Remarks on a certain restricted partition function of Lin

TL;DR

The paper investigates the restricted partition function $b(n)$ counting triples with a distinct odd parts block and two blocks of parts divisible by $4$. It provides elementary $q$-series proofs for Lin's generating functions of $b(3n+1)$ and $b(3n+2)$ using $2$- and $3$-dissections and theta-function identities, thereby deriving $b(3n+2)\equiv 0 \pmod{3}$. The authors further derive infinite families of internal congruences modulo $3$ for $b(n)$ by applying known $3$-dissections of related restricted partitions and theta-function manipulations. These results offer an accessible alternative to modular-form arguments and reveal rich modular structure in restricted partition functions with implications for congruence properties in partition theory.

Abstract

Let $b(n)$ be the number of partition triples $π=(π_1,π_2,π_3)$ of $n$ such that $π_1$ consists of distinct odd parts, and $π_2$ and $π_3$ consist of parts divisible by $4$. Utilizing modular forms, Lin obtained the generating functions for $b(3n+1)$ and $b(3n+2)$, which yields the congruence $b(3n+2)\equiv 0\pmod{3}$ for all $n\geq 0$. We provide in this note elementary proofs of these generating functions by employing $q$-series manipulations and dissection formulas. We also establish infinite families of internal congruences modulo $3$ for $b(n)$.

Theorems & Definitions (10)

• Theorem 1.1: lin
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm11']}
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm12']}