Ideals of quasi-symmetric functions and super-covariant polynomials for [formula omitted

Abstract

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 SH n 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 SH n in terms of number of Dyck paths with a given number of factors.