Preference-restricted parking functions

TL;DR

This paper introduces preference-restricted parking functions, a refined concept that restricts the parking preferences to subsets, leading to new combinatorial insights and interpretations of parking procedures, including prime functions and multi-spot parking.

Contribution

It defines preference-restricted parking functions and explores their combinatorial properties, providing new interpretations and results for various parking scenarios.

Findings

New combinatorial interpretations of parking functions

Reproving Abel's binomial theorem using restricted parking functions

Analysis of parking functions with fewer spots and multiple cars per spot

Abstract

A parking function is a function $\pi:[n]\to [n]$ whose $i$th-smallest output is at most $i,$ corresponding to a parking procedure for $n$ cars on a one-way street. We refine this concept by introducing preference-restricted parking functions, which are parking functions with codomain restricted to some $S\subseteq[n]$. Particular choices of $S$ yield new combinatorial interpretations of previous results about variant parking procedures, and new results too. In particular we consider prime parking functions, parking procedures with fewer spots than cars, and parking functions where each spot has space for multiple cars. We also use restricted parking functions to reprove Abel's binomial theorem.