Arithmetic Properties of Partition Triples With Odd Parts Distinct

Let $\mathrm{pod}_{-3}(n)$ denote the number of partition triples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-3}(n)$ involving the following infinite family of congruences: for any integers $\alpha \ge 1$ and $n\ge 0$, \[\mathrm{pod}_{-3}\Big({{3}^{2\alpha +2}}n+\frac{23\times {{3}^{2\alpha +1}}+3}{8}\Big)\equiv 0 \pmod{9}.\] We also establish some arithmetic relations between $\mathrm{pod}(n)$ and $\mathrm{pod}_{-3}(n)$, as well as some congruences for $\mathrm{pod}_{-3}(n)$ modulo 7 and 11.