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

Erdős, Paul; Nicolas, J. L.; Szalay, M.

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

MSC:

11P81

Elementary theory of partitions

Keywords:

partitions with unequal parts; number of partitions; table

Citations:

Zbl 0667.00007; Zbl 0548.10010