Parking Cars of Different Sizes

Ehrenborg, Richard; Happ, Alex

doi:10.4169/amer.math.monthly.123.10.1045

Cited by 7 publications

(1 citation statement)

References 8 publications

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“…Parking functions have seen many generalizations: G-parking functions [11], u-parking functions [9], parking sequences [4], rational parking functions [1], and those defined on tree-shaped parking lots [2,10]. In this paper, we extend the "drivers searching for a parking spot" analogy from the trees in [10] to general digraphs and give a description that generalizes the set definition of the classical parking functions in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Parking Functions on Directed Graphs and Some Directed Trees

King¹,

Yan

2020

Electron. J. Combin.

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Classical parking functions can be deﬁned in terms of drivers with preferred parking spaces searching a linear parking lot for an open parking spot. We may consider this linear parking lot as a collection of n vertices (parking spots) arranged in a directed path. We generalize this notion to allow for more complicated “parking lots” and deﬁne parking functions on arbitrary directed graphs. We then consider a relationship proved by Lackner and Panholzer between parking functions on trees and “mapping digraphs” and we show that a similar relationship holds when edge orientations are reversed.

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Section: Introductionmentioning

confidence: 99%

Parking Functions on Directed Graphs and Some Directed Trees

King¹,

Yan

2020

Electron. J. Combin.

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Cycle structure of random parking functions

2023

Advances in Applied Mathematics

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Prime parking functions on rooted trees

King

Yan

2019

Journal of Combinatorial Theory, Series A

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For a labeled, rooted tree with edges oriented towards the root, we consider the vertices as parking spots and the edge orientation as a one-way street. Each driver, starting with her preferred parking spot, searches for and parks in the first unoccupied spot along the directed path to the root. If all n drivers park, the sequence of spot preferences is called a parking function. We consider the sequences, called prime parking functions, for which each driver parks and each edge in the tree is traversed by some driver after failing to park at her preferred spot. We prove that the total number of prime parking functions on trees with n vertices is (2n − 2)!. Additionally, we generalize increasing parking functions, those in which the drivers park with a weakly-increasing order of preference, to trees and prove that the total number of increasing prime parking functions on trees with n vertices is (n − 1)!S n−1 , where {S i } i≥0 are the large Schröder numbers.

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Parking Cars of Different Sizes

Cited by 7 publications

References 8 publications

Parking Functions on Directed Graphs and Some Directed Trees

Parking Functions on Directed Graphs and Some Directed Trees

Cycle structure of random parking functions

Prime parking functions on rooted trees

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