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Báo cáo toán học: "Crossings and nestings in set partitions of classical types"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Crossings and nestings in set partitions of classical types. | Crossings and nestings in set partitions of classical types Martin Rubey Institut fur Algebra Zahlentheorie und Diskrete Mathematik Leibniz Universitat Hannover mart in.rubey@math.uni-hannover.de http www.iazd.uni-hannover.de rubey Christian Stump Centre de Recherches Mathmatiques Universite de Montreal and Laboratoire de Combinatoire et d Informatique Mathematique Universite du Quebec a Montreal christian.stump@univie.ac.at http homepage.univie.ac.at Christian.stump Submitted Sep 8 2009 Accepted Jun 14 2010 Published Sep 1 2010 Mathematics Subject Classifications 05E15 05A18 Abstract In this article we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types B and C that interchange crossings and nestings which generalize a construction by Kasraoui and Zeng for type A. On the other hand we generalize a bijection to type B and C that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting as given by Chen Deng Du Stanley and Yan for type A. For type D we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R120 1 Contents 1 Set partitions for classical types 3 2 Crossings and nestings in set partitions of type A 4 3 Crossings and nestings in set partitions of type C 6 4 Crossings and nestings in set partitions of type B 8 5 Non-crossing and non-nesting set partitions in type D 10 6 k-crossing and k-nesting set partitions of type C 13 Introduction The lattice of non-crossing set partitions was first considered by Germain Kreweras in 15 . It was later reinterpreted by Paul Edelman Rodica Simion and Daniel Ullman as a well-behaved sub-lattice of the intersection lattice for the hyperplane .