  \#P-hardness of computing matrix immanants are proved for each member of a
broad class of shapes and restricted sets of matrices. We prove \#P-hardness of
computing $\lambda$-immanants of $0$-$1$ matrices when $\lambda$ has a large
domino-tilable part and satisfying some technical conditions. We also give
hardness proofs of some $\lambda$-immanants of weighted adjacency matrices of
planarly drawable directed graphs, such that the shape $\lambda =
(\mathbf{1}+\lambda_d)$ has size $n$ such that $|\lambda_d| = n^{\varepsilon}$
for some $0<\varepsilon<\frac{1}{2}$, and for some $w$, the shape
$\lambda_d/(w)$ is tilable with $1\times 2$ dominos.
