Distributions of Hook lengths in integer partitions

Abstract

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 ${\mu }_t(n)\sim \frac{\sqrt{6n}}{\pi }-\frac{t}{2}$ and variance ${\sigma }_t^2(n)\sim \frac{({\pi }^2-6)\sqrt{6n}}{2{\pi }^3}.$ 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 $k=(t-1)/2$ and scale $\theta =\sqrt{2/(t-1)}.$

Minor edits and a new first paragraph on previous related work (added at the suggestion of a referee)