Parking Completions and Volumes of Polytopes

Kimberly Hadaway
Graduate Student
Iowa State University

Parking functions correspond with preferences of n cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. We generalize parking functions to parking completions: Here, we are given that some cars have already parked in a set of spots, which are indexed in a vector t. We then consider a preference vector c, where len(t)+len(c)=n. If all cars can park, we say that c is a parking completion. Adeniran, et al. (2020) state an open problem which connects the number of parking completions to the volumes of Pitman-Stanley polytopes by explicit computation on small values of n. In this talk, we provide a partial solution to this open problem by exploring edge cases.