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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 548.10010

**Autor: ** Erdös, Paul; Szalay, M.

**Title: ** On the statistical theory of partitions. (In English)

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

**Review: ** [For the entire collection see Zbl 541.00002.]

Let \Pi = **{**\lambda_{1}+\lambda_{2}+...+\lambda_{m} = n; \lambda_{1} \geq \lambda_{2} \geq ... \geq \lambda_{m} \geq 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 025.10703)] 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 mult(\nu) in \Pi**}** \leq (2\pi)^{-1}(6n)^{ ½} log n+\pi^{-1}(6n)^{ ½} log log log n+\pi^{-1}(6n)^{ ½}c is (\exp(-\pi^{-1}6^{ ½}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}+...+\alpha'_{m} = n; 1 \leq \alpha'_{1} < \alpha'_{2} < ... < \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 \epsilon > 0 such that, for n > n_{0} with the restriction \epsilon^{-1} \leq \mu \leq \epsilon · n^{ ½}, the estimation |\alpha'_{\mu}-2\mu| \leq \eta \mu holds uniformly with the exception of at most \eta q(n) unequal partitions of n.

**Classif.: ** * 11P81 Elementary theory of partitions

11P81 Elementary theory of partitions

**Keywords: ** unequal partitions of integers; distribution of summands

**Citations: ** Zbl 541.00002; Zbl 025.107

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