arXiv 2006.134462020-06math.NTmath.CO

划分中T-钩的分布性质

Distribution properties for t-hooks in partitions

William Craig, Anna Pun

发表信息：: Annals of Combinatorics, Vol. 25 Issue 3, 2021

摘要：

配分、配分函数$p(n)$和它们的费勒斯-杨图的钩长是组合数学、数论和表示论中的重要对象。对于正整数$n$和$t$，我们研究了$p_t^e(n)$（分别为. $p_t^o(n)$）、具有偶数的$n$的分区的数量（分别为奇数）$t$ -钩子的数量。我们研究的限制行为的比例$p_t^e(n)/p(n)$，这也给$p_t^o(n)/p(n)$，因为$p_t^e(n) + p_t^0(n) = p(n)$。对于偶数$t$，我们证明了$$\lim\limits_{n \to \infty} \dfrac{p_t^e(n)}{p(n)} = \dfrac{1}{2},$$，对于奇数$t$，我们建立了非均匀分布$$\lim\limits_{n \to \infty} \dfrac{p^e_t(n)}{p(n)} = \begin{cases} \dfrac{1}{2} + \dfrac{1}{2^{(t+1)/2}} & \text{if } 2 \mid n, \\ \\ \dfrac{1}{2} - \dfrac{1}{2^{(t+1)/2}} & \text{otherwise.} \end{cases}$$。使用Rademacher圆方法，我们找到了$p_t^e(n)$和$p_t^o(n)$的精确公式，并且这个精确公式产生了大$n$的这些分布特性。我们还表明，足够大的$n$，$p_t^e(n) - p_t^o(n)$的迹象是周期性的。

英文摘要：

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 $p_t^e(n)/p(n)$, which also gives $p_t^o(n)/p(n)$ since $p_t^e(n) + p_t^0(n) = p(n)$. For even $t$, we show that $$\lim\limits_{n \to \infty} \dfrac{p_t^e(n)}{p(n)} = \dfrac{1}{2},$$ and for odd $t$ we establish the non-uniform distribution $$\lim\limits_{n \to \infty} \dfrac{p^e_t(n)}{p(n)} = \begin{cases} \dfrac{1}{2} + \dfrac{1}{2^{(t+1)/2}} & \text{if } 2 \mid n, \\ \\ \dfrac{1}{2} - \dfrac{1}{2^{(t+1)/2}} & \text{otherwise.} \end{cases}$$ Using the Rademacher circle method, we find an exact formula for $p_t^e(n)$ 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.

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