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The partition function $pod(n)$ enumerates the partitions of $n$ wherein oddparts are distinct and even parts are unrestricted. Recently, a number ofproperties 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 newproperties for $pod(n)$. In this context, we derive two new infinite familiesof linear inequalities involving the function $pod(n)$ and obtain newidentities of Watson type.

$4$-Regular partitions and the pod function