The University of Michigan Combinatorics Seminar
Fall 2004
October 1, 4:10-5:00, 3866 East Hall



Canonical characters on quasisymmetric functions
and bivariate Catalan numbers

Sam Hsiao

University of Michigan


Abstract

Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character.

The aim of this talk is to give explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasisymmetric functions. They have simple descriptions 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.

The necessary background on quasisymmetric functions and Hopf algebras will be reviewed during the talk.

This is joint work with Marcelo Aguiar.