Asymptotic normality for product-type partition statistics

Establish that every partition statistic whose generating function is built from factors of the forms \((q^a;q^b)_\infty^{-1}\) and \((zq^a;q^b)_\infty^{-1}\) has an asymptotically normal distribution.

Background

The paper proves asymptotic normality for the alternating sum of parts using a two-variable generating function involving the factors (zq;q2)∞−1(zq;q^2)_\infty^{-1} and (q2;q2)∞−1(q^2;q^2)_\infty^{-1}. It then proposes that this behavior may extend to a broad class of partition statistics whose generating functions are assembled from analogous ordinary and statistic-marked infinite-product factors.

This is explicitly presented as the first part of a conjectural framework, so the claimed universality of asymptotic normality remains unresolved.

References

We propose the following: \begin{enumerate} \item A partition statistic with a generating function built from pieces of the shape $ qa; qb _\infty$ and $ zqa;qb _\infty$ will have an asymptotically normal distribution.

— Distribution of Alternating Sums of Parts in Partitions  (2501.17065 - Craig et al., 28 Jan 2025) in Conjecture in Section 5.2, “Implications for distributions”