Arithmetic properties of partition k-tuples with odd parts distinct

Let pod−k(n) denote the number of partition k-tuples of n wherein odd parts are distinct(and even parts are unrestricted). We establish some interesting infinite families of congruences and internal congruences modulo 4, 16, and 5 for pod−2(n), pod−4(n),and pod−6(n), respectively. We also find Ramanujan-type congruences modulo 5 for pod−3(n) and densities of pod−2(n), pod−3(n), pod−4(n), and pod−6(n) modulo 4, 5,16, and 5, respectively