Distribution properties for t-hooks in partitions

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 $p_t^e(n)\mathrm{/}p(n)$, which also gives $p_t^o(n)\mathrm{/}p(n)$ since $p_t^e(n) + p_t^0(n) = p(n)$\mathrm{.}Foreven$t$,weshowthat$$\lim\limits_{n \to \infty} \dfrac{p_t^e(n)}{p(n)} = \dfrac{1}{2},and for odd $t$ we establish the non-uniform distribution\lim\limits_{n \to \infty} \dfrac{p^e_t(n)}{p(n)} = \begin{cases} \dfrac{1}{2} + \dfrac{1}{2^{(t+1)/2}} & $\text{if }$ 2 \mid n, \ \ \dfrac{1}{2} - \dfrac{1}{2^{(t+1)/2}} & $\text{otherwise.}$ \end{cases}$$UsingtheRademachercirclemethod,wefindanexactformulafor$p_t^e(n)$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.