On certain spaces of lattice diagram determinants

The aim of this work is to study some lattice diagram polynomials $\Delta_D(X,Y)$. We recall that $M_D$ denotes the space of all partial derivatives of $\Delta_D$. 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.