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 ≥ 1. Using these identities, we prove that 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 provided complete parity characterizations of p_1,1(n) and p_3,3(n). We prove that p_t,t(n) ≡ C_4t,t(n) (mod 2) for all n ≥ 0 and t ≥ 1, where C_4t,t(n) is 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 C_4,1(n).

We also give elementary proofs of certain congruences already proved by da Silva and Sellers.