On the statistical theory of partitions. (English) Zbl 0548.10010

Erdős, Paul; Szalay, M.

Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 397-450 (1984).

Let \(\Pi =\{\lambda_1+\lambda_2+\ldots+\lambda_m = n\); \(\lambda_1 \ge \lambda_2 \ge\ldots\ge \lambda_m \ge 1\}\) be a generic partition of \(n\) where \(m=m(\Pi)\) and the \(\lambda_\mu\)’s are integers. Let \(p(n)\) denote the number of partitions of \(n\). The first author and J. Lehner [Duke Math. J. 8, 335–345 (1941; Zbl 0025.10703)] determined the distribution of \(\lambda_1\) where \(\lambda_1 = \lambda_1(\Pi) = \max_{\nu\in \Pi}\nu\). The following analogous result is proved for the maximum with multiplicities.
Theorem 1. The number of partitions of \(n\) with the property
\[ \max_{\nu\in\Pi} \{\nu\, \operatorname{mult}(\nu)\text{ in } \Pi\} \leq (2\pi)^{-1}(6n)^{1/2} \log n+\pi^{-1}(6n)^{1/2} \log\log\log n+\pi^{-1}(6n)^{1/2}c \]
is \[(\exp(-\pi^{-1}6^{1/2}e^{-c}) + o(1))p(n). \]
As to the \(\lambda_\mu\)’s, some consequences of earlier results are also discussed. For “unequal” partitions (their number is \(q(n))\), the increasing order \((\alpha'_1 +\ldots+ \alpha'_m = n\); \(1\le\alpha'_1 < \alpha'_2 <\ldots< \alpha'_m)\) is more interesting.
Theorems 2 and 3 state estimates for \(\alpha'_{\mu}\) which yield the following:
Corollary. For arbitrary \(\eta > 0\), there exist \(n_0\) and \(\varepsilon > 0\) such that, for \(n > n_0\) with the restriction \(\varepsilon^{-1} \le \mu \le \varepsilon \cdot n^{1/2}\), the estimation \(|\alpha'_{\mu}-2\mu| \le \eta \mu\) holds uniformly with the exception of at most \(\eta q(n)\) unequal partitions of \(n\).
For the entire collection see Zbl 0541.00002.

Reviewer: Paul Erdős (Budapest)

MSC:

11P81

Elementary theory of partitions

Keywords:

unequal partitions of integers; distribution of summands

Citations:

Zbl 0541.00002; Zbl 0025.10703