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

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.

### MSC:

 11P81 Elementary theory of partitions

### Citations:

Zbl 0541.00002; Zbl 0025.10703