On two partition problems
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 \(0.7\).
scientific article; zbMATH DE number 1225568