Arithmetic Properties of Partition Quadruples With Odd Parts Distinct

Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\alpha \ge 1$ and $n \ge 0$, \[\mathrm{pod}_{-4}\Big({{3}^{\alpha +1}}n+\frac{5\cdot {{3}^{\alpha }}+1}{2}\Big)\equiv 0 \pmod{9}.\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by $\mathrm{pod}_{-4}(n)$.