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Báo cáo toán học: "Enumeration of alternating sign matrices of even size (quasi-)invariant under a quarter-turn rotation"

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: Enumeration of alternating sign matrices of even size (quasi-)invariant under a quarter-turn rotation. | Enumeration of alternating sign matrices of even size quasi- invariant under a quarter-turn rotation Jean-Christophe Aval Philippe Duchon LaBRI Universite Bordeaux 1 CNRS 351 cours de la Liberation 33405 Talence cedex FRANCE aval duchon @labri.fr Submitted Oct 16 2009 Accepted Mar 19 2010 Published Mar 29 2010 Mathematics Subject Classification 05A15 05A99 Abstract The aim of this work is to enumerate alternating sign matrices ASM that are quasi-invariant under a quarter-turn. The enumeration formula conjectured by Duchon involves as a product of three terms the number of unrestricted ASM s and the number of half-turn symmetric ASM s. 1 Introduction An alternating sign matrix is a square matrix with entries in -1 0 1 and such that in any row and column the non-zero entries alternate in sign and their sum is equal to 1. Their enumeration formula was conjectured by Mills Robbins and Rumsey 8 and proved years later by Zeilberger 16 and almost simultaneously by Kuperberg 6 . Kuperberg s proof is based on the study of the partition function of a square ice model whose states are in bijection with ASM s. Kuperberg was able to get an explicit formula for the partition function for some special values of the spectral parameter. To do this he used a determinant representation for the partition function that was obtained by Izergin 4 . Izergin s proof is based on the Yang-Baxter equations and on recursive relations discovered by Korepin 5 . This method is more flexible than Zeilberger s original proof and Kuperberg also used it in 7 to obtain many enumeration or equinumeration results for various symmetry classes of ASM s most of them having been conjectured by Robbins 13 . Among these results can be found the following remarkable one. Both authors are supported by the ANR project MARS BLAN06-2-0193 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R51 1 Theorem 1 Kuperberg . The number AqT 4N of ASM s of size 4N invariant under a quarter-turn QTASM s is related to the number