A Generalization of Parking Functions Allowing Backward Movement

Christensen, Alex; Harris, Pamela E.; Jones, Zakiya; Loving, Marissa; Rodríguez, Andrés Ramos; Rennie, Joseph; Kirby, Gordon Rojas

doi:10.37236/8948

Cited by 6 publications

(4 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let P (n) = (n + 1) n−1 be the number of parking functions. The following recursive formula for P (n) has been proved in [5] and [3], but can also be seen as plugging in p = 0 into our recursive formula (2) for T p,k (n) (because if p = 0, then for α to park, α must be a parking function)…”

Section: Distribution Of Random 1-naplesmentioning

confidence: 99%

“…One could then generalize the concept to k-Naples parking functions, where cars back up up to k spots one by one and take the first empty spot they find behind them, and if no such spot exists, they move forward in search of a parking spot. Christensen et al [3] found the following recursive formula to find the number N k (n + 1) of k-Naples parking functions on n + 1 cars:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalizing Parking Functions with Randomness

Tian

Treviño

2022

Electron. J. Combin.

View full text Add to dashboard Cite

Consider $n$ cars $C_1, C_2, \ldots, C_n$ that want to park in a parking lot with parking spaces $1,2,\ldots,n$ that appear in order. Each car $C_i$ has a parking preference $\alpha_i \in \{1,2,\ldots,n\}$. The cars appear in order, if their preferred parking spot is not taken, they take it, if the parking spot is taken, they move forward until they find an empty spot. If they do not find an empty spot, they do not park. An $n$-tuple $(\alpha_1, \alpha_2, \ldots, \alpha_n)$ is said to be a parking function, if this list of preferences allows every car to park under this algorithm. For an integer $k$, we say that an $n$-tuple is a $k$-Naples parking function if the cars can park with the modified algorithm, where when car $C_i$'s preference is taken, $C_i$ backs up $k$-spaces (one by one) and takes the first empty spot. If there are no empty spots in the (up to) $k$ spaces behind $\alpha_i$, they then try to find a parking spot in front of them. We introduce randomness to this problem in two ways: 1) For the original parking function definition, for each car $C_i$ that has their preference taken, we decide with probability $p$ whether $C_i$ moves forwards or backwards when their preferred spot is taken; 2) For the $k$-Naples definition, for each car $C_i$ that has their preference taken, we decide with probability $p$ whether $C_i$ backs up $k$ spaces or not before moving forward. In each of these models, for an $n$-tuple $\alpha\in\{1,2,\ldots,n\}^n$, there is now a probability that the model ends in all cars parking or not. For each of these random models, we find a formula for the expected value. Furthermore, for the second random model, in the case $k =1$, $p=1/2$, we prove that for any $1\le t\le 2^{n-2}$, there is exactly one $\alpha\in\{1,2,\ldots,n\}^n$ such that the probability that $\alpha$ parks is $(2t-1)/2^{n-1}$.

show abstract

Section: Distribution Of Random 1-naplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalizing Parking Functions with Randomness

Tian

Treviño

2022

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…However they did not include variable movement, nor any other generalization presented here. Cars that can back up to a certain number of spots when finding their preferred parking spot occupied were studied in [5]. This only included looking one parking spot back at a time.…”

Section: Epiloguementioning

confidence: 99%

Parking Functions: Choose Your Own Adventure

Carlson¹,

Christensen²,

Harris³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…al. [3] found the following recursive formula to find the number N k (n + 1) of k-Naples parking functions on n + 1 cars:…”

Section: Introductionmentioning

confidence: 99%

Generalizing parking functions with randomness

Tian¹,

Treviño²

2021

Preprint

View full text Add to dashboard Cite

Consider n cars C1, C2, . . . , Cn that want to park in a parking lot with parking spaces 1, 2, . . . , n that appear in order. Each car Ci has a parking preference αi ∈ {1, 2, . . . , n}. The cars appear in order, if their preferred parking spot is not taken, they take it, if the parking spot is taken, they move forward until they find an empty spot. If they do not find an empty spot, they do not park. An n-tuple (α1, α2, . . . , αn) is said to be a parking function, if this list of preferences allows every car to park under this algorithm. For an integer k, we say that an n-tuple is a k-Naples parking function if the cars can park with the modified algorithm, where park Ci backs up k-spaces (one by one) if their spot is taken before trying to find a parking spot in front of them. We introduce randomness to this problem in two ways: 1) For the original parking function definition, for each car Ci that has their preference taken, we decide with probability p whether Ci moves forwards or backwards when their preferred spot is taken; 2) For the k-Naples definition, for each car Ci that has their preference taken, we decide with probability p whether Ci backs up k spaces or not before moving forward. In each of these models, for an n-tuple α ∈ {1, 2, . . . , n} n , there is now a probability that the model ends in all cars parking or not. For each of these random models, we find a formula for the expected value. Furthermore, for the second random model, in the case k = 1, p = 1/2, we prove that for any 1 ≤ t ≤ 2 n−2 , there is exactly one α ∈ {1, 2, . . . , n} n such that the probability that α parks is (2t − 1)/2 n−1 .

show abstract

A Generalization of Parking Functions Allowing Backward Movement

Cited by 6 publications

References 6 publications

Generalizing Parking Functions with Randomness

Generalizing Parking Functions with Randomness

Parking Functions: Choose Your Own Adventure

Generalizing parking functions with randomness

Contact Info

Product

Resources

About