Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
Combinatorics
2016-11-08 v1
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.
Keywords
Cite
@article{arxiv.math/0202071,
title = {Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n},
author = {J. -C. Aval and F. Bergeron and N. Bergeron},
journal= {arXiv preprint arXiv:math/0202071},
year = {2016}
}
Comments
LaTeX, 3 figures, 12 pages