We consider some problems concerning the maximum number of (strong)
dominating sets in a regular graph, and their weighted analogues. Our primary
tool is Shearer's entropy lemma. These techniques extend to a reasonably broad
class of graph parameters enumerating vertex colorings satisfying conditions on
the multiset of colors appearing in (closed) neighborhoods. We also generalize
further to enumeration problems for what we call existence homomorphisms. Here
our results are substantially less complete, though we do solve some natural
problems
