It is well known that exponentiating the EGF(exponential generating function) for cycles gives the EGF for permutations: link here. This is something summarized by the slogan all = exp(connected).
I wonder if it is possible to give a lie-theoretic explanation to this phenomenon where we have group = exp(algebra).
Is there some way to relate the "counting" done by the exponential generating function to an actual exponential between the Lie group and its algebra? Perhaps there is some way to use the representation theory of $S_n$ to establish some connection? Is this connection one of those near-misses that holds nothing deeper?