Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for Sn
Abstract
The 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. We prove here that the dimension of Rn is given by Cn, the n-th Catalan number. This is also the dimension of the space SHn of super-covariant polynomials, that is 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.
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