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

Abstract: Let

l a m b d a

be a partition of the positive integer

n

chosen umiformly at random among all such partitions. Let

L_{n} = L_{n} (l a m b d a)

and

M_{n} = M_{n} (l a m b d a)

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

f r a c 12 l o g n

We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.