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Journal Articles Advances in Mathematics Year : 2004

Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n

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. We prove here that the dimension of R_n is given by C_n, the n-th Catalan number. This is also the dimension of the space SH_n of super-covariant polynomials, that is 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.

Dates and versions

hal-00185495 , version 1 (06-11-2007)

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Jean-Christophe Aval, F. Bergeron, N. Bergeron. Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n. Advances in Mathematics, 2004, 181 (2), pp.353-367. ⟨hal-00185495⟩
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