  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.
