On the Largest Part Size and Its Multiplicity of a Random Integer Partition
Abstract
Let $\lambda$ be a partition of the positive integer $n$ chosen umiformly at random among all such partitions. Let $L_n=L_n(\lambda)$ and $M_n=M_n(\lambda)$ be the largest part size and its multiplicity, respectively. For large $n$, we focus on a comparison between the partition statistics $L_n$ and $L_n M_n$. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of $L_n M_n L_n$ grows as fast as $\frac{1}{2}\log{n}$ We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.03233
 arXiv:
 arXiv:1712.03233
 Bibcode:
 2017arXiv171203233M
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics;
 05A17;
 11P82;
 60C05
 EPrint:
 14 pages, 1 figure