A Generalization of Parking Functions Allowing Backward Movement

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

doi:10.48550/arxiv.1908.07658

Cited by 3 publications

(13 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was also proved in [3] that |B n,k | = (n + 1) n−1 independent of k, using a modified version of Pollack's technique [5]. Since this proof was not bijective, the authors posed the open problem of finding a bijection between B n,k and the set P F n,0 of classical (n, n)-parking functions that "preserves" in some manner certain parking function statistics such as ascents, descents, and ties.…”

Section: Introductionmentioning

confidence: 72%

“…They have been generalized in various directions, including parking functions on digraphs [7] and Naples parking functions [1]. In this paper, we will explore a generalization of the latter called the k-Naples parking functions [3].…”

Section: Introductionmentioning

confidence: 99%

“…In the case m = n, denote P F n,k := P F (n, n, k) and B n,k := B(n, n, k), and |P F n,k | has the following recursive formula [3]:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Connecting $k$-Naples parking functions and obstructed parking functions via involutions

Tian

2020

Preprint

View full text Add to dashboard Cite

Parking functions were classically defined for n cars attempting to park on a one-way street with n parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set P F n,k of k-Naples parking functions have cars who can drive backward a maximum of k steps before driving forward. A recursive formula for |P F n,k | has been obtained, though deriving a closed formula for |P F n,k | appears difficult. In addition, an important subset B n,k of P F n,k , called the contained k-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set P Fn of classical parking functions, independent of k.In this paper, we study k-Naples parking functions in the more general context of m cars and n parking spots, for any m ≤ n. We use various parking function involutions to establish a bijection between the contained k-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of k-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.

show abstract

Section: Introductionmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Connecting $k$-Naples parking functions and obstructed parking functions via involutions

Tian

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Parking functions have connections to many other areas of combinatorics, such as hyperplane arrangements [5] and the lattice of non-crossing partitions [6]. We refer the reader to the survey of Yan [9] for an elegant proof of (1) and a more in depth study of parking functions. Many generalizations and variants of parking functions have been studied, such as x-parking functions [8] and G-parking functions [4].…”

Section: Introductionmentioning

confidence: 99%

“…One can also generalize the parking analogy. An example of this is to allow cars to park a few spaces before their preferred spot if this is already taken, which has been studied recently [1].…”

Section: Introductionmentioning

confidence: 99%

Subset Parking Functions

Spiro

2019

Preprint

View full text Add to dashboard Cite

A parking function (c 1 , . . . , c n ) can be viewed as having n cars trying to park on a one-way street with n parking spots, where car i tries to park in spot c i , and otherwise he parks in the leftmost available spot after c i . Another way to view this is that each car has a set C i of "acceptable" parking spots, namely C i = [c i , n], and that each car tries to park in the leftmost available spot that they find acceptable.Motivated by this, we define a subset parking function (C 1 , . . . , C n ), with each C i a subset of {1, . . . , n}, by having the ith car try to park in the leftmost available element of C i . We further generalize this idea by restricting our sets to be of size k, intervals, and intervals of length k. In each of these cases we provide formulas for the number of such parking functions.

show abstract

Connecting $k$-Naples Parking Functions and Obstructed Parking Functions via Involutions

Tian

2022

Electron. J. Combin.

View full text Add to dashboard Cite

Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$. In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.

show abstract

A Generalization of Parking Functions Allowing Backward Movement

Cited by 3 publications

References 0 publications

Connecting $k$-Naples parking functions and obstructed parking functions via involutions

Connecting $k$-Naples parking functions and obstructed parking functions via involutions

Subset Parking Functions

Connecting $k$-Naples Parking Functions and Obstructed Parking Functions via Involutions

Contact Info

Product

Resources

About