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

DETERMINANT IDENTITIES AND A GENERALIZATION OF THE NUMBER OF TOTALLY SYMMETRIC SELF-COMPLEMENTARY PLANE PARTITIONS C Krattenthaler Institut făr Mathematik der Universităt Wien, u a 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) × (2n) × (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 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 + + 2j) instead of (m + + 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 † 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 AMS-TEX the electronic journal of combinatorics (1997), #R27 Conjecture Let x and n be nonnegative integers Then 0≤i