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There is a natural [[partial order]] on partitions given by inclusion of Young diagrams. This partially ordered set is known as ''[[Young's lattice]]''. The lattice was originally defined in the context of [[representation theory]], where it is used to describe the [[irreducible representation]]s of [[symmetric group]]s ''S''<sub>''n''</sub> for all ''n'', together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a [[differential poset]].
== Random
There is a deep theory of random partitions chosen according to the uniform probability distribution on the [[symmetric group]] via the [[Robinson–Schensted correspondence]]. In 1977, Logan and Shepp, as well as Vershik and Kerov, showed that the Young diagram of a typical large partition becomes asympototically close to the graph of a certain analytic function minimizing a certain functional. In 1988, Baik, Deift and Johansson extended these results to determine the distribution of the longest increasing subsequence of a random permutation in terms of the [[Tracy–Widom distribution]].<ref>{{Cite book |last=Romik |first=Dan |title=The surprising mathematics of longest increasing subsequences |date=2015 |publisher=Cambridge University Press |isbn=978-1-107-42882-9 |series=Institute of Mathematical Statistics Textbooks |___location=New York}}</ref> [[Andrei Okounkov|Okounkov]] related these results to the combinatorics of [[Riemann surface
== See also ==
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