Generalized Pak-Stanley labeling for multigraphical hyperplane arrangements for n=3

Abstract

In 1998 Pak and Stanley defined the original Pak-Stanley labeling as a bijective map from the set of regions of an extended Shi arrangement to the set of parking functions. This map was later generalized to other arrangements: Sam Hopkins and David Perkinson considered Pak-Stanley labeling on bigraphical arrangements, and Mikhail Mazin generalized the labeling to arrangements associated with directed multigraphs. In this generalized setting the labeling always provides a surjective map from the set of regions of the arrangement to the set of graphical parking functions. However, this map often failed to be injective. This lead to a natural question, what graphs admit arrangements with a bijective labeling? In this paper we present a necessary, but not sufficient, condition for the injectivity of the generalized Pal-Stanley labeling. Moreover, for n=3 we show that even if an arrangement has duplicate labels, then the closure of the union of regions with the duplicate label is connected. Lastly, we present ways to construct bijective arrangements for several families of graphs in n=3, and present examples showing that the conditions are not sufficient.