The braid and the Shi arrangements and the Pak-Stanley labelling

Abstract

In this article we study a construction, due to Pak and Stanley, with which every region R of the Shi arrangement is (bijectively) labelled with a parking function lambda(R). In particular, we construct an algorithm that returns Rout of lambda(R). This is done by relating lambda to another bijection, that labels every region S of the braid arrangement with r(S), the unique central parking function f such that lambda(-1)(f) subset of S. We also prove that lambda maps the bounded regions of the Shi arrangement bijectively onto the prime parking functions. Finally, we introduce a variant (that we call "s-parking") of the parking algorithm that is in the very origin of the term "parking function". S-parking may be efficiently used in the context of our new algorithm, but we show that in some (well defined) cases it may even replace it.