Partitions and the maximal excludant

Abstract

For each nonempty integer partition $π$, we define the maximal excludant of $π$ to be the largest nonnegative integer smaller than the largest part of $π$ that is not a part of $π$. Let $σ\!\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $σ\!\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews \textit{et al.} and Cohen. Further, we show that, as $n\to \infty$, $σ\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts of all partitions of $n$. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of $n$ is shown to converge to $1$ as $n\to \infty$.