  We study descriptive complexity of counting complexity classes in the range
from #P to #$\cdot$NP. A corollary of Fagin's characterization of NP by
existential second-order logic is that #P can be logically described as the
class of functions counting satisfying assignments to free relation variables
in first-order formulae. In this paper we extend this study to classes beyond
#P and extensions of first-order logic with team semantics. These team-based
logics are closely related to existential second-order logic and its fragments,
hence our results also shed light on the complexity of counting for extensions
of FO in Tarski's semantics. Our results show that the class #$\cdot$NP can be
logically characterized by independence logic and existential second-order
logic, whereas dependence logic and inclusion logic give rise to subclasses of
#$\cdot$NP and #P , respectively. Our main technical result shows that the
problem of counting satisfying assignments for monotone Boolean
$\Sigma_1$-formulae is #$\cdot$NP-complete as well as complete for the function
class generated by dependence logic.
