Fixed Points of Parking Functions

2019/01/09 by McCammond, Jon, Thomas, Hugh, Williams, Nathan
#05A05 #05A19 #05E10 #55M20 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.02906

Abstract

We define an action of words in [m]ⁿ on 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.

Related

• Rational parking functions and (m, n)-invariant sets
• Fixed points and cycles of parking functions
• Type C parking functions and a zeta map
• On vector parking functions and q-analogue
• A Generalization of Parking Functions Allowing Backward Movement