  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 $$ F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \
F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\text {and}}\
\ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). $$
