Fixed points of parking functions
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- by Jon McCammond, Hugh Thomas and Nathan Williams;
- Trans. Amer. Math. Soc. 377 (2024), 1807-1849
- DOI: https://doi.org/10.1090/tran/8994
- Published electronically: January 10, 2024
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Abstract:
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 [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]).References
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Bibliographic Information
- Jon McCammond
- Affiliation: University of California, Santa Barbara
- MR Author ID: 311045
- ORCID: 0000-0001-5969-7138
- Email: jon.mccammond@math.ucsb.edu
- Hugh Thomas
- Affiliation: LaCIM, Université du Québec à Montréal, Canada
- MR Author ID: 649257
- ORCID: 0000-0003-1177-9972
- Email: hugh.ross.thomas@gmail.com
- Nathan Williams
- Affiliation: University of Texas at Dallas
- MR Author ID: 986170
- ORCID: 0000-0003-2084-6428
- Email: nathan.f.williams@gmail.com
- Received by editor(s): February 13, 2020
- Received by editor(s) in revised form: May 13, 2023
- Published electronically: January 10, 2024
- Additional Notes: The second author was partially supported by the Canada Research Chair grant CRC-2014-00042 and NSERC Discovery Grant RGPIN-2016-04872. This research was supported in part by the National Science Foundation under Grant No. NSF PHY17-48958. The third author was partially supported by Simons Foundation grant 585380 and NSF grant 2246877.
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 1807-1849
- MSC (2020): Primary 05A19, 55M20, 05E10, 05A05
- DOI: https://doi.org/10.1090/tran/8994
- MathSciNet review: 4744742