Congruences modulo $4$ for Rogers--Ramanujan--Gordon type overpartitions

TL;DR

This paper establishes new congruence identities modulo 4 for the count of Rogers--Ramanujan--Gordon type overpartitions, extending previous work on congruences modulo 3 and connecting to overpartition analogues of classical partition theorems.

Contribution

It introduces novel congruence identities modulo 4 for overpartition counts related to Rogers--Ramanujan--Gordon type overpartitions, expanding the understanding of their arithmetic properties.

Findings

Derived congruences modulo 4 for overpartition counts

Connected overpartition congruences to classical partition theorems

Extended previous modulo 3 results to modulo 4

Abstract

In a recent work, Andrews defined the singular overpartitions with the goal of presenting an overpartition analogue to the theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo $3$ for the number of singular overpartitions prescribed by parameters $k=3$ and $i=1$. It should be noticed that this number equals the number of the Rogers--Ramanujan--Gordon type overpartitions with $k=i=3$ which come from the overpartition analogue of Gordon's Rogers--Ramanujan partition theorem introduced by Chen, Sang and Shi. In this paper, we derive numbers of congruence identities modulo $4$ for the number of Rogers--Ramanujan--Gordon type overpartitions.