On the largest part size and its multiplicity of a random integer partition

Let λ be a partition of the positive integer n chosen uniformly at random among all such partitions. Let L_n = L_n(λ) and M_n = M_n(λ) 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_nM_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_nM_n – L_n grows as fast as 12logn{1 \over 2}\log n. We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.