##### Department of Mathematics,

University of California San Diego

****************************

### Math 264 - Combinatorics

## Sam Hsiao

#### University of Michigan

## Canonical characters on quasisymmetric functions and bivariate Catalan numbers

##### Abstract:

In a recent preprint ``Combinatorial Hopf algebras and generalized Dehn-Sommerville relations" \noindent(math.CO/0310016), Aguiar, Bergeron, and Sottile set up a framework for studying combinatorial invariants encoded by quasisymmetric functions. They show that every character (i.e., multiplicative linear functional) on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character, both of which can in theory be computed from the even and odd parts of the ``universal character" on the Hopf algebra of quasisymmetric functions. \vskip .1in \noindent In my talk I will introduce some of these ideas and then go on to give explicit formulas for the even and odd parts of the universal character. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$C(m,n)=(2m)!(2n)!/(m!(m+n)!n!).$$ I will explain how properties of characters and of quasisymmetric functions can be used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients. \vskip .1in \noindent This work is joint with Marcelo Aguiar.

Host: Fan Chung Graham

### November 9, 2004

### 3:00 PM

### AP&M 7321

****************************