Yep! You might also be interested in my take on a more complete solution here. The main takeaway is that we can get away with considering a bit less structure than a species, namely a groupoid equipped with a “weight function.”

]]>and the species T of transitive actions of π.

An action is definitely nothing more than a disjoint union of transitive ones, so the the exponential formula says A(z) = exp(T(z)). Let’s show that that is precisely the formula you want.

First A(z) is the LHS of your formula: the number of actions of π on a set with n elements is exactly |Hom(π,S_n)|.

Now for the RHS to match, we must show the number t_n of transitive actions on a set with n elements is given by (n-1)! a_n, where a_n is the number of index n subgroups of π.

This is because if X has n elements there is a bijection between pairs (transitive action of π on X, point x in X) and pairs (index n subgroup G of π, bijection between X and π/G). Given a pair of the first kind, you can take the stabilizer of x as the subgroup G in a pair of the second kind, and use the action to get the required bijection. In the opposite direction, the bijection lets you transport the action on π/G to X, and you can pick x to be the image of the coset of the identity.

That bijection shows that n t_n = n! a_n, and so t_n = (n-1)! a_n as desired.

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