A new generalization of parking functions: ρ-Naples parking functions
Abstract
We introduce a new model of parking functions, called ρ-Naples parking functions, in which each car has a certain number of spots to check preceding its preferred one before looking for available spots by moving forward. This notion, where each car has its own number of previous spots to check, generalizes that of Naples parking function, where this number is the same for all cars. We start by showing that, for a given sequence of preferences for each car, a successful parking strategy (that is, an assignment of how many backward spots each car is allowed to check) remains successful if we increase the number of backwards spots to check for some of the cars. This means that the set of successful strategies for a given parking preference is an up-set in the lattice of parking strategies, which raises the intriguing question of characterizing such up-sets. Though we are not able to completely solve this problem, we provide a characterization of those successful strategies which minimize the total number of allowed backward steps for a given parking preference, and we determine a lower bound for the number of cars which are allowed to move backwards (also showing that it can be attained). To conclude, we analyze the specific situation in which each car can take at most one backward step, and we completely characterize the set of successful parking preferences in this case.
Related
- Characterizing k-Naples parking functions and refining bounds via obstructed parking functions
- Pattern restricted parking functions with limited preferences
- A total variation threshold for the first coordinates of uniform parking functions via Lorentzian polynomials
- Symmetries on vector parking functions via bounded lattice paths
- Real-rootedness and interlacing for parking functions and Chow polynomials
BibTeX
@misc{ferrari2026generalization,
title = {{A new generalization of parking functions: $ρ$-Naples parking functions}},
author = {Luca Ferrari and Francesco Verciani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20267},
eprint = {2610.20267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20267},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A new generalization of parking functions: ρ-Naples parking functions
AU - Luca Ferrari
AU - Francesco Verciani
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20267
UR - https://arxiv.org/abs/2610.20267
AB - We introduce a new model of parking functions, called ρ-Naples parking functions, in which each car has a certain number of spots to check preceding its preferred one before looking for available spots by moving forward. This notion, where each car has its own number of previous spots to check, generalizes that of Naples parking function, where this number is the same for all cars. We start by showing that, for a given sequence of preferences for each car, a successful parking strategy (that is, an assignment of how many backward spots each car is allowed to check) remains successful if we increase the number of backwards spots to check for some of the cars. This means that the set of successful strategies for a given parking preference is an up-set in the lattice of parking strategies, which raises the intriguing question of characterizing such up-sets. Though we are not able to completely solve this problem, we provide a characterization of those successful strategies which minimize the total number of allowed backward steps for a given parking preference, and we determine a lower bound for the number of cars which are allowed to move backwards (also showing that it can be attained). To conclude, we analyze the specific situation in which each car can take at most one backward step, and we completely characterize the set of successful parking preferences in this case.
ER -