285 research outputs found
Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
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.Comment: LaTeX, 3 figures, 12 page
On certain spaces of lattice diagram polynomials
The aim of this work is to study some lattice diagram determinants
. We recall that denotes the space of all partial
derivatives of . In this paper, we want to study the space
which is defined as the sum of spaces where the lattice
diagrams are obtained by removing cells from a given partition, these
cells being in the ``shadow'' of a given cell in a fixed Ferrers
diagram. We obtain an upper bound for the dimension of the resulting space
, that we conjecture to be optimal. This dimension is a
multiple of and thus we obtain a generalization of the conjecture.
Moreover, these upper bounds associated to nice properties of some special
symmetric differential operators (the ``shift'' operators) allow us to
construct explicit bases in the case of one set of variables, i.e. for the
subspace consisting of elements of 0 -degree
- …