On the Coincidence of Limit Shapes for Integer Partitions and Compositions,and a Slicing of Young Diagrams

We consider the slicing of Young diagrams into slices associated with summands that have equal multiplicities. It is shown that for the uniform measure on all partitions of an integer n, as well as for the uniform measure on partitions of an integer n into m summands with m ∼ An^α, α ≤ 1/2, all slices after rescaling concentrate around their limit shapes. The similar problem is solved for compositions of an integer n into m summands. These results explain why the limit shapes of partitions and compositions coincide in the case α < 1/2. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 266–280.