On the coincidence of limit shapes for integer partitions and compositions, and a slicing of Young diagrams

We consider a 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, $m\sim An^\alpha$, $\alpha\le1/2$, 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 are applied to explain why limit shapes of partitions and compositions coincide in the case $\alpha<1/2$.