Parking functions and Łukasiewicz paths

Abstract

We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and Łukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the corresponding Łukasiewicz path, and the total displacement to the area of the path. We also study this bijection restricted to two specific families of parking-functions: unit-interval parking functions, and prime parking functions.

11 pages, 5 figures. Comments welcome!