In a parking function, a lucky car is a car that parks in its preferred parking spot and the parking outcome is the permutation encoding the order in which the cars park on the street. We give a characterization for the set of parking outcomes arising from parking functions with a fixed set of lucky cars. This characterization involves the descent bottom set of a permutation, and we use the characterization to we give a formula for the number of parking functions with a fixed set of lucky cars. Our work includes the cases where the number of cars is equal to the number of parking spots, and where there are more spots than cars. We also give product formulas for the number of weakly increasing parking functions having a fixed set of lucky cars, and when the number of cars equals the number of spots this is a product of Catalan numbers.
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.
Recently, the authors extended the notion of parking functions to parking sequences, which include cars of different sizes, and proved a product formula for the number of such sequences. We here give a refinement of that result involving parking the cars after a trailer. The proof of the refinement uses a multi-parameter extension of the Abel--Rothe polynomial due to Strehl.
In 1980, G. Kreweras gave a recursive bijection between forests and parking functions. In this paper we construct a nonrecursive bijection from forests onto parking functions, which answers a question raised by R. Stanley. As a by-product, we obtain a bijective proof of Gessel and Seo's formula for lucky statistic on parking functions.
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 preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $n$-tuple containing the cars' parking preferences a parking function. In this paper, we introduce a generalization of the …
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 th…
We introduce a new parking procedure called MVP parking in which $n$ cars sequentially enter a one-way street with a preferred parking spot from the $n$ parking spots on the street. If their preferred spot is empty, they park there. Otherwise, they park there and the car parked in that spot is bumped to the next unoccupied spot on the street. If all cars can park under this parking procedure, we say the list of preferences of the $n$ cars is an MVP parking function of length $n$. We show that the set of (classical) parking functions is exactly the set of MVP parking functions although the park…
The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the cars to have different sizes and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. A preference sequence in which all the cars are able to park is called a parking sequence. In this paper, we study increasing parking sequences and count them via bijections to lattice paths with right boundaries. Then we st…
Parking sequences (a generalization of parking functions) are defined by specifying car lengths and requiring that a car attempts to park in the first available spot after its preference. If it does not fit there, then a collision occurs and the car fails to park. In contrast, parking assortments generalize parking sequences (and parking functions) by allowing cars (also of assorted lengths) to seek forward from their preference to identify a set of contiguous unoccupied spots in which they fit. We consider both parking sequences and parking assortments and establish that the number of prefere…
Parking functions, classically defined in terms of cars with preferred parking spots on a directed path attempting to park there, arise in many combinatorial situations and have seen various generalizations. In particular, parking functions have been defined for general digraphs, which yields many more enumeration problems. For example, in a directed tree whose edges are orientated away from the root, it is unknown in general how the number of parking functions on it changes once the orientation is reversed, even in the case when the tree is a star. We show that this orientation reversal resul…
We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised by assigning a tier (or colour) to each of the cars,…
We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $σ^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $Σ^{2}$ conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ Θ:= (1-m)^2- Σ^2 (σ^2+m^2-m).$$ Specifically, when $m \leq 1$, if $ Θ>0,$ then all but (possibly) a few cars will manage to park, whereas if $Θ<0$, then a positive fracti…
Parking Cascades: From the Simplest Sequence to Motzkin and Catalan
Fitchburg State University（菲奇堡州立大学）；Missouri Western State University（密苏里西部州立大学）；University of Wisconsin-Milwaukee（威斯康星大学密尔沃基分校）；Hollins University（霍金斯大学）；University of Colorado Denver（科罗拉多大学丹佛分校）
In this paper, we investigate a parking process on a uniform random rooted plane tree with $n$ vertices. Every vertex of the tree has a parking space for a single car. Cars arrive at independent uniformly random vertices of the tree. If the parking space at a vertex is unoccupied when a car arrives there, it parks. If not, the car drives towards the root and parks in the first empty space it encounters (if there is one). We are interested in asymptotics of the probability of the event that all cars can park when $\lfloor αn \rfloor$ cars arrive, for $α> 0$. We observe that there is a phase tra…
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 $α_i \in \{1,2,\ldots,n\}$. 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 $(α_1, α_2, \ldots, α_n)$ 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 …
A parking function $(c_1,\ldots,c_n)$ 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 $(C_1,\ldots,C_n)$, with each $C_i$ a subset of $\{1,\ldots,n\}$, by having the $i$th car try to park in…
In the Page parking (or packing) model on a discrete interval (also known as the discrete R{é}nyi packing problem or the unfriendly seating problem), cars of length two successively park uniformly at random on pairs of adjacent places, until only isolated places remain. We give a probabilistic proof of the (known) fact that the proportion of the interval covered by cars goes to 1-exp(-2) , when the length of the interval goes to infinity. We obtain some new consequences, and also study a version of this process defined on the infinite line.
Models of parking in which cars are placed randomly and then move according to a deterministic rule have been studied since the work of Konheim and Weiss in the 1960s. Recently, Damron, Gravner, Junge, Lyu, and Sivakoff introduced a model in which cars are both placed and move at random. Independently at each point of a Cayley graph $G$, we place a car with probability $p$, and otherwise an empty parking space. Each car independently executes a random walk until it finds an empty space in which to park. In this paper we introduce three new techniques for studying the model, namely the space-ba…
In a recent paper J. Haglund showed that a certain symmetric function expresion enumerates by t^{area} q^{dinv} of the parking functions whose diagonal word is in the shuffle of 12...j and j+1...j+n with k of the cars j+1,...,j+n in the main diagonal including car j+n in the cell (1,1). In view of some recent conjectures of Haglund-Morse-Zabrocki it is natural to conjecture that replacing E_{n,k} by the modified Hall-Littlewood functions would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting cars j+1…
Given a strictly increasing sequence $\mathbf{t}$ with entries from $[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with $|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\le i\}|+|\{c\in \mathbf{c}\mid c\le i\}|\ge i$ for all $i$ in $[n]$. We can think of $\mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $\mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion correspond…
We recall that the $k$-Naples parking functions of length $n$ (a generalization of parking functions) are defined by requiring that a car which finds its preferred spot occupied must first back up a spot at a time (up to $k$ spots) before proceeding forward down the street. Note that the parking functions are the specialization of $k$ to $0$. For a fixed $0\leq k\leq n-1$, we define a function $φ_k$ which maps a $k$-Naples parking function to the permutation denoting the order in which its cars park. By enumerating the sizes of the fibers of the map $φ_k$ we give a new formula for the number o…
Consider a uniform rooted Cayley tree $T_{n}$ with $n$ vertices and let $m$ cars arrive sequentially, independently, and uniformly on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and parks as soon as possible. Lackner & Panholzer (arXiv:1504.04972) established a phase transition for this process when $ m \approx \frac{n}{2}$. In this work, we couple this model with a variant of the classical Erdös-Rényi random graph process. This enables us to describe the phase transition for the size of the components of…
We construct an action of the braid group on $n$ strands on the set of parking functions of $n$ cars such that elementary braids have orbits of length 2 or 3. The construction is motivated by a theorem of Lyashko and Looijenga stating that the number of the distinguished bases for $A_n$ singularity equals $(n+1)^{n-1}$ and thus equals the number of parking functions. We construct an explicit bijection between the set of parking functions and the set of distinguished bases, which allows us to translate the braid group action on distinguished bases in terms of parking functions.
Consider a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number $m \le n$ of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider $m =[αn]$ and let $A_{n,α}$ denote the event that all $[αn]$ c…
The original Shuffle Conjecture of Haglund et al. has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as $<\nabla e_n, h_μ>$ where \nabla is the Macdonald polynomial eigen-operator of Bergeron and Garsia and $h_μ$ is the homogeneous basis indexed by $μ=(μ_1,μ_2,...,μ_k)$ partitions of n. The combinatorial side q,t-enumerates a family of Parking Functions whose reading word is a shuffle of k successive segments of 1,2,3,...,n of respective lengths $μ_1,μ_2,...,μ_k$. It can be shown that for t=1/q the symmetric function side reduces to a pr…
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 investigate various probabilistic properties of a uniform parking function. Through a combinatorial construction termed a parking function multi-shuffle, we give a formula for the law of multiple coordinates in the g…
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 between parking functions $\text{PF}(m, n)$ with $m$ cars and $n$ spots and spanning forests $\mathscr{…
We enumerate a class of fully parked trees. In a probabilistic context, this means computing the partition function $F(x,y)$ of the parking process where an i.i.d. number of cars arrives at each vertex of a Galton-Watson tree with a geometric offspring distribution, conditioned to leave no vertex unoccupied at the end. The variables $x$ and $y$ count the number of vertices in the tree and the number of cars exiting from the root, respectively. For any car arrival distribution $\mathbf b$, we obtain an explicit parametric expression of $F(x,y)$ in terms of the probability generating function $B…
