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

In this article we study a construction, due to Pak and Stanley, with which every region RR of the Shi arrangement is (bijectively) labelled with a parking function λ(R)λ(R). In particular, we construct an algorithm that returns RR out of λ(R)λ(R). This is done by relating λλ to another bijection, that labels every region SS of the braid arrangement with r(S)r(S), the unique central parking function ff such that λ−1(f)⊆Sλ−1(f)⊆S. We also prove that λλ 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.