Dyson's Partition Ranks and their Multiplicative Extensions

We study the Dyson rank function $N(r,3;n)$, the number of partitions of $n$ with rank $\equiv r(\mathrm{m}\mathrm{o}\mathrm{d}3)$. We investigate the convexity of these functions. We extend $N(r,3;n)$ multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size $n$.