2023/05/05 by Snider, Lauren, Yan, Catherine
#05C30 #05C57 #05E18 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2305.03651
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 \boldsymbolu-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 \mathfrakSn, 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 \boldsymbolU-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 (\mathfrakSp × \mathfrakSq)-invariant, where p+q+1 is the order of G.