Arithmetic Properties of Partition Triples With Odd Parts Distinct

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