We give a dichotomy theorem for the problem of counting homomorphisms to
directed acyclic graphs. H is a fixed directed acyclic graph.
The problem is, given an input digraph G, how many homomorphisms are there
from G to H. We give a graph-theoretic classification, showing that
for some digraphs H, the problem is in P and for the rest of the digraphs H
the problem is \#P-complete. An interesting feature of the dichotomy, which is absent
from related dichotomy results, is
that
there is a rich supply of tractable graphs~H with complex structure.