Distributions on partitions arising from Hilbert schemes and hook lengths

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of $n$ points on $\mathbb{C}^2.$ For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty.$ For $t$-hooks, we prove distributions which are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results which are of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\text {and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right).$$