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 $\lambda $ be a partition selected uniformly at random from among all such $p(n)$ partitions. It is well known that each partition $\lambda $ 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 $\lambda $. For large $n$, we study the asymptotic behavior of the hook length $Z_n=Z_n(\lambda ,c)$ of the cell $c$ of a random partition $\lambda $. This two-step sampling procedure suggests a product probability measure, which assigns the probability $1/np(n)$ to each pair $(\lambda ,c)$. With respect to this probability measure, we show that the random variable $\pi Z_n/\sqrt {6n}$ converges weakly, as $n\to \infty $, to a random variable whose probability density function equals $6y/(\pi ^2(e^y-1))$ if $0<y<\infty $, and zero elsewhere. Our method of proof is based on Hayman's saddle point approach for admissible power series.