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.
Categories: math.NT math.CO