Related papers: Statistics on $\ell$-interval parking functions
An \emph{$(r,k)$-parking function} of length $n$ may be defined as a sequence $(a_1,\dots,a_n)$ of positive integers whose increasing rearrangement $b_1\leq\cdots\leq b_n$ satisfies $b_i\leq k+(i-1)r$. The case $r=k=1$ corresponds to…
Parking accurately and safely in highly constrained spaces remains a critical challenge. Unlike structured driving environments, parking requires executing complex maneuvers such as frequent gear shifts and steering saturation. Recent…
In the parking model on $\mathbb{Z}^d$, each vertex is initially occupied by a car (with probability $p$) or by a vacant parking spot (with probability $1-p$). Cars perform independent random walks and when they enter a vacant spot, they…
Using measured data we demonstrate that there is an amazing correspondence among the statistical properties of spacings between parked cars and the distances between birds perching on a power line. We show that this observation is easily…
We show that the number of parking functions of length $n$ with zero secondary dinv is equal to the number of ordered cycle decompositions of permutations of $[n]$.
Let $\ell_j:=-\frac{d^2}{dx^2}+k^2q_j(x),$ $k=const>0, j=1,2,$ $0<c_0\leq q_j(x)\leq c_1,$ %$q\in BV([0,1])$, $q$ has finitely many discontinuity points $x_m\in [0,1],$ and is real-analytic on the intervals $[x_m,x_{m+1}]$ between these…
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…
With the number of vehicles continuously increasing, parking monitoring and analysis are becoming a substantial feature of modern cities. In this study, we present a methodology to monitor car parking areas and to analyze their occupancy in…
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…
In this paper, we infer the statuses of a taxi, consisting of occupied, non-occupied and parked, in terms of its GPS trajectory. The status information can enable urban computing for improving a city's transportation systems and land use…
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…
Warning. The reading of this paper will send you down many winding roads toward new and exciting research topics enumerating generalized parking functions. Buckle up!
The main purpose of this paper is to provide threshold functions for the events that a random subset of the points of a finite vector space has certain properties related to point-flat incidences. Specifically, we consider the events that…
A pair $(G,K)$ of a group and its subgroup is called a Gelfand pair if the induced trivial representation of $K$ on $G$ is multiplicity free. Let $(a_j)$ be a sequence of positive integers of length $n$, and let $(b_i)$ be its…
We compute the correlation function in the equilibrium version of R\'enyi's {\sl parking problem}. The correlation length is found to diverge as $2^{-1}\pi^{-2}(1-\rho)^{-2}$ when $\rho\nearrow1$ (maximum density) and as…
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…
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…
Motivated by questions in lossy data compression and by theoretical considerations, we examine the problem of estimating the rate-distortion function of an unknown (not necessarily discrete-valued) source from empirical data. Our focus is…
This paper provides an exploration of parking functions, a classical combinatorial object. We present two viewpoints on their structure and properties: through poset of noncrossing partitions and polytopes.