On the number of partitions with a fixed least part
For natural numbers \(j\) and \(n\) with \(j\leq n\), let \(F_j(n)\) denote the number of partitions of \(n\) whose least part is \(j\), and let \(f_j(n)\) denote the number of partitions of \(n\) into distinct parts whose least part is \(j\). In this paper, several properties of \(F_j(n)\) and \(f_j(n)\) are presented. On writing \(q(n)\) for the number of partitions of \(n\) into distinct parts, the author proves, amongst other results, that \(f_1(n)\leq q(n)/2\), and that for any fixed \(j\), one has \(f_j(n)\sim 2^{-j}q(n)\) as \(n\rightarrow\infty\). All the proofs rely on elementary ideas. As regards \(q(n)\), \textit{P. Hagis} proved that \(q(n)\sim 18^{-1/4}(24n+1)^{-3/4}\exp(\pi\sqrt{48n+2}/12)\) as \(n\rightarrow\infty\) [Trans. Am. Math. Soc. 112, 401-415 (1964; Zbl 0131.04203)].
scientific article; zbMATH DE number 1744066