A new approach to Naples parking functions through complete parking preferences, and its enumerative consequences

Naples parking functions were introduced as a generalization of classical parking functions, in which cars are allowed to park backwards, by checking up to a fixed number of previous spots, before proceeding forward as usual. In this work we introduce the notion of a complete parking preference, through which we are able to give some information on the combinatorics of Naples parking functions. Roughly speaking, a complete parking preference is a parking preference such that, for any index j, there are more cars with preference at least j than spots available from j onward. We provide a characterization of Naples parking functions in terms of certain complete subsequences of them. As a consequence of this result we derive a characterization of permutation-invariant Naples parking functions which turns out to be equivalent to the one given by (Carvalho et al., 2021), but using a totally different approach (and language). Concerning enumeration, we propose an effective approach to enumerate per-mutation-invariant Naples parking functions and complete Naples parking functions which is based on some natural combinatorial decompositions. We thus obtain for-mulas depending on some (generally simpler) quantities, which are of interest in their own right, and that can be described in a recursive fashion.