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: Counting the number of elements in the ˜ mutation classes of An −quivers. | Counting the number of elements in the mutation classes of An-quivers Janine Bastian Institut fur Algebra Zahlentheorie und Diskrete Mathematik Leibniz Universitat Hannover Welfengarten 1 D-30167 Hannover Germany bastian@ Thomas Prellberg School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS United Kingdom Martin Rubey Institut fuur Algebra Zahlentheorie und Diskrete Mathematik Leibniz Universituat Hannover Welfengarten 1 D-30167 Hannover Germany mart Christian Stump Laboratoire de combinatoire et d informatique mathematique Universite du Quebec a Montreal President-Kennedy Montreal Quebec H2X 3Y7 Canada Submitted Oct 17 2010 Accepted Apr 20 2011 Published Apr 29 2011 Mathematics Subject Classification 05A15 16G20 Abstract In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type An in the derived equivalence classes. In particular we obtain the number of elements in the mutation classes of quivers of type An. As a by-product this provides an alternative proof for the number of quivers mutation equivalent to a quiver of Dynkin type Dn which was first determined by Buan and Torkildsen in 5 . THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R00 1 1 Introduction Quiver mutation is a central element in the recent theory of cluster algebras introduced by Fomin and Zelevinsky in 10 . It is an elementary operation on quivers which generates an equivalence relation. The mutation class of a quiver Q is the class of all quivers which are mutation equivalent to Q. The mutation class of quivers of type An is the class containing all quivers mutation equivalent to a quiver whose underlying graph is the Dynkin diagram of type An shown in Figure 1 a . This mutation class was described by Caldero Chapoton and Schiffler 7 in terms of triangulations. An explicit characterisation of the quivers .
