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

TL;DR

This paper explores the properties of three classes of integer partitions—$s$-modular, $s$-congruent, and $s$-duplicate—generalizing existing identities and linking them to combinatorial enumeration.

Contribution

It introduces new classes of partitions and generalizes known series expansions and identities, connecting them to combinatorial enumeration of partitions.

Findings

Generalized Alladi's series expansion for $ ext{mypod}(n)$

Connected Andrews' Göllnitz-Gordon identities to $s$-congruent and $t$-distinct partitions

Established new combinatorial interpretations of partition classes

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).