  We study the complexity of answer counting for ontology-mediated queries and
for querying under constraints, considering conjunctive queries and unions
thereof (UCQs) as the query language and guarded TGDs as the ontology and
constraint language, respectively. Our main result is a classification
according to whether answer counting is fixed-parameter tractable (FPT),
W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent, lifting a
recent classification for UCQs without ontologies and constraints due to Dell
et al. The classification pertains to various structural measures, namely
treewidth, contract treewidth, starsize, and linked matching number.
  Our results rest on the assumption that the arity of relation symbols is
bounded by a constant and, in the case of ontology-mediated querying, that all
symbols from the ontology and query can occur in the data (so-called full data
schema).
  We also study the meta-problems for the mentioned structural measures, that
is, to decide whether a given ontology-mediated query or constraint-query
specification is equivalent to one for which the structural measure is bounded.
