 
Summary: CANONICAL CHARACTERS ON QUASISYMMETRIC FUNCTIONS
AND BIVARIATE CATALAN NUMBERS
MARCELO AGUIAR AND SAMUEL K. HSIAO
Abstract. Every character on a graded connected Hopf algebra decomposes uniquely as
a product of an even character and an odd character [2]. We obtain explicit formulas for
the even and odd parts of the universal character on the Hopf algebra of quasisymmetric
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) =
(2m)!(2n)!
m!(m + n)!n!
.
Properties of characters and of quasisymmetric functions are then used to derive several
interesting identities among bivariate Catalan numbers and in particular among Catalan
numbers and central binomial coefficients.
Contents
1. Introduction 2
2. Even and odd characters 4
3. The canonical characters of QSym on the monomial basis 6
4. Application: Identities for Catalan numbers and central binomial coefficients 9
