Moments for Restricted Sum of Reciprocal Parts of Partitions

Abstract

We study the distribution of a partition statistic $\widetilde{\operatorname{srp}}(\lambda)$, which is the sum of reciprocals of the different parts of the partition $\lambda$ via its moments. The generating functions for the moments are related to polylogarithms, and using Wright's circle method, we derive asymptotic formulas for the moments. In particular, we show that as $n \to \infty$,
\[
\mathbb{E}\left[ \widetilde{\operatorname{srp}} (\lambda) : \lambda \in \mathcal{P}_n \right] = \left(\log\frac{\sqrt{6n}}{\pi}\right) \left( 1+ O\left(n^{-\frac{1}{4}}\right)\right) \\
\]
and
\[
\operatorname{Var}\left[ \widetilde{\operatorname{srp}} (\lambda) : \lambda \in \mathcal{P}_n \right] = O\left(n^{-\frac{1}{4}}(\log n)^2\right),
\]
where $\mathcal{P}_n$ is the set of ordinary partitions of $n$.