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

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

2002-02-08

arxiv.org/abs/math/0202071v1

Adv. Math. 181 (2004), no. 2, 353--367.

Mathematics

Combinatorics

LaTeX, 3 figures, 12 pages

Scientific paper

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.

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