Related papers: Generalizing parking functions with randomness
We continue the study of parking assortments, a generalization of parking functions introduced by Chen, Harris, Mart\'{i}nez, Pab\'{o}n-Cancel, and Sargent. Given $n$ cars of lengths $\mathbf{y}=(y_1,y_2,\dots,y_n) \in \mathbb{N}^n$, we…
Let $(A_u : u \in \mathbb{B})$ be i.i.d.~non-negative integers that we interpret as car arrivals on the vertices of the full binary tree $ \mathbb{B}$. Each car tries to park on its arrival node, but if it is already occupied, it drives…
Finding a parking space nowadays becomes an issue that is not to be neglected, it consumes time and energy. We have used computer vision techniques to infer the state of the parking lot given the data collected from the University of The…
Proportionality is an attractive fairness concept that has been applied to a range of problems including the facility location problem, a classic problem in social choice. In our work, we propose a concept called Strong Proportionality,…
We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\mathbb{R}^d$ and of point sets that are (almost) subadditive in their first…
We prove here that the polynomial <nabla(C_p(1)), e_a h_b h_c> q, t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p of size a + b + c and…
We settle a conjecture of B\'ona regarding the log-concavity of a certain statistic on parking functions by utilizing recent log-concavity results on matroids. This result allows us to also prove that connected, labeled graphs graded by…
Let $W$ be an irreducible finite complex reflection group acting on a complex vector space $V$. For a positive integer $k$, we consider a class function $\varphi_k$ given by $\varphi_k(w) = k^{\dim V^w}$ for $w \in W$, where $V^w$ is the…
We investigate the problem of predicting driver behavior in parking lots, an environment which is less structured than typical road networks and features complex, interactive maneuvers in a compact space. Using the CARLA simulator, we…
Smart city research envisions a future in which data-driven solutions and sustainable infrastructure work together to define urban living at the crossroads of urbanization and technology. Within this framework, smart parking systems play an…
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 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…
We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and {\L}ukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the…
Autonomous parking systems start with the detection of available parking slots. Parking slot detection performance has been dramatically improved by deep learning techniques. Deep learning-based object detection methods can be categorized…
An improvement in technology is linearly related to time and time-relevant problems. It has been seen that as time progresses, the number of problems humans face also increases. However, technology to resolve these problems tends to improve…
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.…
Kreweras proved that the reversed sum enumerator for parking functions of length $n$ is equal to the inversion enumerator for labeled trees on $n+1$ vertices. Recently, Perkinson, Yang, and Yu gave a bijective proof of this equality that…
In urban environments, parking has proven to be a significant source of congestion and inefficiency. In this study, we propose a methodology that offers a systematic solution to minimize the time spent by drivers in finding parking spaces.…
We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper…
To complete a previous paper, the probability density functions of the center-of-gravity as positioning algorithm are derived with classical methods. These methods, as suggested by the textbook of Probability, require the preliminary…