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 t -hooks in the partitions of n n . We prove that the limiting distribution is normal with mean \[ μ t ( n ) ∼ 6 n π − t 2 μ _t(n)~ {√6n}{π }-t/2 \] and variance \[ σ t 2 ( n ) ∼ ( π 2 − 6 ) 6 n 2 π 3 . σ _t^2(n)~ {(π ^2-6)√6n}{2π ^3}. \] Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed t ≥ 4 t≥ 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 ) θ =√2/(t-1) .