($\mathfrak{S}_p \times \mathfrak{S}_q$)-Invariant Graphical Parking Functions

Graphical parking functions, or $G$-parking functions, are a generalization of classical parking functions which depend on a connected multigraph $G$ having a distinguished root vertex. Gaydarov and Hopkins characterized the relationship between $G$-parking functions and another vector-dependent generalization of parking functions, the $\boldsymbol{u}$-parking functions. The crucial component of their result was their classification of all graphs $G$ whose $G$-parking functions are invariant under action by the symmetric group $\mathfrak{S}_n$, where $n+1$ is the order of $G$. In this work, we present a 2-dimensional analogue of Gaydarov and Hopkins' results by characterizing the overlap between $G$-parking functions and 2-dimensional $\boldsymbol{U}$-parking functions, i.e., pairs of integer sequences whose order statistics are bounded by certain weights along lattice paths in the plane. Our key result is a total classification of all $G$ whose set of $G$-parking functions is $(\mathfrak{S}_p \times \mathfrak{S}_q)$-invariant, where $p+q+1$ is the order of $G$.