The Limiting Distribution of the Hook Length of a Randomly Chosen Cell in a Random Young Diagram

Let $$p(n)$$ be the number of all integer partitions of the positive integer n , and let λ be a partition selected uniformly at random from among all such $$p(n)$$ partitions. It is well known that each partition λ has a unique graphical representation composed of n non-overlapping cells in the plane, called a Young diagram. As a second step of our sampling experiment, we select a cell c uniformly at random from among the n cells of the Young diagram of the partition λ . For large n , we study the asymptotic behavior of the hook length Zₙ=Zₙ(λ,c) of the cell c of a random partition λ . This two-step sampling procedure suggests a product probability measure, which assigns the probability $$1/np(n)$$ to each pair (λ,c) . With respect to this probability measure, we show that the random variable π Zₙ/√6n converges weakly, as n→∞ , to a random variable whose probability density function equals 6y/(π²(eʸ-1)) if 0<y<∞ , and zero elsewhere. Our method of proof is based on Hayman’s saddle point approach for admissible power series.