Related papers: Symmetry in Tree Parking Distributions

Stirling permutations are parking functions, and we investigate two parking function statistics in the context of these objects: lucky cars and displacement. Among our results, we consider two extreme cases: extremely lucky Stirling…

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…

We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the…

We introduce a rather natural family of non-uniform distributions on $PF_n$, $n\in\mathbb{N}$, the set of parking functions of length $n$. One of the motivations for this comes from a similar situation in the context of integer partitions.…

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…

Modeling of headway/spacing between two consecutive vehicles has many applications in traffic flow theory and transport practice. Most known approaches only study the vehicles running on freeways. In this paper, we propose a model to…

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…

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…

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…

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…

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of…

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…

Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This…

In this article, we establish new results on the probabilistic parking model (introduced by Durm\'ic, Han, Harris, Ribeiro, and Yin) with $m$ cars and $n$ parking spots and probability parameter $p\in[0,1]$. For any $ m \leq n$ and $p \in…

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyi's parking problem, alternatively called blocking…

We study the asymptotic distribution of integers sharing the same rooted-tree structure that encodes their complete prime factorization tower. For each tree we derive an explicit density formula depending only on a pair $(m,k)$, the density…

During the attempt to line up into a dense traffic people have necessarily to share a limited space under turbulent conditions. From the statistical point view it generally leads to a probability distribution of the distances between the…

Parking functions are a widely studied class of combinatorial objects, with connections to several branches of mathematics. On the algebraic side, parking functions can be identified with the standard monomials of $M_n$, a certain monomial…

In this paper we analyze the spatial distribution of elements around points of interest. Based on a spatial exclusion principle we model the system by means of a Fermi-Dirac distribution defined by two easily interpretable parameters. By…

We introduce a generalization of parking functions called $t$-metered $(m,n)$-parking functions, in which one of $m$ cars parks among $n$ spots per hour then leaves after $t$ hours. We characterize and enumerate these sequences for $t=1$,…