Annals of Combinatorics ( IF 0.6 ) Pub Date : 2021-07-03 , DOI: 10.1007/s00026-021-00547-2 William Craig 1 , Anna Pun 1
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.
中文翻译：
分区中 t-Hook 的分布特性
分区、分区函数p ( n ) 和他们的 Ferrers-Young 图的钩子长度是组合学、数论和表示论中的重要对象。为正整数Ñ和吨，我们研究\（P_T ^ \ mathrm {E}（N）\）（相应地，\（P_T ^ \ mathrm {Ó}（N）\） ），的分区的数量Ñ与偶数（或奇数）t钩。我们研究了比率\(p_t^\mathrm{e}(n)/p(n)\)的极限行为，这也给出了\(p_t^\mathrm{o}(n)/p(n)\)，因为\(p_t^\mathrm{e}(n) + p_t^\mathrm{o}(n) = p(n)\)。对于偶数t，我们证明
$$\begin{aligned} \lim \limits _{n \rightarrow \infty } \dfrac{p_t^\mathrm{e}(n)}{p(n)} = \dfrac{1}{2}, \结束{对齐}$$对于奇数t，我们建立非均匀分布
$$\begin{aligned} \lim \limits _{n \rightarrow \infty } \dfrac{p^\mathrm{e}_t(n)}{p(n)} = {\left\{ \begin{array }{ll} \dfrac{1}{2} + \dfrac{1}{2^{(t+1)/2}} &{} \text {if } 2 \mid n, \\ \\ \dfrac {1}{2} - \dfrac{1}{2^{(t+1)/2}} &{} \text {otherwise.} \end{array}\right. } \end{对齐}$$使用 Rademacher 圆法，我们找到了\(p_t^\mathrm{e}(n)\)和\(p_t^\mathrm{o}(n)\) 的精确公式，这个精确公式产生了这些分布特性对于大n。我们还表明，对于足够大的n，\(p_t^\mathrm{e}(n) - p_t^\mathrm{o}(n)\) 的符号是周期性的。
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