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Báo cáo toán học: "The Generating Function of Ternary Trees and Continued Fractions"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The Generating Function of Ternary Trees and Continued Fractions. | The Generating Function of Ternary Trees and Continued Fractions Ira M. Gessel and Guoce XiT Department of Mathematics Brandeis University Waltham MA 02454-9110 gessel@brandeis.edu Department of Mathematics Brandeis University Waltham MA 02454-9110 guoce.xin@gmail.com Submitted May 4 2005 Accepted Feb 1 2006 Published Jun 12 2006 Mathematics Subject Classification 05A15 05A10 05A17 30B70 33C05 Abstract Michael Somos conjectured a relation between Hankel determinants whose entries 2F 1 T count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss s continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos s Hankel determinants to known determinants and we obtain up to a power of 3 a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof in terms of nonintersecting paths of determinant identities involving the number of ternary trees and more general determinant identities involving the number of r-ary trees. Partially supported by NSF Grant DMS-0200596. iBoth authors wish to thank the Institut Mittag-Leffler and the organizers of the Algebraic Combinatorics program held there in the spring of 2005 Anders Bjorner and Richard Stanley. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R53 1 1 Introduction Let an 2 1 3n i b 1 be the number of ternary trees with n vertices and dehne the Hankel determinants U det ai j ij -1 V det ai j 1 0 i j -1 2 W det a i j 1 0 itj n-i 3 where we take ak to be 0 if k is not an integer. We also interpret determinants of 0 X 0 matrices as 1. The hrst few values of these