On the combinatorics of $k$-Naples parking functions and parking strategies

We propose a characterization of $k$-Naples parking functions in terms of subsequences with the structure of a complete $k$-Naples parking function. We define complete parking preferences by requiring that for all $j=2,\dots,n$, the number of cars having preference at least $j$ is strictly greater than the number of spots in $[j,n]$. We also provide a characterization of permutation invariant $k$-Naples parking functions. Finally, we introduce a generalization of the parking problem where each car is given its own parking rule, by defining parking strategies as vectors of rules that allow all cars to park. Given a parking preference, we also investigate ways to find parking strategies that minimize certain natural parameters, such as the total number of backward steps, or the number of cars that need to drive backwards.