Related papers: Type C parking functions and a zeta map
In this extended abstract, we show how a bijection between parking functions and regions of the Shi arrangement from [Athanasiadis, Linusson '99] (in type $A_n$) and [Armstrong, Reiner, Rhoades '15] (in type $B_n, C_n, D_n$) allows for the…
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted…
We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.
We give a description of the well known Zeta map on Dyck paths which sends the dinv,area statistics to the area,bounce statistics. Our description uses Dyck paths area sequences and can be implemented easily.
We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson.…
Parking functions of length $n$ are well known to be in correspondence with both labelled trees on $n+1$ vertices and factorizations of the full cycle $\sigma_n=(0\,1\,\cdots\,n)$ into $n$ transpositions. In fact, these correspondences can…
We study three natural types of restrictions on Fubini rankings and unit interval parking functions, which are motivated by their correspondence with ordered set partitions. For each restriction type, we define the corresponding subset of…
In this paper, we obtain a q-exponential generating function for inversions on parking functions via symmetric function theory and also through a direct bijection to rooted labeled forests. We then apply these techniques to unit interval…
A depth-first search version of Dhar's burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the…
For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j…
This work builds on the notion of record of rooted trees. We provide an alternative definition of parking functions, derive from it a record-preserving bijection between rooted trees and parking functions, and establish a join…
We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of $m$-parking functions of length $n$. This bijection…
Classical parking functions are a generalization of permutations that appear in many combinatorial structures. Prime parking functions are indecomposable components such that any classical parking function can be uniquely described as a…
In parking problems, a given number of cars enter a one-way street sequentially, and try to park according to a specified preferred spot in the street. Various models are possible depending on the chosen rule for collisions, when two cars…
We apply the concept of parking functions to rooted labelled trees and functional digraphs of mappings (i.e., functions $f : [n] \to [n]$) by considering the nodes as parking spaces and the directed edges as one-way streets: Each driver has…
The original Pak-Stanley labeling was defined by Pak and Stanley as a bijective map from the set of regions of an extended Shi arrangement to the set of parking functions. This map was later generalized to other arrangements associated with…
The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product…
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…
Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their…
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 some elegant, known generating functions. It…