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.
Categories: math.NT math.CO