Partitions into parts which are unequal and large. (English) Zbl 0679.10013

Number theory, Proc. 15th Journ. Arith., Ulm/FRG 1987, Lect. Notes Math. 1380, 19-30 (1989).
[For the entire collection see Zbl 0667.00007.]
Let \(q(n)\) be the number of partitions of \(n\) into unequal parts, and let \(\rho(n,m)\) be the number of partitions of \(n\) into unequal parts \(\geq m\). The first and third authors have previously shown that \(\rho(n,m)=(1+o(1))q(n)/2^{m-1}\) for \(m=o(n^{1/5})\) [Colloq. Math. Soc. János Bolyai 34, 397-450 (1984; Zbl 0548.10010)]. Three additional theorems giving estimates for \(\rho(n,m)\) are now obtained.
Theorem 1: For all \(n\geq 1\) and \(m\) such that \(1\leq m\leq n\), we have \((i)\quad q(n)/2^{m-1}\leq \rho(n,m)\leq q(n+m(m-1)/2)/2^{m-1}\) and \((ii)\quad \rho(n,m)\leq q(n+[m(m-1)/4])/2^{m-2},\) where \([x]\) is the integral part of \(x\).
Theorem 2: When n tends to infinity, and \(m=o(n/\log n)^{1/3}\), we have \[ \rho (n,m)=(1+o(1))q(n+[m(m-1)/4])/2^{m-1}. \] Theorem 3: For fixed \(\epsilon\), with \(0<\epsilon <10^{-2}\) and for \(m=m(n)\), \(1\leq m\leq n^{3/8-\epsilon}\), and \(n\to \infty\), \[ \rho(n,m)=(1+o(1))q(n)/\prod_{1\leq j\leq m-1}(1+\exp(-\pi j/2\sqrt{3n})). \] The paper concludes with a table of values for \(\rho(n,m)\) with \(1\leq n\leq 100\) and \(1\leq m\leq \min (n,12)\).
Reviewer: B.Garrison


11P81 Elementary theory of partitions