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Let $\lambda$ be a partition of the positive integer $n$ chosen umiformly atrandom 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 $n$, wefocus on a comparison between the partition statistics $L_n$ and $L_n M_n$. Interms 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 thisexpectation and conclude with an open problem arising from this study.

On the Largest Part Size and Its Multiplicity of a Random Integer Partition