Distributions of Hook lengths in integer partitions

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 $\mu_t(n)\sim \frac{\sqrt{6n}}{\pi}-\frac{t}{2}$andvariance$\sigma_t^2(n)\sim \frac{(\pi^2-6)\sqrt{6n}}{2\pi^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\ge 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)\mathrm{/}2$ and scale $\theta =\sqrt{2\mathrm{/}(t-1)}\mathrm{.}$