Related papers: Generalizing parking functions with randomness

In the Page parking (or packing) model on a discrete interval (also known as the discrete R{\'e}nyi packing problem or the unfriendly seating problem), cars of length two successively park uniformly at random on pairs of adjacent places,…

Consider the following simple parking process on $\Lambda_n := \{-n, \ldots, n\}^d,d\ge1$: at each step, a site $i$ is chosen at random in $\Lambda_n$ and if $i$ and all its nearest neighbor sites are empty, $i$ is occupied. Once occupied,…

We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = -1 into this generalized polynomial…

Current navigation systems conflate time-to-drive with the true time-to-arrive by ignoring parking search duration and the final walking leg. Such underestimation can significantly affect user experience, mode choice, congestion, and…

Initially a car is placed with probability p at each site of the two-dimensional integer lattice. Each car is equally likely to be East-facing or North-facing, and different sites receive independent assignments. At odd time steps, each…

There are so many vehicles in the world and the number of vehicles is increasing rapidly. To alleviate the parking problems caused by that, the smart parking system has been developed. The parking planning is one of the most important parts…

Searching for a parking spot can waste time and gasoline. This waste can be reduced by assigning drivers to parking lots based on their destination and arrival time. In such a system, drivers could request a parking spot in advance and be…

In this paper, we study the total displacement statistic of parking functions from the perspective of cooperative game theory. We introduce parking games, which are coalitional cost-sharing games in characteristic function form derived from…

We extend the notion of parking function polytopes and study their geometric and combinatorial structure, including normal fans, face posets, and $h$-polynomials, as well as their connections to other classes of polytopes. To capture their…

We introduce Lehmer parking functions and study their set of parking outcomes. Our main results establish that the number of outcomes of Lehmer parking functions of length $n$ is given by a Bell number, which is exactly the number of set…

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…

If the moments of a probability measure on $\R$ are interpreted as a specialization of complete homogeneous symmetric functions, its free cumulants are, up to sign, the corresponding specializations of a sequence of Schur positive symmetric…

We recall the occupancy problem introduced by Konheim & Weiss in 1966 and we consider parking functions as hash maps. Each car $c_i$ prefers parking space $p_i$ (the hash map $c_i \mapsto p_i$ with $c_i$ is a key and $p_i$ an index into an…

We present a queuing model of parking dynamics and a model-based prediction method to provide real-time probabilistic forecasts of future parking occupancy. The queuing model has a non-homogeneous arrival rate and time-varying service time…

We define a "shifted analogue" $\mathrm{SH}_n$ of the parking function symmetric function $\mathrm{PF}_n$. The expansion of $\mathrm{SH}_n$ in terms of three bases for shifted symmetric functions is explicitly described. We don't know a…

In smart cities, it is common practice to define a maximum length of stay for a given parking space to increase the space's rotativity and discourage the usage of individual transportation solutions. However, automatically determining…

We propose a generalized car parking problem where either a car of size $\sigma$ or of size $m\sigma$ ($m>1$) is sequentially parked on a line with probability $q$ and $(1-q)$, respectively. The free parameter $q$ interpolates between the…

A parking function of length n is a sequence (b_1, b_2,..., b_n) of nonnegative integers whose nondecreasing rearrangement (a_1, a_2,...,a_n) has the property that a_i < i for every i. A well-known result about parking functions is that the…

Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0$, $\Delta_{ij}\leq 0\text{ for }i\neq j,$ and there exists a vector ${\bf r}=(r_1,\ldots,r_n)>0$ such that ${\bf r}\Delta \geq 0$. Here the…

A defective $(m,n)$-parking function with defect $d$ is a parking function with $m$ cars attempting to park on a street with $n$ parking spots in which exactly $d$ cars fail to park. We establish a way to compute the defect of a defective…