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 \(b(3n+2)\equiv 0\pmod {3}\) 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).
