  We present a framework for the complexity classification of parameterized
counting problems that can be formulated as the summation over the numbers of
homomorphisms from small pattern graphs H_1,...,H_l to a big host graph G with
the restriction that the coefficients correspond to evaluations of the M\"obius
function over the lattice of a graphic matroid. This generalizes the idea of
Curticapean, Dell and Marx [STOC 17] who used a result of Lov\'asz stating that
the number of subgraph embeddings from a graph H to a graph G can be expressed
as such a sum over the lattice of partitions of H. In the first step we
introduce what we call graphically restricted homomorphisms that, inter alia,
generalize subgraph embeddings as well as locally injective homomorphisms. We
provide a complete parameterized complexity dichotomy for counting such
homomorphisms, that is, we identify classes of patterns for which the problem
is fixed-parameter tractable (FPT), including an algorithm, and prove that all
other pattern classes lead to #W[1]-hard problems. The main ingredients of the
proof are the complexity classification of linear combinations of homomorphisms
due to Curticapean, Dell and Marx [STOC 17] as well as a corollary of Rota's
NBC Theorem which states that the sign of the M\"obius function over a
geometric lattice only depends on the rank of its arguments. We use the general
theorem to classify the complexity of counting locally injective homomorphisms
as well as homomorphisms that are injective in the r-neighborhood for constant
r. Furthermore, we show that the former has "real" FPT cases by considering the
subgraph counting problem restricted to trees on both sides. Finally we show
that the dichotomy for counting graphically restricted homomorphisms readily
extends to so-called linear combinations.
