  The partition function $pod(n)$ enumerates the partitions of $n$ wherein odd
parts are distinct and even parts are unrestricted. Recently, a number of
properties for $pod(n)$ have been established. In this paper, for $k\in\{0,2\}$
we consider the partitions of $n$ into distinct parts not congruent to $k$
modulo $4$ and the $4$-regular partitions of $n$ in order to obtain new
properties for $pod(n)$. In this context, we derive two new infinite families
of linear inequalities involving the function $pod(n)$ and obtain new
identities of Watson type.
