1,117 research outputs found

    History of Catalan numbers

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    We give a brief history of Catalan numbers, from their first discovery in the 18th century to modern times. This note will appear as an appendix in Richard Stanley's forthcoming book on Catalan numbers.Comment: 10 page

    The area of cyclic polygons: Recent progress on Robbins' Conjectures

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    In his works [R1,R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments.Comment: To appear in Advances Applied Math. (special issue in memory of David Robbins

    A short proof of rigidity of convex polytopes

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    We present a much simplified proof of Dehn's theorem on the infinitesimal rigidity of convex polytopes. Our approach is based on the ideas of Trushkina and Schramm.Comment: to appear in Siberian Journal of Mathematics; 5 pages 2 figure

    Non-commutative extensions of the MacMahon Master Theorem

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    We present several non-commutative extensions of the MacMahon Master Theorem, further extending the results of Cartier-Foata and Garoufalidis-Le-Zeilberger. The proofs are combinatorial and new even in the classical cases. We also give applications to the β\beta-extension and Krattenthaler-Schlosser's qq-analogue.Comment: 28 pages, 6 figure

    On the complexity of computing Kronecker coefficients

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    We study the complexity of computing Kronecker coefficients g(λ,μ,ν)g(\lambda,\mu,\nu). We give explicit bounds in terms of the number of parts \ell in the partitions, their largest part size NN and the smallest second part MM of the three partitions. When M=O(1)M = O(1), i.e. one of the partitions is hook-like, the bounds are linear in logN\log N, but depend exponentially on \ell. Moreover, similar bounds hold even when M=eO()M=e^{O(\ell)}. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in O(logN)O(\log N) time for a bounded number \ell of parts and without restriction on MM. Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of SnS_n are also considered.Comment: v3: incorporated referee's comments; accepted to Computational Complexit
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