### arXiv:1901.02906 [math.CO]AbstractReferencesReviewsResources

#### Fixed Points of Parking Functions

Jon McCammond, Hugh Thomas, Nathan Williams

Published 2019-01-09Version 1

We define an action of words in $[m]^n$ on $\mathbb{R}^m$ to give a new characterization of rational parking functions---they are exactly those words whose action has a fixed point. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani's zeta map on rational parking functions when m and n are coprime, and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington's sweep map on rational Dyck paths.

**Comments:**37 pages

**Categories:**math.CO

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