A central limit theorem for integer partitions into small powers

Gabriel F. Lipnik, Manfred G. Madritsch, Robert F. Tichy
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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.

将整数分割为小幂的中心极限定理
众所周知的分区函数 p(n) 是计算 \(n = a_{1} + \dots + a_{\ell }\) 的解的个数,而 \(1 \le a_{1} \le \dots \le a_{\ell }\) 在数论和组合学中有着悠久的历史。在本文中,我们研究一种变体,即把整数分割成 $$begin{aligned} n=left\floor a_1^alpha\right\rfloor +\cdots +\left\lfloor a__\ell ^alpha\right\rfloor\end{aligned}$$with\(1\le a_1<;\cdots<a_ell)和一些固定的(0<\alpha<1)。特别是,我们利用鞍点法证明了这种分区中求和数的中心极限定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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