A conjecture of Baruah and Begum on the smallest parts function of restricted overpartitions

TL;DR

This paper confirms three families of congruences related to the smallest parts function of restricted overpartitions, advancing understanding of its modular properties and supporting a conjecture by Baruah and Begum.

Contribution

It proves three families of congruences for the smallest parts function of restricted overpartitions, validating parts of Baruah and Begum's conjecture.

Findings

Confirmed three families of congruences modulo powers of 5.

Validated parts of Baruah and Begum's conjecture.

Enhanced understanding of the modular behavior of the smallest parts function.

Abstract

In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_\omega(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts are less than twice the smallest part. Recently, Baruah and Begum established several internal congruences and congruences modulo small powers of $5$ for $\overline{\textrm{spt}}_\omega(n)$. Moreover, they conjectured a family of internal congruences modulo any powers of $5$ and two families of congruences modulo any even powers of $5$. In this paper, we confirm three families of congruences due to Baruah and Begum.