Parking Functions: Choose Your Own Adventure

Carlson, Joshua; Christensen, Alex; Harris, Pamela E.; Jones, Zakiya; Rodríguez, Andrés Ramos

doi:10.1080/07468342.2021.1943115

Cited by 5 publications

(5 citation statements)

References 11 publications

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“…We now introduce the Coin Problem, which involves a new parking protocol, motivated by Carlson et al in [2]. We have n cars in a queue to enter a one-way street with n parking spots numbered from 1 to n. Let a i denote the preference of car i, and let α = (a 1 , a 2 , .…”

Section: The Coin Problemmentioning

confidence: 99%

“…Many generalizations of parking functions exist and include considering the case where there are more parking spots than cars, considering cases where cars prefer an interval of parking spots rather than a single parking spot, and others include changing the parking procedure and allowing cars to do something different whenever they find their preferred spot occupied. For an expository article with recent finding and many open problems related to parking functions we recommend [2].…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic Parking Functions

Durmić,

Han,

Harris

et al. 2023

Electron. J. Combin.

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We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodríguez. Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of special interest is when $p=1/2$, where we demonstrate a sharp transition in some parking statistics. We also present several interesting combinatorial consequences of the parking protocol. In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots. Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.

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Section: The Coin Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Probabilistic Parking Functions

Durmić,

Han,

Harris

et al. 2023

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…In a playful paper by Carlson, Christensen, Harris, Jones, and Ramos Rodríguez [2], the authors introduce many variants of parking functions and mention many open problems. In particular, chapter 1.9 of the paper suggests introducing randomness to Naples parking functions.…”

Section: Introductionmentioning

confidence: 99%

“…. We have that f (0) is the number of tuples that are not 1-Naples parking functions 2. We wrote the code to compute this table (and other tables) in Python.…”

mentioning

confidence: 99%

Generalizing Parking Functions with Randomness

Tian

Treviño

2022

Electron. J. Combin.

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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 $(2t-1)/2^{n-1}$.

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“…The Coin Problem. We now introduce the Coin Problem, which involves a new parking protocol, motivated by Carlson et al in [2]. We have n cars in a queue to enter a one-way street with n parking spots numbered from 1 to n. Let a i denote the preference of car i, and let α = (a 1 , a 2 , .…”

Section: Introducing Probabilitymentioning

confidence: 99%

Probabilistic Parking Functions

Durmić¹,

Han²,

Harris³

et al. 2022

Preprint

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We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper "Parking Functions: Choose your own adventure," (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodríguez. Among our results, we prove that the probability of obtaining a parking function, from a length n preference vector, is independent of the probabilistic parameter p. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter p. Of special interest is when p = 1/2, where we demonstrate a sharp transition in some parking statistics. We also present several interesting combinatorial consequences of the parking protocol. In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots, which solves an open problem of Novelli and Thibon posed in 2020 (arXiv:1209.5959). Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.

show abstract

Parking Functions: Choose Your Own Adventure

Cited by 5 publications

References 11 publications

Probabilistic Parking Functions

Probabilistic Parking Functions

Generalizing Parking Functions with Randomness

Probabilistic Parking Functions

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