Distribution properties for t-hooks in partitions

Abstract

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 $$\underset{n\to \infty }{\lim }\frac{p_t^e(n)}{p(n)}=\frac{1}{2},$$ and for odd $t$ we establish the non-uniform distribution $$\underset{n\to \infty }{\lim }\frac{p_t^e(n)}{p(n)}=\{\begin{array}{ll}\frac{1}{2}+\frac{1}{2^{(t+1)/2}}& \text{if }2\mid n,\\ & \\ \frac{1}{2}-\frac{1}{2^{(t+1)/2}}& \text{otherwise.}\end{array}$$ 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.