Remarks on a certain restricted partition function of Lin

Let $b(n)$ be the number of partition triples $\pi =({\pi }_1,{\pi }_2,{\pi }_3)$ of $n$ such that ${\pi }_1$ consists of distinct odd parts, and ${\pi }_2$ and ${\pi }_3$ 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 $b(3n+2)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}3)$ for all $n\ge 0$. 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 $b(n)$.