Fixed points of parking functions

Abstract

We define an action of words in [ m ] n [m]^n on R m {\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 m and n n are coprime [Trans. Amer. Math. Soc. 368 (2016), pp. 8403–8445], and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington’s sweep map on rational Dyck paths (see D. Armstrong, N. A. Loehr, and G. S. Warrington [Adv. Math. 284 (2015), pp. 159–185; E. Gorsky, M. Mazin, and M. Vazirani [Electron. J. Combin. 24 (2017), p. 29; H. Thomas and N. Williams, Selecta Math. (N.S.) 24 (2018), pp. 2003–2034]).