Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts

TL;DR

The paper studies arithmetic properties of colored partitions restricted by parity of the parts, introducing the two-parameter family $a_{r,s}(n)$ with generating function $\sum_{n\ge0} a_{r,s}(n) q^n = \dfrac{f_2^{s-r}}{f_1^s}$. It develops a toolkit based on Ramanujan theta functions, 3-dissections, and a lifting lemma to derive extensive congruences modulo $3$ and various primes, providing both elementary proofs and systematic theorems. The main contributions include multiple infinite families of congruences for many $(r,s)$-pairs and a unified method to obtain such congruences via $q$-series manipulations and residue-class extractions, culminating in results valid for primes $p$ under prescribed residue conditions. This work significantly extends prior results on parity-restricted colored partitions by offering a general framework and numerous new congruences for the generalized $a_{r,s}(n)$.

Abstract

Let $a_{r,s}(n)$ denote the number of mutlicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. The paper aims to study arithmetic properties satisfied by $a_{r,s}(n)$, using elementary generating function manipulations and classical $q$-series techniques.

Theorems & Definitions (51)

• Theorem 1.1: hs1
• Theorem 1.2: thfa
• Theorem 1.3: thfa
• Theorem 1.4: sellers
• Theorem 1.5: ajsi
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10