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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?

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    $\begingroup$ Some other questions on somewhat similar "exponential formula": mathoverflow.net/questions/272045/… $\endgroup$
    – Alexander Chervov
    Commented Jun 18, 2020 at 11:05
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    $\begingroup$ I think if there is a deep answer to this question, it might involve a combinatorial interpretation of the BCH formula. $\endgroup$
    – Mark Wildon
    Commented Jun 18, 2020 at 11:55

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