Ideals of quasi-symmetric functions and super-covariant polynomials for Sn

The aim of this work is to study the quotient ring R-n of the ring Q[x(1), . . . x(n)] over the ideal J(n) generated by non-constant homogeneous quasi-symmetric functions. This article is a sequel of Aval and Bergeron (Proc. Amer. Math. Soc., to appear), in which we investigated the case of infinitely many variables. We prove here that the dimension of R-n is given by C-n, the nth Catalan number. This is also the dimension of the space SHn of super-covariant polynomials, defined as the orthogonal complement of J(n) with respect to a given scalar product. We construct a basis for R-n whose elements are naturally indexed by Dyck paths. This allows us to understand the Hilbert series of SHn in terms of number of Dyck paths with a given number of factors. (C) 2003 Elsevier Science (USA). All rights reserved.