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

AbstractThe aim of this work is to study the quotient ring Rn of the ring Q[x1,…,xn] over the ideal Jn 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 Rn is given by Cn, the nth Catalan number. This is also the dimension of the space SHn of super-covariant polynomials, defined as the orthogonal complement of Jn with respect to a given scalar product. We construct a basis for Rn 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