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 \[ \mu _t(n)\sim \frac {\sqrt {6n}}{\pi }-\frac {t}{2} \] 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\geq 4 in partitions of n converge to a shifted Gamma distribution with parameter k=(t-1)/2 and scale \theta =\sqrt {2/(t-1)}. [ABSTRACT FROM AUTHOR]