## 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 = {\lambda1+\lambda2+...+\lambdam = n;   \lambda1 \geq \lambda2 \geq ... \geq \lambdam \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 \lambda1 where \lambda1 = \lambda1(\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) ½c

is (\exp(-\pi-16 ½e-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 n0 and \epsilon > 0 such that, for n > n0 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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