Partitions and the maximal excludant

Shane Chern¹

¹The Pennsylvania State University

The electronic journal of combinatorics, Tome 28 (2021) no. 3

Résumé

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ as the largest nonnegative integer smaller than the largest part of $\pi$ that is not itself a part. 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, Dyson and Hickerson, and Cohen, respectively. Further, we show that, as $n\to \infty$, $\sigma\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts over 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 $n\to \infty$.

Détail

DOI : 10.37236/8736

Classification : 05A17, 05A19, 05A15, 11P84, 11P81
Mots-clés : maximal excludant of an integer partition, generating function