\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
\nopagenumbers
\noindent
%
%
{\bf Marcelo Aguiar and Samuel K. Hsiao}
%
%
\medskip
\noindent
%
%
{\bf Canonical Characters on Quasi-Symmetric Functions and Bivariate Catalan Numbers}
%
%
\vskip 5mm
\noindent
%
%
%
%
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, math.CO/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
{\it bivariate Catalan numbers:}
$$C(m,n)={(2m)!(2n)!\over m!(m+n)!n!}\,.$$ 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.
\bye