Enumerative counting is hard

An n-variable Boolean formula can have anywhere from 0 to 2ⁿ satisfying assignments. The question of whether a polynomial-time machine, given such a formula, can reduce this exponential number of possibilities to a small number of possibilities is explored. Such a machine is called an enumerator, and it is proved that if there is a good polynomial-time enumerator for #P (i.e. one where the small set has at most O(|f|^1-e) numbers), then P=NP=P^#P and probabilistic polynomial time equals polynomial time. Furthermore, #P and enumerating #P are polynomial-time Turing equivalent