Complexity of counting output patterns of logic circuits

Let C be a logic circuit consisting of s gates g1, g2, gs, then the output pattern of C for an input x ε {0, 1}ⁿ is defined to be a vector (g1(x), g2(x), gs(x)) ε {0, 1}s of the outputs of g1, g2, gs for x. For each f : {0, 1}² → {0, 1}, we define an f-circuit as a logic circuit where every gate computes f, and investigate computational complexity of the following counting problem: Given an f-circuit C, how many output patterns arise in C? We then provide a dichotomy result on the counting problem: We prove that the problem is solvable in polynomial time if f is PARITY or any degenerate function, while the problem is #P-complete even for constant-depth f-circuits if f is one of the other functions, such as AND, OR, NAND and NOR.