  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.
