# 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.

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## Author information

Authors

### Corresponding author

Correspondence to William Craig.