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

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