Multigraph Hyperplane Arrangements and Parking Functions

Abstract

Pak and Stanley introduced a labeling of the regions of a k-Shi arrangement by k-parking functions and proved its bijectivity. Duval, Klivans, and Martin considered a modification of this construction associated with a graph G. They introduced the G-Shi arrangement and a labeling of its regions by G-parking functions. They conjectured that their labeling is surjective, i.e., that every G-parking function appears as a label of a region of the G-Shi arrangement. Later Hopkins and Perkinson proved this conjecture. In particular, this provided a new proof of the bijectivity of Pak-Stanley labeling in the k = 1 case. We generalize Hopkins-Perkinson’s construction to the case of arrangements associated with oriented multigraphs. In particular, our construction provides a simple straightforward proof of the bijectivity of the original Pak-Stanley labeling for arbitrary k.