Bigraphical Arrangements
arXiv:1212.4398 · doi:10.1090/tran/6341
Abstract
We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement.
Added Remark 19 addressing arbitrary G-parking functions; minor revisions