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    Enumeration of Standard Young Tableaux

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    A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction

    Random Sorting Networks

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    A sorting network is a shortest path from 12...n to n...21 in the Cayley graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n->infinity the space-time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Holder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.Comment: 38 pages, 12 figure

    Skew Howe duality and random rectangular Young tableaux

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    We consider the decomposition into irreducible components of the external power Λp(CmCn)\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n) regarded as a GLm×GLn\operatorname{GL}_m\times\operatorname{GL}_n-module. Skew Howe duality implies that the Young diagrams from each pair (λ,μ)(\lambda,\mu) which contributes to this decomposition turn out to be conjugate to each other, i.e.~μ=λ\mu=\lambda'. We show that the Young diagram λ\lambda which corresponds to a randomly selected irreducible component (λ,λ)(\lambda,\lambda') has the same distribution as the Young diagram which consists of the boxes with entries p\leq p of a random Young tableau of rectangular shape with mm rows and nn columns. This observation allows treatment of the asymptotic version of this decomposition in the limit as m,n,pm,n,p\to\infty tend to infinity.Comment: 17 pages. Version 2: change of title, section on bijective proofs improve
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