Title: Distribution of hooks in self-conjugate partitions Show Chinese title

Title: 自共轭分拆中的钩子分布

Abstract: We confirm the speculation that the distribution of $t$-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length $t$ among the size $n$ self-conjugate partitions is asymptotically normally distributed with mean $\mu_t(n) \sim \frac{\sqrt{6n}}{\pi} + \frac{3}{\pi^2} - \frac{t}{2}+\frac{\delta_t}{4}$ and variance $\sigma_t^2(n) \sim \frac{(\pi^2 - 6) \sqrt{6n}}{\pi^3},$ where $\delta_t:=1$ if $t$ is odd, and is 0 otherwise. Show Chinese abstract

Abstract: 我们确认了关于无限制整数分拆中$t$-钩子分布的猜测，这本质上退化为自共轭分拆。 即，我们证明了在大小为$n$的自共轭分拆中，长度为$t$的钩子数量渐近服从均值为$\mu_t(n) \sim \frac{\sqrt{6n}}{\pi} + \frac{3}{\pi^2} - \frac{t}{2}+\frac{\delta_t}{4}$且方差为$\sigma_t^2(n) \sim \frac{(\pi^2 - 6) \sqrt{6n}}{\pi^3},$的正态分布，其中当$t$为奇数时$\delta_t:=1$为某个值，否则为 0。