Jean-Christophe Aval

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

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.

Catalan paths and quasi-symmetric functions

We investigate the quotient ring

Monomial bases related to the n ! conjecture

R

Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

of the ring of formal power series

Dyck tableaux

\Q[[x_1,x_2,...]]

Vanishing Ideals of Lattice Diagram Determinants

over the closure of the ideal generated by non-constant quasi-\break symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from

Combinatorics of labelled parallelogram polyominoes

(0,0)

The symmetry of the partition function of some square ice models

and above the line

Spaces of lattice diagram polynomials in one set of variables

y=x-k

On Sets Represented by Partitions

. We investigate as well the quotient ring