  A lattice diagram is a finite set $L=\{(p_1,q_1),... ,(p_n,q_n)\}$ of lattice
cells in the positive quadrant. The corresponding lattice diagram determinant
is $\Delta_L(\X;\Y)=\det \| x_i^{p_j}y_i^{q_j} \|$. The space $M_L$ is the
space spanned by all partial derivatives of $\Delta_L(\X;\Y)$. We denote by
$M_L^0$ the $Y$-free component of $M_L$. For $\mu$ a partition of $n+1$, we
denote by $\mu/ij$ the diagram obtained by removing the cell $(i,j)$ from the
Ferrers diagram of $\mu$. Using homogeneous partially symmetric polynomials, we
give here a dual description of the vanishing ideal of the space $M_\mu^0$ and
we give the first known description of the vanishing ideal of $M_{\mu/ij}^0$.
