Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Canonical characters on quasi-symmetric functions and bivariate Catalan numbers. | Canonical characters on quasi-symmetric functions and bivariate Catalan numbers Marcelo Aguiar Department of Mathematics Texas A M University College Station TX 77843 USA maguiar@ Samuel K. Hsiao Department of Mathematics University of Michigan Ann Arbor MI 48109 USA shsiao@ Submitted Sep 4 2004 Accepted Dec 31 2004 Published Feb 21 2005 Mathematics Subject Classifications 05A15 05E05 16W30 16W50. Keywords Hopf algebra character quasi-symmetric function central binomial coefficient Catalan number bivariate Catalan number peak of a permutation. Abstract Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character Aguiar Bergeron and Sottile 0310016 . We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre s beta function evaluated at half-integers or in terms of bivariate Catalan numbers C m n m m n n Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients. Work supported in part by NSF grant DMS-0302423 and by the NSF Postdoctoral Research Fellowship. We benefited from discussions with Ira Gessel and from the expertise of Francois Jongmans who generously helped us search the 19th century literature in pursuit of a hard-to-find article by Catalan. We also thank the referees for interesting remarks and suggestions. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2005 R15 1 1 Introduction The numbers z. 1 2m 2n C m n -7 7 m m n n 2m 2n m n 1 appeared in work of Catalan 4 pp. 14-15 5 p. 207 6 Sections CV and CCXIV 7 pp. 110-113 von Szily 25 pp. 89-91 Riordan 19 Chapter 3 Exercise 9 p. 120 and recent work of Gessel 12 . We call them bivariate Catalan numbers. They are integers and except for C 0 0 1 .
