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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Correspondence to William Craig.

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Communicated by Marni Mishna

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Craig, W., Pun, A. Distribution Properties for t-Hooks in Partitions. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00547-2

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