Partitions and the maximal excludant

TL;DR

This paper introduces the concept of the maximal excludant for integer partitions, explores its generating function's connection to mock theta functions, and analyzes its asymptotic behavior and expected differences in large partitions.

Contribution

It defines the maximal excludant for partitions, links its generating function to mock theta functions, and studies its asymptotic properties and expectations.

Findings

Generating function related to a mock theta function

Asymptotic equivalence to sum of largest parts

Expected difference converges to 1

Abstract

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