arXiv subjects

Jean-Christophe Aval

Publications and source records attributed to Jean-Christophe Aval.

44 records · Page 3Linked to original sources

arXiv2007

On certain spaces of lattice diagram determinants

The aim of this work is to study some lattice diagram polynomials $Δ_D(X,Y)$. We recall that $M_D$ denotes the space of all partial derivatives of $Δ_D$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is the sum of $M_D$ spaces where the lattice diagrams $D$ are obtained by removing $k$ cells from a given partition, these cells being in the ``shadow'' of a given cell $(i,j)$ of the Ferrers diagram. We obtain an upper bound for the dimension of the resulting space $M^k_{i,j}(X,Y)$, that we conjecture to be optimal. These upper bounds allow us to construct explicit bases for the subspace $M^k_{i,j}(X)$ consisting of elements of 0 $Y$-degree.

math.CO↗

arXiv2007

Ideals and quotients of B-quasisymmetric functions

The space $QSym_n(B)$ of $B$-quasisymmetric polynomials in 2 sets of $n$ variables was recently studied by Baumann and Hohlweg. The aim of this work is a study of the ideal $ $ generated by $B$-quasisymmetric polynomials without constant term. In the case of the space $QSym_n$ of quasisymmetric polynomials in 1 set of $n$ variables, Aval, Bergeron and Bergeron proved that the dimension of the quotient of the space of polynomials by the ideal $ $ is given by Catalan numbers $C_n=\frac 1 {n+1} {2n \choose n}$. In the case of $B$-quasisymmetric polynomials, our main result is that the dimension of the analogous quotient is equal to $\frac{1}{2n+1}{3n\choose n}$, the numbers of ternary trees with $n$ nodes. The construction of a Gröbner basis for the ideal, as well as of a linear basis for the quotient are interpreted by a bijection with lattice paths. These results are finally extended to $p$ sets of variables, and the dimension is in this case $\frac{1}{pn+1}{(p+1)n\choose n}$, the numbers of $p$-ary trees with $n$ nodes.

math.CO↗

arXiv2007

Multivariate Fuss-Catalan numbers

Catalan numbers $C(n)=\frac{1}{n+1}{2n\choose n}$ enumerate binary trees and Dyck paths. The distribution of paths with respect to their number $k$ of factors is given by ballot numbers $B(n,k)=\frac{n-k}{n+k}{n+k\choose n}$. These integers are known to satisfy simple recurrence, which may be visualised in a ``Catalan triangle'', a lower-triangular two-dimensional array. It is surprising that the extension of this construction to 3 dimensions generates integers $B_3(n,k,l)$ that give a 2-parameter distribution of $C_3(n)=\frac 1 {2n+1} {3n\choose n}$, which may be called order-3 Fuss-Catalan numbers, and enumerate ternary trees. The aim of this paper is a study of these integers $B_3(n,k,l)$. We obtain an explicit formula and a description in terms of trees and paths. Finally, we extend our construction to $p$-dimensional arrays, and in this case we obtain a $(p-1)$-parameter distribution of $C_p(n)=\frac 1 {(p-1)n+1} {pn\choose n}$, the number of $p$-ary trees.

math.CO↗

arXiv2007

Quasi-invariant and super-coinvariant polynomials for the generalized symmetric group

The aim of this work is to extend the study of super-coinvariant polynomials, to the case of the generalized symmetric group $G_{n,m}$, defined as the wreath product $C_m\wr§_n$ of the symmetric group by the cyclic group. We define a quasi-symmetrizing action of $G_{n,m}$ on $\Q[x_1,...,x_n]$, analogous to those defined by Hivert in the case of $§_n$. The polynomials invariant under this action are called quasi-invariant, and we define super-coinvariant polynomials as polynomials orthogonal, with respect to a given scalar product, to the quasi-invariant polynomials with no constant term. Our main result is the description of a Gröbner basis for the ideal generated by quasi-invariant polynomials, from which we dedece that the dimension of the space of super-coinvariant polynomials is equal to $m^n C_n$ where $C_n$ is the $n$-th Catalan number.

math.CO↗

arXiv2007

Polynômes quasi-invariants et super-coinvariants pour le groupe symétrique généralisé

A classical result of Artin states that the ideal generated by symmetric polynomials in $n$ variables is of codimension $n!$. The author, F. Bergeron and N. Bergeron have recently obtained a surprising analogous in the case of quasi-symmetric polynomials. In this case, the ideal is of codimension given by $C_n$, the $n$-th Catalan number. Quasi-symmetric polynomials are the invariants of a certain action of the symmetric group $S_n$ defined by F. Hivert. The aim of this work is to generalize these results to the wreath product $S_n\wr \Z_m$, also known as the generalized symmetric group $G\nm$. We first define a quasi-symmetrizing action of $G\nm$ on $\C[x_1,...,x_n]$, then obtain a description of the invariants and the codimension of the associated ideal, which is $m^n C_n$.

math.CO↗

arXiv2001

Schur Partial Derivative Operators

A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is Δ_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of Δ_L(X;Y). The ``shift operators'', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on lattice diagrams with distinct cells in the positive quadrant.

math.CO↗

arXiv2001

Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces

We investigate the quotient ring $R$ of the ring of formal power series $\Q[[x_1,x_2,...]]$ 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 $(0,0)$ and above the line $y=x-k$. We investigate as well the quotient ring $R_n$ of polynomial ring in $n$ variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of $R_n$ is bounded above by the $n$th Catalan number.

math.CO↗

arXiv2001

Lattice Diagram polynomials in one set of variables

The space $M_{μ/i,j}$ spanned by all partial derivatives of the lattice polynomial $Δ_{μ/i,j}(X;Y)$ is investigated in math.CO/9809126 and many conjectures are given. Here, we prove all these conjectures for the $Y$-free component $M_{μ/i,j}^0$ of $M_{μ/i,j}$. In particular, we give an explicit bases for $M_{μ/i,j}^0$ which allow us to prove directly the central {\sl four term recurrence} for these spaces.

math.CO↗