## More on Catalan and Quasi-symmetric Functions

**Nantel Bergeron**

(York University)

Last September we have discussed:

Catalan number that classically enumerate Dyck paths,

and investigate the quotient ring $R_n$ of the ring of polynomials

$\Q[x_1,x_2,\ldots,x_n]$ over the the ideal generated by non-constant

quasi-symmetric polynomials.

We expected the dimension of $R_n$ to be the $n$th Catalan number.

Now we can prove it all, and say even more...