论文标题
整数分区中钩长的分布
Distributions of Hook lengths in integer partitions
论文作者
论文摘要
在数学上长度扮演的许多角色的动机，我们研究了$ n $的分区中$ t $ hooks数量的分布。我们证明，限制分布是正常的，平均$μ_t（n）\ sim \ frac {\ sqrt {\ sqrt {6n}}π-\ frac {t} {2} $ and差异$σ_t^2（n）此外，我们证明，$ n $的分区中固定的$ t \ geq 4 $的挂钩长度数的分布收敛到带有参数$ k =（t-1）/2 $和比例$θ= \ sqrt {2/（t-1/（t-t-1）}的gamma分布，
Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $μ_t(n)\sim \frac{\sqrt{6n}}π-\frac{t}{2}$ and variance $σ_t^2(n)\sim \frac{(π^2-6)\sqrt{6n}}{2π^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $θ=\sqrt{2/(t-1)}.$
