In this post we will give a heuristic derivation of a result of Vershik, describing the shape of a random partition of a large integer . (Vershik’s Russian original is available here; English translation is pay-walled.)

By a partition of , we mean positive integers with . We draw a partition as a collection of boxes: For example, this is :

Suppose we let , select partitions of uniformly at random and rescale the size of the boxes by , so that the diagram of the partition always has area . What is the shape of the most likely diagram?