Minimal excludant over partitions into distinct parts

TL;DR

This paper investigates the sum of minimal excludants over partitions into distinct parts, revealing connections with Ramanujan's function and strengthening previous results in partition theory.

Contribution

It introduces a new study of the sum of minimal excludants for partitions into distinct parts and links it to Ramanujan's function, extending prior work.

Findings

Connection between minimal excludant sum and Ramanujan's function σ(q)

Derived a stronger version of Uncu's result

Enhanced understanding of partition gaps and their properties

Abstract

The average size of the "smallest gap" of a partition was studied by Grabner and Knopfmacher in 2006. Recently, Andrews and Newman, motivated by the work of Fraenkel and Peled, studied the concept of the "smallest gap" under the name "minimal excludant" of a partition and rediscovered a result of Grabner and Knopfmacher. In the present paper, we study the sum of the minimal excludants over partitions into distinct parts, and interestingly we observe that it has a nice connection with Ramanujan's function $\sigma(q)$. As an application, we derive a stronger version of a result of Uncu.