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

• Main Author: Duarte, Rui
• Publication Date: 2015
• Other Authors: Guedes de Oliveira, António
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10773/15081
• topic: Parking functions; Shi arrangement; Braid arrangement

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). In particular, we construct an algorithm that returns R out of λ(R). This is done by relating λ to another bijection, that labels every region S of the braid arrangement with r(S), the unique central parking function f such that λ−1(f)⊆S. We also prove that λ 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.