Congruences for an analogue of Lin's partition function

We study certain arithmetic properties of an analogue $B(n)$ of Lin’s restricted partition function that counts the number of partition triples $\pi =({\pi }_1,{\pi }_2,{\pi }_3)$ of $n$ such that ${\pi }_1$ and ${\pi }_2$ comprise distinct odd parts and ${\pi }_3$ consists of parts divisible by $4$. With the help of elementary $q$-series techniques and modular functions, we establish Ramanujan-type congruences modulo $2,3,5,7$, and $9$ for certain sums involving $B(n)$.