  In a recent paper, Andrews and Newman introduced certain families of
partition functions using the minimal excludant or "mex" function. In this
article, we study two of the families of functions Andrews and Newman
introduced, namely $p_{t,t}(n)$ and $p_{2t,t}(n)$. We establish identities
connecting the ordinary partition function $p(n)$ to $p_{t,t}(n)$ and
$p_{2t,t}(n)$ for all $t\geq 1$. Using these identities, we prove that the
Ramanujan's famous congruences for $p(n)$ are also satisfied by $p_{t,t}(n)$
and $p_{2t,t}(n)$ for infinitely many values of $t$. Very recently, da Silva
and Sellers provide complete parity characterizations of $p_{1,1}(n)$ and
$p_{3,3}(n)$. We prove that $p_{t,t}(n)\equiv \overline{C}_{4t,t}(n) \pmod{2}$
for all $n\geq 0$ and $t\geq 1$, where $\overline{C}_{4t,t}(n)$ is the Andrews'
singular overpartition function. Using this congruence, the parity
characterization of $p_{1,1}(n)$ given by da Silva and Sellers follows from
that of $\overline{C}_{4,1}(n)$. We also give elementary proofs of certain
congruences already proved by da Silva and Sellers.
