  Partitions, the partition function $p(n)$, and the hook lengths of their
Ferrers-Young diagrams are important objects in combinatorics, number theory
and representation theory. For positive integers $n$ and $t$, we study
$p_t^e(n)$ (resp. $p_t^o(n)$), the number of partitions of $n$ with an even
(resp. odd) number of $t$-hooks. We study the limiting behavior of the ratio
$p_t^e(n)/p(n)$, which also gives $p_t^o(n)/p(n)$ since $p_t^e(n) + p_t^0(n) =
p(n)$. For even $t$, we show that $$\lim\limits_{n \to \infty}
\dfrac{p_t^e(n)}{p(n)} = \dfrac{1}{2},$$ and for odd $t$ we establish the
non-uniform distribution $$\lim\limits_{n \to \infty} \dfrac{p^e_t(n)}{p(n)} =
\begin{cases} \dfrac{1}{2} + \dfrac{1}{2^{(t+1)/2}} & \text{if } 2 \mid n, \\
\\ \dfrac{1}{2} - \dfrac{1}{2^{(t+1)/2}} & \text{otherwise.} \end{cases}$$
Using the Rademacher circle method, we find an exact formula for $p_t^e(n)$ and
$p_t^o(n)$, and this exact formula yields these distribution properties for
large $n$. We also show that for sufficiently large $n$, the signs of $p_t^e(n)
- p_t^o(n)$ are periodic.
