698 research outputs found

### 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 \Delta_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 \Delta_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.Comment: 8 pages, LaTe

### On certain spaces of lattice diagram polynomials

The aim of this work is to study some lattice diagram determinants
$\Delta_L(X,Y)$. We recall that $M_L$ denotes the space of all partial
derivatives of $\Delta_L$. In this paper, we want to study the space
$M^k_{i,j}(X,Y)$ which is defined as the sum of $M_L$ spaces where the lattice
diagrams $L$ are obtained by removing $k$ cells from a given partition, these
cells being in the ``shadow'' of a given cell $(i,j)$ in a fixed 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. This dimension is a
multiple of $n!$ and thus we obtain a generalization of the $n!$ 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 $M^k_{i,j}(X)$ consisting of elements of 0 $Y$-degree

### 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

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