Limiting shapes of birth-and-death processes on Young diagrams

We consider a family of birth processes and birth-and-death processes on Young diagrams of integer partitions of n. This family incorporates three famous models from very different fields: Rostʼs totally asymmetric particle model (in discrete time), Simonʼs urban growth model, and Moranʼs infinite alleles model. We study stationary distributions and limit shapes as n tends to infinity, and present a number of results and conjectures.