Related papers: Counting Parking Sequences and Parking Assortments…
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 1966, Konheim and Weiss [33] introduced a now classical parking protocol. The deterministic process and its resultant objects, known as parking functions, have since become a favorite object of study in enumerative combinatorics. In our…
A person is given a numbered sequence of positions on a sheet of paper. The person is asked, "Which will be the next (or the next after that) position?" Everyone has an opinion as to how he or she would proceed. There are regular sequences…
The fundamental question considered in algorithms on strings is that of indexing, that is, preprocessing a given string for specific queries. By now we have a number of efficient solutions for this problem when the queries ask for an exact…
Out-of-distribution generalization capabilities of sequence-to-sequence models can be studied from the lens of two crucial forms of generalization: length generalization -- the ability to generalize to longer sequences than ones seen during…
In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]=\{1,...,$n$\}. In 2020, Yue Cai and Catherine H.…
A consecutive pattern in a permutation $\pi$ is another permutation $\sigma$ determined by the relative order of a subsequence of contiguous entries of $\pi$. Traditional notions such as descents, runs and peaks can be viewed as particular…
Smart-parking solutions use sensors, cameras, and data analysis to improve parking efficiency and reduce traffic congestion. Computer vision-based methods have been used extensively in recent years to tackle the problem of parking lot…
In a totally ordered set the notion of sorting a finite sequence is defined through a suitable permutation of the sequence's indices. In this paper we prove a simple formula that explicitly describes how the elements of a sequence are…
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 investigate parking in a one-dimensional lot, where cars enter at a rate $\lambda$ and each attempts to park close to a target at the origin. Parked cars also depart at rate 1. An entering driver cannot see beyond the parked cars for…
We define a class of sequences ${a_n}$ by $a_1=a$ and $a_{n+1}=P(a_n)$, where $P(x)$ is a polynomial with real coefficients. We then find out for which values $a$ and for which polynomials $P(x)$ these sequences will be constant after a…
A Skolem sequence is a linear arrangement of the multiset, {1, 1, 2, 2, ..., n, n} such that if r in [n] appears in positions i and j, then |i-j| = r. We first translate the problem to a particular set of perfect matchings, then apply the…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
Autonomous parking remains a critical yet challenging task in intelligent driving systems, particularly within constrained urban environments where maneuvering space is limited and precise control is essential. While recent advances in…
Searching for available parking spots in high-density urban centers is a stressful task for drivers that can be mitigated by systems that know in advance the nearest parking space available. To this end, image-based systems offer cost…
We determine the maximal number of steps required to sort $n$ labeled points on a circle by adjacent swaps. Lower bounds for sorting by all swaps, not necessarily adjacent, are given as well.
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.…
We explore the fundamental problem of sorting through the lens of learning-augmented algorithms, where algorithms can leverage possibly erroneous predictions to improve their efficiency. We consider two different settings: In the first…
Shared Memory is a mechanism that allows several processes to communicate with each other by accessing -- writing or reading -- a set of variables that they have in common. A Consistency Model defines how each process observes the state of…