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. Using the Rademacher circle method, we find an exact formula for $p_t^e(n)$ and $p_t^o(n)$.
In this talk, we will discuss how we use this exact formula to show the distribution properties of $p_t^e(n)$ and $p_t^o(n)$ which is far from uniform, and the signs of $p_t^e(n) - p_t^o(n)$ for large $n$.