In this paper, we decompose $\overline {D}(a,M)$ into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let $\overline {p}(n)$ be the number of partitions of n and $\overline {N}(a,M,n)$ be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average:

$$ \begin{align*} \overline{D}(a,M) &=\sum\limits_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n}. \end{align*} $$

Let $\overline{t}(n)$ be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for $\overline{t}(n)$ modulo small powers of $2$ and $3$ have been established. We present two infinite families of congruences modulo $5$ and $27$ for $\overline{t}(n)$, the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, Proc. Math. Sci. 129 (2019), Article 31].

Ramanujan famously found congruences like $p(5n\,+\,4)\,\equiv \,0$ mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on ${{\Gamma }_{1}}(4)$ that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored $F$-partitions.

Using the framework of overpartitions, we give a combinatorial interpretation and proof of the q-Bailey identity. We then deduce from this identity a couple of facts about overpartitions. We show that the method of proof of the q-Bailey identity also applies to the (first) q-Gauss identity.