Thanks for the comments, I refined the question.
What is the complexity relationship between counting and enumeration problems? If a counting problem is #P-complete, it means that the enumeration version is #P-hard.
How about problems that can be solved in polynomial time? For example, do counting triangles and enumerating triangles in a graph have the same complexity lower/upper bound?
Given an algorithm that can solve the counting problem in $O(f(n))$ time, can we also solve the corresponding enumeration problem in $O(f(n))$ time? Assume that the output size of enumeration problem is $O(f(n))$ or somehow can be compressed using $O(f(n))$ storage.
For example, we may use cliques instead of triangles to represent the output results of triangle enumeration to reduce the output size. It may be not an appropriate example since it may not reduce the worst case space complexity. But for some enumeration problems, I think it may be possible.
Could anyone give any explanations or references?