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 t -hooks in the partitions of n n . We prove that the limiting distribution is normal with mean \[ μ t ( n ) ∼ 6 n π − t 2 \mu _t(n)\sim \frac {\sqrt {6n}}{\pi }-\frac {t}{2} \] and variance \[ σ t 2 ( n ) ∼ ( π 2 − 6 ) 6 n 2 π 3 . \sigma _t^2(n)\sim \frac {(\pi ^2-6)\sqrt {6n}}{2\pi ^3}. \] Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed t ≥ 4 t\geq 4 in partitions of n n converge to a shifted Gamma distribution with parameter k = ( t − 1 ) / 2 k=(t-1)/2 and scale θ = 2 / ( t − 1 ) \theta =\sqrt {2/(t-1)} .