s-Modular, s-congruent and s-duplicate partitions

In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) s-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to 0 or 1 modulo s, (2) s-congruent partitions, which generalize Sellers’ partitions into parts not congruent to 2 modulo 4, and (3) s-duplicate partitions, of which the partitions having distinct odd parts and enumerated by the function \({{\,\textrm{pod}\,}}(n)\) $\text{pod}(n)$ are a special case. In this vein, we generalize Alladi’s series expansion for the product generating function of \({{\,\textrm{pod}\,}}(n)\) $\text{pod}(n)$ and show that Andrews’ generalization of Göllnitz–Gordon identities coincides with the number of partitions into parts simultaneously s-congruent and t-distinct (parts appearing fewer than t times).