Parking Functions: Choose Your Own Adventure

Abstract

Warning. The reading of this paper will send you down many winding roads toward new and exciting research topics enumerating generalized parking functions. Buckle up! PrefaceLet n ∈ N := {1, 2, 3, . . .} and consider a parking lot consisting of n consecutive parking spots along a one-way street. Suppose n cars want to park one at a time in the parking lot and each car has a preferred parking spot. Each car coming into the lot initially tries to park in its preferred spot. However, if a car's preferred spot is already occupied, then it will park in the next available spot. Since the parking lot is along a one-way street, it is not guaranteed that every car will be able to park before driving past the parking lot. This dilemma leads to the idea of a parking function.Let us make this definition precise. For n ∈ N, let [n] := {1, . . . , n}. Formally, suppose the parking spots are labeled 1, 2, . . . , n, in order, along the one-way street and the cars are labeled according to the order in which they try to park. In other words, for each i ∈ [n], car c i is the i th car to try to park and prefers spot a i ∈ [n]. Note that more than one car can have the same preference. To park, cars first drive to their preferred spot and park in it if it is available. If their preferred spot is occupied then they drive forward and park in the next available spot. If all n cars can park in the parking lot under these conditions, then the preference list (a 1 , a 2 , . . . , a n ) is called a parking function (of length n). For example, (1, 2, 4, 2, 2) is a parking function, but (1, 2, 2, 5, 5) is not. Naturally, the first question that arises is: "For any n ∈ N, how many parking functions are there?" Konheim and Weiss showed that the number of parking functions of length n is (n + 1) n−1 [12].In this paper, enumerating the set of preference lists that allow all cars to park satisfying certain conditions is what we call the parking problem. In the adventure that follows, we will make a variety of choices to define new parking problems. For instance, we could change the structure of the parking lot, the way cars prefer parking spots, or the way cars drive down the street. Regardless of the choices we make, the quest in the first part of our adventure is to count the number of preference lists that satisfy the conditions of the parking problem at hand. With this enumeration, we then begin a new part of the adventure where we study properties of the set of parking functions arising in these parking problems. This gives rise to an entire new avenue of open problems. Let us begin! 1 Part 1: The adventure begins "Two roads diverged in a wood, and I -I took the one less traveled by, And that has made all the difference." -Robert Frost1 This adventure will be an ad free experience. We withhold references until our concluding remarks in Part 3, where we direct the reader to literature related to the problems presented.