Corpus ID: 232428060

Parking functions: From combinatorics to probability

  title={Parking functions: From combinatorics to probability},
  author={R. Kenyon and Mei Yin},
  journal={arXiv: Combinatorics},
Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $m=n$; we study here combinatorial and probabilistic aspects of this generalized case. We construct a family of bijections… Expand

Figures from this paper

Parking functions: Interdisciplinary connections
Abstract. Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwiseExpand
Asymptotic behaviour of the first positions of uniform parking functions
In this paper we study the asymptotic behavior of a random uniform parking function πn of size n. We show that the first kn places πn(1), . . . , πn(kn) of πn are asymptotically i.i.d. and uniform onExpand


Parking functions and noncrossing partitions
  • R. Stanley
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 1997
A parking function is a sequence $(a_1,\dots,a_n)$ of positive integers such that, if $b_1\leq b_2\leq \cdots\leq b_n$ is the increasing rearrangement of the sequence $(a_1,\dots, a_n),$ thenExpand
Generalized Parking Functions, Tree Inversions, and Multicolored Graphs
  • C. Yan
  • Mathematics, Computer Science
  • Adv. Appl. Math.
  • 2001
It is shown that the sum enumerator of complements of x-parking functions is identical to the inversion enumerators of sequences of rooted b-forests by generating function analysis. Expand
Mappings of acyclic and parking functions
The two functions in question are mappings: [n]→[n], with [n] = {1, 2,⋯,n}. The acyclic function may be represented by forests of labeled rooted trees, or by free trees withn + 1 points; the parkingExpand
Multiparking functions, graph searching, and the Tutte polynomial
The bijection induced by the breadth-first search leads to a new characterization of external activity, and hence a representation of Tutte polynomial by the reversed sum of G-multiparking functions is constructed. Expand
Probabilizing parking functions
We explore the link between combinatorics and probability generated by the question "What does a random parking function look like?" This gives rise to novel probabilistic interpretations of someExpand
Tutte polynomials and G-parking functions
This paper gives a new expression for the Tutte polynomial of a general connected graph G in terms of statistics of G-parking functions and proves that T"G(x,y))=@?"f"@?"P, where P"G is the set of G. Expand
A family of bijections between G-parking functions and spanning trees
A family of bijective maps is constructed between the set PG of G-parking functions and the set JG of spanning trees of G rooted at 0, thus providing a combinatorial proof of |PG| = |JG|. Expand
A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron
In the second subdivision of Πn(x), the chambers are indexed in a natural way by rooted binary trees with n+1 vertices, and the configuration of these chambers provides a representation of another polytope with many applications, the associahedron. Expand
Enumeration of (p, q)-parking functions
It is shown that parking functions can be interpreted as recurrent configurations in the sandpile model for some graphs and established a correspondence with a Lukasiewicz language, which enables to enumerate (p1,..., pk)-parking functions as well as increasing ones. Expand
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
This paper provides tight bounds for the moments of the width of rooted labeled trees with $n$ nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the manyExpand