In symbolic method, one often considers two operators on ordinary generating functions, namely
$$ \operatorname{PSET}F(x) = \exp\left(F(x)-\frac{F(x^2)}{2}+\frac{F(x^3)}{3}-\dots\right), $$
and
$$ \operatorname{MSET}F(x) = \exp\left(F(x)+\frac{F(x^2)}{2}+\frac{F(x^3)}{3}+\dots\right). $$
These operators allowed to enumerate sets (for $\operatorname{PSET}$) or multisets (for $\operatorname{MSET}$) constructed of unlabeled $F$-structures.
One may note that $\operatorname{MSET} F(x)$ is, essentially, unlabeled generating functions for the species $E \circ F$, where $E$ is the species of sets and $\circ$ denotes the composition of species. Correspondingly, the relevant exponential generating function would be $e^{F(x)}$.
Is there any meaningful way to interpret $\operatorname{PSET}$ in the terms of labeled species as well? Ideally, to get some formula for the exponential generating function of the labeled version of $\operatorname{PSET}$.
