$4$-Regular partitions and the pod function

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 ∈ {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.

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