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

Let $\text{cR}=\{r+jm:j\ge 0\}$ and $\text{cS}=\{s+jm:j\ge 0\}$ 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 $\text{cR}$ is larger than the number of parts in $\text{cS}$. More precisely, if $a=r\mathrm{/}m$ and $b=s\mathrm{/}m$, then this probability converges to $I_{1\mathrm{/}2}(a,b)$, the regularized incomplete beta function. We prove the stronger distributional statement that, after multiplication by the partition saddle-point parameter, the difference of the two numbers of parts converges to $m^{-1}(\log G_b-\log G_a)$, where $G_a$ and $G_b$ are independent gamma variables. The proof uses the Boltzmann conditioning device for random partitions, a $q$-gamma evaluation of the unconditioned characteristic function, and a uniform Fourier estimate showing that conditioning on the total size does not change the limit. We also evaluate the finite-truncation limit studied by Li and Zhao. The two evaluations coincide, proving their conjecture on the commutation of the truncation and large-size limits.