse Mapping flajolet_gourdon_dumas1995:mellin_transforms_and
• Lemma 4
• proof
• Lemma 5
• Lemma 6: Li--Chen li_chen2018:r_th_root
• Lemma 7
• proof