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Summary: CANONICAL CHARACTERS ON QUASI-SYMMETRIC 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 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) =
(2m)!(2n)!
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.
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
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