  We study the problem of counting answers to unions of conjunctive queries
(UCQs) under structural restrictions on the input query. Concretely, given a
class C of UCQs, the problem #UCQ(C) provides as input a UCQ Q in C and a
database D and the problem is to compute the number of answers of Q in D.
  Chen and Mengel [PODS'16] have shown that for any recursively enumerable
class C, the problem #UCQ(C) is either fixed-parameter tractable or hard for
one of the parameterised complexity classes W[1] or #W[1]. However, their
tractability criterion is unwieldy in the sense that, given any concrete class
C of UCQs, it is not easy to determine how hard it is to count answers to
queries in C. Moreover, given a single specific UCQ Q, it is not easy to
determine how hard it is to count answers to Q.
  In this work, we address the question of finding a natural tractability
criterion: The combined conjunctive query of a UCQ $\varphi_1 \vee \dots \vee
\varphi_\ell$ is the conjunctive query $\varphi_1 \wedge \dots \wedge
\varphi_\ell$. We show that under natural closure properties of C, the problem
#UCQ(C) is fixed-parameter tractable if and only if the combined conjunctive
queries of UCQs in C, and their contracts, have bounded treewidth. A contract
of a conjunctive query is an augmented structure, taking into account how the
quantified variables are connected to the free variables. If all variables are
free, then a conjunctive query is equal to its contract; in this special case
the criterion for fixed-parameter tractability of #UCQ(C) thus simplifies to
the combined queries having bounded treewidth.
  Finally, we give evidence that a closure property on C is necessary for
obtaining a natural tractability criterion: We show that even for a single UCQ
Q, the meta problem of deciding whether #UCQ({Q}) can be solved in time
$O(|D|^d)$ is NP-hard for any fixed $d\geq 1$.
