Geodesic
On the coincidence of limit shapes for integer partitions and compositions, and a~slicing of Young diagrams
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 266-280

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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 $\alpha1/2$. 
@article{ZNSL_2004_307_a8,
     author = {Yu. V. Yakubovich},
     title = {On the coincidence of limit shapes for integer partitions and compositions, and a~slicing of {Young} diagrams},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {266--280},
     publisher = {mathdoc},
     volume = {307},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_307_a8/}
}
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Yu. V. Yakubovich. On the coincidence of limit shapes for integer partitions and compositions, and a~slicing of Young diagrams. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 266-280. http://geodesic.mathdoc.fr/item/ZNSL_2004_307_a8/