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...