On the sets represented by the partitions of an integer \(n\)
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.
scientific article; zbMATH DE number 1334438
scientific article