On the sets represented by the partitions of an integer \(n\)
type: Item
instance of: scholarly article
title: On the sets represented by the partitions of an integer \(n\)
zbMATH Open document ID: 0976.11048
DOI: 10.1016/S0012-365X(98)00330-6
author: Marc Deléglise
author: Jean-Louis Nicolas
author: P. Erdős
published in: Discrete Mathematics
publication date: 2002-01-13T00:00:00Z
review text: Let \(N^*\) be the set of all natural numbers. If \(n\in N^*\), let a partition, \(\pi\), of \(n\) be given by: \(n=n_1+n_2 +\cdots +n_j\), where \(n_1\geq n_2\geq \cdots\geq n_j\). Say that \(\pi\) represents the integer \(r\) if \(r\) equals a sumset of some of the parts of \(\pi\). Let \(E(\pi)\) be the set of all integers represented by \(\pi\). If \(A\subset N^*\), define \(p_A(n)\) as the number of partitions of \(n\) with parts in \(A\), \(\hat p_A(n)\) as the number of distinct sets represented by these partitions. If \(k\geq 1\), let \(q(A,n,k)\) denote the number of ''\(k\)-reduced'' partitions of \(n\) with parts in \(A\), that is, the number of such partitions in which each part occurs at most \(k\) times. (Other authors have referred to such partitions as \((k-1)\)-regular.) This paper is largely devoted to obtaining estimates for \(\hat p_A(n)\) in the cases: (i) \(A=N^*\), (ii) A is the set of all powers of 2. An interesting feature of this paper is the use of a Tauberian theorem by Ingham to obtain the classic asymptotic estimates for \(p(n)\) (the unrestricted partition function) and \(q(n,k)\) (the number of \(k\)-reduced partitions of \(n)\). For \(q(n,1)=q(n)\), this estimate had been obtained by P. Hagis some time ago.
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 1334438
zbMATH Keywords: \(k\)-reduced partitions
zbMATH Keywords: unrestricted partitions
zbMATH Keywords: asymptotic estimate
zbMATH Keywords: number of partitions
zbMATH Keywords: Tauberian theorem
Wikidata QID: Q105857720
reviewed by: Neville Robbins
MaRDI profile type: Publication
cites work: Q5786798
cites work: Q3360966
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full work available at URL: s0012-365x(98)00330-6
OpenAlex ID: W2051081366
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description: scientific article; zbMATH DE number 1334438