On two partition problems
type: Item
instance of: scholarly article
title: On two partition problems
zbMATH Open document ID: 0902.05003
DOI: 10.1023/A:1006587723678
published in: Acta Mathematica Hungarica
publication date: 1998-11-22T00:00:00Z
review text: Let \(\rho(n,x)\) denote the number of partitions of \(n\) into distinct parts \(\geq x\). In the first part of the paper, it is proved that \(\log\rho(n,\lambda \sqrt n)\sim h(\lambda)\sqrt n\), where \(h(\lambda)\) is a function of \(\lambda\) explicitly described through a functional equation it satisfies. The second part of the paper is concerned with asymptotics of sets represented by partitions. Given a partition \(\Pi\) of \(n\), \(n=n_1+n_2+\dots+n_k\), we say that \(a\) is represented by \(\Pi\) if \(a\) can be written as \(a=\varepsilon_1n_1+\varepsilon_2n_2+\dots+\varepsilon_kn_k\) with \(\varepsilon_i=0\) or 1. For each \(\Pi\) denote the set of these integers by \(T(\Pi)\). For fixed \(n\), let \(\hat p(n)\) be the number of different sets \(T(\Pi)\) as \(\Pi\) varies over all partitions of \(n\). The authors prove that there are constants \(c_1\) and \(c_2\), \(0<c_1\leq c_2<1\), such that \(p(n)^{c_1}\leq \hat p(n)\leq p(n)^{c_2}\) for \(n\) large enough. Here, as usual, \(p(n)\) denotes the total number of (unrestricted) partitions of \(n\). It is also shown that one can choose \(c_1=0.361\) and \(c_2=0.948\). Empirical data, which are listed in the paper, suggest that the ``right'' exponent may be somewhat less than \(0.7\).
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A16
Mathematics Subject Classification ID: 11P82
zbMATH DE Number: 1225568
zbMATH Keywords: integer partitions
author: Jean-Louis Nicolas
author: András Sárközy
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
full work available at URL: a:1006587723678
OpenAlex ID: W1916170788
description: scientific article; zbMATH DE number 1225568