Unit-interval parking functions are subset of parking functions in which cars park at most one spot away from their preferred parking spot. In this paper, we characterize unit-interval parking functions by understanding how they decompose into prime parking functions and count unit-interval parking functions when exactly k<n cars do not park in their preference. This count yields an alternate proof of a result of Hadaway and Harris establishing that unit-interval parking functions are enumerated by the Fubini numbers. Then, our main result, establishes that for all integers 0≤k<n, the unit-interval parking functions of length n with displacement k are in bijection with the k-dimensional faces of the permutohedron of order n. We conclude with some consequences of this result.
Cite
@article{arxiv.2305.15554,
title = {Unit-Interval Parking Functions and the Permutohedron},
author = {Lucas Chaves Meyles and Pamela E. Harris and Richter Jordaan and Gordon Rojas Kirby and Sam Sehayek and Ethan Spingarn},
journal= {arXiv preprint arXiv:2305.15554},
year = {2023}
}