Congruences for the Andrews--Uncu partition function ${\mathcal{EO}}_u(n)$

Abstract

In 2018, Andrews defined and studied the partition function $\mathcal{EO}(n)$, which counts the number of partitions of $n$ where each even part is less than each odd part. Thereafter, Uncu considered a different subset of such partitions, namely those partitions counted by $\mathcal{EO}(n)$ where there are no repeated even parts (we denote the number of such partitions by ${\mathcal{EO}}_u(n)$). In this paper, we prove several congruences modulo powers of $2$ for ${\mathcal{EO}}_u(n)$. We also prove some infinite families of congruences as well as a recurrence for the number of this class of partitions. Our methods involve elementary, algorithmic, and modular form techniques.

arXiv commentThis work was completed in the summer of 2024 and no new results have since been added to the manuscript