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: On Zeilberger’s Constant Term for Andrews’ TSSCPP Theorem. | On Zeilberger s Constant Term for Andrews TSSCPP Theorem Guoce Xin Department of Mathematics Capital Normal University Beijing 100048 PR China Submitted Aug 17 2010 Accepted Jun 1 2011 Published Jun 11 2011 Mathematics Subject Classifications 05A15 05A19 Dedicated to Doron Zeilberger on the occasion of his 60th birthday Abstract This paper studies Zeilberger s two prized constant term identities. For one of the identities Zeilberger asked for a simple proof that may give rise to a simple proof of Andrews theorem for the number of totally symmetric self complementary plane partitions. We obtain an identity reducing a constant term in 2k variables to a constant term in k variables. As applications Zeilberger s constant terms are converted to single determinants. The result extends for two classes of matrices the sum of all of whose full rank minors is converted to a single determinant. One of the prized constant term problems is solved and we give a seemingly new approach to Macdonald s constant term for root system of type BC. 1 Introduction In 1986 6 Mills Robbins and Rumsey defined a class of objects called totally symmetric self complementary plane partitions denoted TSSCPP for short and conjectured that the number tn of TSSCPPs of order n is given by t A. FT 3i 1 1 tn A ý n i 1 The author would like to thank Doron Zeilberger for suggesting this subject and thank the referee for valuable suggestions improving this exposition. Part of this work was done during the author s stay at the Center for Combinatorics Nankai University. This work was supported by the Natural Science Foundation of China. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P11 1 which also counts the number of alternating sign matrices a famous combinatorial struc- ture of order n. In 1994 Andrews 1 proved the conjecture by using Stembridge s Pfaffian representation 8 derived from Doran s combinatorial characterization 2 of tn. At the same time Zeilberger suggested a .
