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 $\operatorname{PSET}$.



Representing PSET as species
See Gilbert Labelle, *[On asymmetric structures](https://doi.org/10.1016/0012-365X(92)90371-L)*, Discrete Math. 99 (1992), 141–164.


