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 $p_t^\mathrm{e}(n)$ (resp. $p_t^\mathrm{o}(n)$), the number of partitions of n with an even (resp. odd) number of t-hooks. We study the limiting behavior of the ratio $p_t^\mathrm{e}(n)/p(n)$, which also gives $p_t^\mathrm{o}(n)/p(n)$, since $p_t^\mathrm{e}(n) + p_t^\mathrm{o}(n) = p(n)$. For even t, we show that

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } \dfrac{p_t^\mathrm{e}(n)}{p(n)} = \dfrac{1}{2}, \end{aligned}$$

and for odd t, we establish the non-uniform distribution

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } \dfrac{p^\mathrm{e}_t(n)}{p(n)} = {\left\{ \begin{array}{ll} \dfrac{1}{2} + \dfrac{1}{2^{(t+1)/2}} &{} \text {if } 2 \mid n, \\ \\ \dfrac{1}{2} - \dfrac{1}{2^{(t+1)/2}} &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Using the Rademacher circle method, we find an exact formula for $p_t^\mathrm{e}(n)$ and $p_t^\mathrm{o}(n)$, and this exact formula yields these distribution properties for large n. We also show that for sufficiently large n, the sign of $p_t^\mathrm{e}(n) - p_t^\mathrm{o}(n)$ is periodic.