Partition function (number theory): Difference between revisions

Revision as of 11:41, 24 February 2024

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:<math> p_A(n) = \left(\prod_{a \in A} a^{-1}\right) \cdot \frac{n^{k-1}}{(k-1)!} + O(n^{k-2}) . </math>

=== Random Partitions ===

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 Schepp, as well as Vershik and Kerov, showed that 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 permutiation, 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|Riemann surfaces]] and representation theory.<ref>{{Cite journal |last=Okounkov |first=Andrei |date=2000 |title=Random matrices and random permutations |url=https://academic.oup.com/imrn/article-lookup/doi/10.1155/S1073792800000532 |journal=International Mathematics Research Notices |volume=2000 |issue=20 |pages=1043 |doi=10.1155/S1073792800000532}}</ref><ref>{{Cite journal |last=Okounkov |first=A. |date=2001-04-01 |title=Infinite wedge and random partitions |url=https://doi.org/10.1007/PL00001398 |journal=Selecta Mathematica |language=en |volume=7 |issue=1 |pages=57 |doi=10.1007/PL00001398 |issn=1420-9020}}</ref>

==References==