Pak–Stanley labeling for central graphical arrangements

The original Pak–Stanley labeling was defined by Pak and Stanley 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 associated with graphs and directed multigraphs. In these more general cases the map is no longer bijective. However, it was shown that it is surjective to the set of the $G$-parking functions, where $G$ is the multigraph associated with the arrangement.

This leads to a natural question: when is the generalized Pak–Stanley map bijective? In this paper we answer this question in the special case of central hyperplane arrangements, i.e. the case when all the hyperplanes of the arrangement pass through a common point.

Keywords

parking functions, hyperplanes

2010 Mathematics Subject Classification

Primary 05A17. Secondary 05C30, 52C35.

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The first-named author is supported by the Simons Foundation Collaboration Grant 524324.

Received 15 May 2020

Accepted 17 November 2020

Published 31 January 2022