It is well known that exponentiating the EGF(exponential generating function) for cycles gives the EGF for permutations: link here. Usually summarized under the catchy slogan `all = exp(connected)`

.

I wonder if it is possible to give a lie-theoretic explanation to this phenomenon: The similarity to `group = exp(algebra)`

is tantalizing.

Is there some way to relate the counting done by the EGF function to an *actual* exponential between the 'algebra of cycles' and the group $S_n$? Perhaps there is some way to use the representation theory of $S_n$ to establish some connection? Is this one of those near-misses that holds no deep content?