Journal of Integer Sequences, Vol. 24 (2021), Article 21.6.3 |

Department of Mathematics

Indian Institute of Technology

Guwahati-781039

India

**Abstract:**

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.

Received February 22 2021; revised versions received February 23 2021; May 17 2021; May 18 2021.
Published in *Journal of Integer Sequences*,
May 18 2021.

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