On the parity of some partition functions

Recently, Andrews carried out a thorough investigation of integer partitions in which all parts of a given parity are smaller than those of the opposite parity. Further, considering a subset of this set of partitions, he obtains several interesting arithmetic and combinatorial properties and its connections to the third order mock theta function $\nu(q)$. In fact, he shows the existence of a Dyson-type crank that explains a mod $5$ congruence in this subset. At the end of his paper, one of the problems he poses is to undertake a more extensive investigation on the properties of the subset of partitions. Since then there have been several investigations in various ways, including works of Jennings-Shaffer and Bringmann (Ann. Comb. 2019), Barman and Ray (2019), and Uncu (2019). In this paper, we study certain congruences satisfied by the above set of partitions (and the subset above) along with a certain subset of partitions (of Andrews' partitions above) studied by Uncu and also establish a connection between one of Andrews' partition function above with $p(n)$, the number of unrestricted partitions of $n$. Besides, we provide a combinatorial description of Uncu's partition function.