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

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.