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

Abstract

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 $\mypod(n)$ are a special case. In this vein, we generalize Alladi's series expansion for the product generating function of $\mypod(n)$ and show that Andrews' generalization of G\"ollnitz-Gordon identities coincides with the number of partitions into parts simultaneously $s$-congruent and $t$-distinct (parts appearing fewer than $t$ times).