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Báo cáo toán học: "DETERMINANT IDENTITIES AND A GENERALIZATION OF THE NUMBER OF TOTALLY SYMMETRIC SELF-COMPLEMENTARY PLANE PARTITIONS"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: DETERMINANT IDENTITIES AND A GENERALIZATION OF THE NUMBER OF TOTALLY SYMMETRIC SELF-COMPLEMENTARY PLANE PARTITIONS. | DETERMINANT IDENTITIES AND A GENERALIZATION OF THE NUMBER OF TOTALLY SYMMETRIC SELF-COMPLEMENTARY PLANE PARTITIONS C. Krattenthaler Institut fur Mathematik der Universitat Wien Strudlhofgasse 4 A-1090 Wien Austria. e-mail KRATT@Pap.Univie.Ac.At WWW http radon.mat.univie.ac.at People kratt Submitted September 16 1997 Accepted November 3 1997 Abstract. We prove a constant term conjecture of Robbins and Zeilberger J. Combin. Theory Ser. A 66 1994 17-27 by translating the problem into a determinant evaluation problem and evaluating the determinant. This determinant generalizes the determinant that gives the number of all totally symmetric self-complementary plane partitions contained in a 2n X 2n X 2n box and that was used by Andrews J. Combin. Theory Ser. A 66 1994 28-39 and Andrews and Burge Pacific J. Math. 158 1993 1-14 to compute this number explicitly. The evaluation of the generalized determinant is independent of Andrews and Burge s computations and therefore in particular constitutes a new solution to this famous enumeration problem. We also evaluate a related determinant thus generalizing another determinant identity of Andrews and Burge loc. cit. . By translating some of our determinant identities into constant term identities we obtain several new constant term identities. 1. Introduction. I started work on this paper originally hoping to find a proof of the following conjecture of Robbins and Zeilberger 16 Conjecture C B caution in the quotient defining B it should read m 1 2j instead of m 1 j which we state in an equivalent form. 1991 Mathematics Subject Classification. Primary 05A15 15A15 Secondary 05A17 33C20. Key words and phrases. determinant evaluations constant term identities totally symmetric self-complementary plane partitions hypergeometric series. 1 Supported in part by EC s Human Capital and Mobility Program grant CHRX-CT93-0400 and the Austrian Science Foundation FWF grant P10191-MAT Typeset by A A-T E-X 1 THE ELECTRONIC JOURNAL OF .