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There has been a great effort in giving machine-independent, algebraic characterizations of complexity classes, especially of functions. Astonishingly, no satisfactory characterization of the prominent class #P is known up to now. Here, we characterize #P as the closure of a set of simple arithmetical functions under summation and weak product. Based on that result, the hierarchy of counting functions, which is the closure of #P under substitution, is characterized, remarkably without using thedoi:10.1016/0304-3975(95)00237-5 fatcat:3esk7336bbh3rpznmclz6enr7a