On Arithmetic-Progression Biases in Random Partitions: Limiting Laws and the Li-Zhao Conjecture

Let $\cR=\{r+jm:j\geq0\}$ and $\cS=\{s+jm:j\geq0\}$ be disjoint arithmetic progressions of positive integers, and choose a partition of $n$ uniformly at random. We determine the limiting probability that the number of parts in $\cR$ is larger than the number of parts in $\cS$. More precisely, if $a=