  Hausel and Rodriguez-Villegas recently observed that work of G\"ottsche,
combined with a classical result of Erd\H{o}s and Lehner on integer partitions,
implies that the limiting Betti distribution for the Hilbert schemes
$(\mathbb{C}^2)^{[n]}$ on $n$ points, as $n\rightarrow +\infty,$ is a
\textit{Gumbel distribution}. In view of this example, they ask for further
such Betti distributions. We answer this question for the quasihomogeneous
Hilbert schemes $((\mathbb{C}^2)^{[n]})^{T_{\alpha,\beta}}$ that are cut out by
torus actions. We prove that their limiting distributions are also of Gumbel
type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on
these Hilbert schemes with our generalization of the result of Erd\H{o}s and
Lehner, which gives the distribution of the number of parts in partitions that
are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_k(A;n)$ denotes
the number of partitions of $n$ with exactly $k$ parts that are multiples of
$A$, then we obtain the asymptotic
  $$
  p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac
k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac
k2-\frac14}k!A^{k+\frac12}(2\pi)^k}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}},
  $$ a result which is of independent interest.
