Asymptotic Gumbel laws for doubly marked product-type statistics

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

Background

The paper contrasts the proposed normal-distribution class with a second class of statistics whose generating functions contain factors marked simultaneously by the statistic variable and the progression variable. The authors suggest that this different product structure should lead to Gumbel rather than Gaussian limiting behavior.

The claim is explicitly labeled as part of a conjecture and is not proved in the paper; it is intended as a broad direction for future work on distributions of partition statistics.

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. \item A partition statistic with a generating function built from pieces of the shape $ qa; qb _\infty$ and $ zqa;zqb _\infty$ will have an asymptotically Gumbel distribution. \end{enumerate}

— 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”