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We study descriptive complexity of counting complexity classes in the range from #P to # • 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 existentialdoi:10.4230/lipics.mfcs.2019.19 dblp:conf/mfcs/HaakKMVY19 fatcat:msle7mcok5cyndnisqgsnzi4ii