Parking Functions Research Articles

Delta and Theta Operator Expansions

Abstract We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking functions. Here, $\Delta _F$ and $\Pi $ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$ . Our main results, in turn, give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta _{Fe_1} \Theta _{G} J$ whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis $\{\Pi e_\lambda ^\ast \}_\lambda $ . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.

Lucky Cars: Expected Values and Generating Functions

We count lucky cars in parking functions of a given length, and provide elementary derivations for the generating functions, expected values and variances. We consider not only classical parking functions, but also problems with additional parking slots, and with cars of different sizes.

Abel-Gontcharoff polynomials, parking trajectories and ruin probabilities

Abstract The central mathematical tool discussed is a non-standard family of polynomials, univariate and bivariate, called Abel-Goncharoff polynomials. First, we briefly summarize the main properties of this family of polynomials obtained in the previous work. Then, we extend the remarkable links existing between these polynomials and the parking functions which are a classic object in combinatorics and computer science. Finally, we use the polynomials to determine the non-ruin probabilities over a finite horizon for a bivariate risk process, in discrete and continuous time, assuming that claim amounts are dependent via a partial Schur-constancy property.

Generalized Parking Function Polytopes

A classical parking function of length n is a list of positive integers (a1,a2,…,an)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(a_1, a_2, \\ldots , a_n)$$\\end{document} whose nondecreasing rearrangement b1≤b2≤⋯≤bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b_1 \\le b_2 \\le \\cdots \\le b_n$$\\end{document} satisfies bi≤i\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b_i \\le i$$\\end{document}. The convex hull of all parking functions of length n is an n-dimensional polytope in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{x}}$$\\end{document}-parking functions for x=(a,b,⋯,b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{x}}=(a,b,\\dots ,b)$$\\end{document}, which we refer to as x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{x}}$$\\end{document}-parking function polytopes. We explore connections between these x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{x}}$$\\end{document}-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{x}}$$\\end{document}-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

Future Public Parks: Integrating Facilities for Locative Augmented Reality Games

Contemporary youth culture has caused public parks to be used as locations for them to play Locative Augmented Reality Games (LARG). Nevertheless, the existing public parks were not designed for LARG, which requires wide space for youth to have rigorous physical activities and movement. Therefore, there is a need to integrate facilities for players of LARG in the development of future public parks. This research aims to evaluate the characteristics and components of public parks often chosen by youngsters to play locative augmented reality games and to recommend additional facilities to be integrated into future public parks for all-inclusiveness. This research uses Vlog and On-site observation to evaluate components of public parks. It also employs an open-ended questionnaire survey for collecting primary data on needed additional facilities for future public parks. For reason of Movement Control Order (MCO) during the Covid19 pandemic, the research could evaluate only three (3) public parks that are located in Kuala Lumpur, Malaysia. The finding shows that public parks need design improvement to integrate spaces and facilities for Locative Augmented Reality games players (LARG). The research implies that the function of public parks evolves based on demands from visitors. This research indicates that public parks should be designed to cater for digital recreational activities, triggered by contemporary youth culture and relevant to the digital future.

Optimization of wastewater treatment systems for growing industrial parks

Wastewater treatment is one of the crucial functions of industrial parks as wastewater from industrial facilities usually contains toxic compounds that can cause damage to the environment. To control their environmental loads, industrial parks make investment decisions for wastewater treatment plants. For this, they need to consider technical and economic factors as well as future growth projections as substantial construction and operational costs of wastewater treatment plants have to be shared by all companies in an industrial park. In this paper, we consider the long-term capacity planning problem for wastewater treatment facilities of a stochastically growing industrial park. By explicitly modeling randomness in the arrival of new tenants and their random wastewater discharges, our model calculates the future mean and variance of wastewater flow in the industrial park. Mean and variance are used in a Mixed Integer Programming Model to optimize wastewater treatment plant selection over a long planning horizon (30 years). By fitting our first model to empirical data from an industrial park in Turkey, we find that considering the variance of wastewater load is critical for long-term planning. Also, we quantify the economic significance of lowering wastewater discharges which can be achieved by water recycling or interplant water exchange.

Developing a Pocket Park Prescription Program for Human Restoration: An Approach That Encourages Both People and the Environment.

Healing through nature has long been confirmed as an efficient way to improve human physical and psychological health in contemporary urban life. This concept evolved into the well-known Park Prescription Program. However, the psychological restoration imparted by nature was not particularly emphasized in the original Park Prescription Program; it primarily addresses the regulation of physical activities. The quality of urban parks may affect how well people pursue these prescriptions, but the program rarely includes designers among its stakeholders. This study is inspired by the Park Prescription Program, and its intent is to develop a Pocket Park Prescription Program that encourages usage by active people and proper landscape design. The inclusion of designers has been found to be extremely important for pocket parks since they are limited in size but have the advantages of high flexibility and accessibility, and their restorative potential needs be maximized with the limited resources available. Ten pocket parks with distinct functional and landscape attributes were selected in Shanghai as research sites. The Restorative Component Scale was designed into a questionnaire-based survey to measure how people perceived restorative experiences in each site. The onsite survey also incorporated questions investigating people's behaviour characteristics of using these pocket parks. Site photos were taken and analysed with semantic image segmentation to indicate the landscape compositions of each site. The relations between people perceived restorativeness and parks' using patterns and landscape characteristics were then explored with correlation analysis to provide cues on instructing how people's visiting behavior and park landscape design can be improved. The results suggest there are better restoration results when people stay longer in pocket parks, and when people visit their neighborhood pocket parks two to three times a week, these benefits are further enhanced. This study also found that when these prescribed health behaviors are uncertain, the restorative experiences perceived by people can be improved with design interventions in regard to landscape elements such as vegetation, person, decorative lamps, pavement and terrain. These interventions should be made also in consideration of specific park functions. The research outcome intends to show that designers should be included as stakeholders in the Pocket Park Prescription Program, and it is expected to guide individuals towards effectively using pocket parks for restoration. This is to ensure that both the design and the people's perspectives will be strengthened through the implementation of this program.

Probabilistic Parking Functions

We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodríguez. Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of special interest is when $p=1/2$, where we demonstrate a sharp transition in some parking statistics. We also present several interesting combinatorial consequences of the parking protocol. In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots. Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.

The brownfield revival: experts’ perceptions on the decommissioning of San Juan Memorial Park, La Union, Philippines

The public’s demand for more open spaces as brought about by the effects of the pandemic has resulted in increased interest in an existing brownfield in La Union. The San Juan Municipal Cemetery is located along the National Highway, near the existing tourism destination of the municipality. Its presence has raised concerns about it being a visual blight as it is visible from the highway. According to the Comprehensive Land-Use Plan (CLUP) of the Municipality of San Juan, La Union (SJLA), the existing Municipal (People’s) Park is 11,500 square meters (sqm). The 2022 Area Requirement for SJLU’s recreational spaces is estimated at 30,907 square meters (sqm). However, the lack of recreational function and facilities of the San Juan Municipal Cemetery may lead to its removal from the total open space for recreational spaces, resulting in the reduction of the total area to only 19,407 square meters (sqm). This may affect the capacity of the municipality to provide for the residents’ recreational needs, forcing the residents to travel to other cities for their recreational needs. This leads to the examination of the function of memorial parks in their everyday function, their significance in the economic development, environmental risks, and tourism value for the residents. Using a descriptive methodology in gathering insights from the relevant Experts on culture, environment, culture, and tourism, the research found the necessity of converting this memorial park into a culturally-relevant space that reflects the identity of the residents while at the same time improving the tourism image of the municipality.

Research on the Healing Potential of Urban Parks from the Perspective of Audio-Visual Integration: A Case Study of Five Urban Parks in Chengdu

In recent decades, rapid urbanization has been linked to negative impacts on people’s mental health. However, the healing potential of urban parks as central features of cityscapes has not been fully explored from the perspective of audio-visual integration. This gap limits designers’ ability to leverage parks’ healing systems to their full potential. To address this gap, this study used the Mindwave monitoring and recovery component scale to evaluate the healing function of urban parks in Chengdu, specifically focusing on audio-visual integration. Based on audio-visual scene combination samples collected through shooting and recording, we found that: (1) The visual and audio qualities of birdsong (the sky above the blue space and the green space) can significantly differ. (2) Birdsong and conversation seem to provide healing effects from seemingly contradictory dimensions of “quiet” and “social”, with gender differences as the primary influencing factor. (3) Visible children laughing at low levels (30% and 50%) has a more negative impact than invisible traffic noise at the same sound pressure level. (4) Audio-visual interaction does not always have a positive effect, with the visibility of the sound source as the primary influencing factor. (5) An increase in audio proportion did not necessarily correspond to a linear increase in the healing effect. Through exploring the influence of audio-visual combination scenes on healing effects in urban parks, this study provides an empirical basis for park design and planning that considers audio-visual healing effects. It offers insights into designing healing systems for parks and urban areas as well.

To Evaluate the Recreational Functions of Botanical Parks with User Participation

Kentsel açık yeşil alanların vazgeçilemez unsurlarından birisi olan parklar, insanların fiziksel ve ruhsal açıdan rahatlamalarını sağlamakla birlikte rekreasyon ihtiyaçlarını karşılamaktadır. Kentlerin imajı güçlendiren ve yaşam kalitesini yükselten parklar, sosyal ve ekonomik açıdan da yarar sağlarlar. Parklar içerisinde yer alan botanik parklar insanlara bitkisel çeşitliliği tanıtarak farkındalık sağlayan kentsel açık yeşil alanlar olarak hizmet etmekle birlikte hem kent ekosistemine katkı sağlamaları hem de rekreasyonel aktivitelere olanak yaratmaları açısından önemli bir konumdadırlar. Bu çalışmada Türkiye’nin en önemli Metropol kenti olan İstanbul’un Bakırköy ilçe sınırları içerisinde yer alan Bakırköy Botanik Parkının rekreasyonel işlevleri kullanıcı katılımı ile değerlendirilmiştir. Çalışmada anket yöntemi uygulanmıştır. Anket çalışmasında kullanıcılar Bakırköy Botanik Parkını bir kent parkı olarak algıladıkları belirtmiş olup çoğunlukla gençlerin (26-35 yaş) parkı ziyaret ettiği ve 1-3 saat zaman geçirdikleri belirlenmiştir. Konfor ve imaj açısından kalabalıktan uzaklaşmak (%78) için gelinen bir park olması yanı sıra, çocuk oyun alanlarının varlığının önemli (%66,80) olduğu görülmüştür. Parkın donatı elemanları ve bakım yönünden bir takım eksiklikleri bulunmakla birlikte eğitim işlevini yeterince yerine getiremediği görülmüştür. Sonuç olarak, Bakırköy Botanik Parkı’nın hem rekrasyonel hem de eğitsel işlevlerini iyi bir biçimde sürdürebilmesi için gereken düzenlemelerin yapılması gerekliliği yönünde çözümler önerilmiştir.

The Number of Prime Parking Functions

The Number of Prime Parking Functions

Generalized Riemann Functions, Their Weights, and the Complete Graph

By a Riemann function we mean a function $f:\mathbb{Z}^n\to\mathbb{Z}$ such that $f(\mathbf{d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$, for $d_1+\cdots+d_n$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions.
 To each Riemann function we associate a related function $W: \mathbb{Z}^n\to \mathbb{Z}$ via Möbius inversion that we call the weight of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function $f$ satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for complete graphs.
 For complete graphs, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to a new linear time algorithm to compute the Baker-Norine rank, via a new formula likely related to one of Cori and Le Borgne, but seemingly simpler for general $\mathbf{d}\in \mathbb{Z}^n$, namely$$r_{{\rm BN},K_n}(\mathbf{d}) =-1+\biggl| \biggl\{ i=0,\ldots,\deg(\mathbf{d}) \ \Bigm|\ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le \deg(\mathbf{d})-i\biggr\} \biggr|.$$However, the formula of Cori and Le Borgne, which requires $\mathbf{d}\in \mathbb{Z}^n$ to be a sorted parking function, is easier to evaluate for such $\mathbf{d}$.
 Our study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions.

Research on Art Design of Urban Wetland Park Planning Based on the Concept of Sustainable Development

Abstract With the development of the social economy, the scale of cities has expanded rapidly, and the urban ecological imbalance has become increasingly severe, which has caused many urban problems. People gradually realise that cities and human beings develop more harmoniously and healthily by building and developing cities on the basis of natural ecology and seeking the harmonious coexistence of ecology and humanity. The results of the sustainable landscape ecological evaluation of Anhui Shilonghu National Wetland Park show that the park has good ecological, social, and economic benefits and functions. The urban ecological environment and social and economic development have a good supporting effect on the park, and its overall evaluation grade is good. Compared with other waterscape parks, the ecological function of urban wetland parks is the most basic and essential function of the landscape. It has multiple functions, such as popular science education and being close to nature and entertainment. Developing wetland ecological experience tourism activities benefits wetland protection and ecologically coordinated, sustainable development. It can also comprehensively improve wetland parks’ ecological, social, and economic benefits.

On Parking Functions and The Tower of Hanoi

The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length n with displacement one is in bijection with the set of ideal states in the famous Tower of Hanoi game with n + 1 disks and n + 1 pegs, both sets being enumerated by the Lah numbers.

Relieve Fatigue by Socializing in Sidoarjo City Park

City parks are one of the green open public spaces. The development of city parks gives meaning to the community because city people who are busy working very rarely interact. The existence of a city park as a public space provides movement for the community to be able to interact with other individuals. In Sidoarjo, there is a City Park called Abhirama Park. The Park is strategically located with easy access. This research aims to analyze the development of city parks as a space for interaction in the Sidoarjo community. The research results are regarding the situation of the Sidoarjo city park, as for the views of the people of Sidoarjo regarding the existence of parks and the various functions of city parks. The method used is qualitative, with data collection through interviews, observation, and documentation. This study uses the theory of Herbert Mead regarding symbolic interaction.

Parking Functions, Multi-shuffle, and Asymptotic Phenomena

Given a positive integer-valued vector $${\textbf{u}}=(u_1, \dots , u_m)$$ with $$u_1<\cdots <u_m$$ , a $${\textbf{u}}$$ -parking function of length m is a sequence $$\varvec{\pi }=(\pi _1, \dots , \pi _m)$$ of positive integers whose non-decreasing rearrangement $$(\lambda _1, \dots , \lambda _m)$$ satisfies $$\lambda _i\le u_i$$ for all $$1\le i\le m$$ . We introduce a combinatorial construction termed a parking function multi-shuffle to generic $${\textbf{u}}$$ -parking functions and obtain an explicit characterization of multiple parking coordinates. As an application, we derive various asymptotic probabilistic properties of a uniform $${\textbf{u}}$$ -parking function of length m when $$u_i=cm+ib$$ . The asymptotic scenario in the generic situation $$c>0$$ is in sharp contrast with that of the special situation $$c=0$$ .

VARIABLES AND INDICATORS TO MEASURE THE PERFORMANCE OF SUSTAINABLE CONSTRUCTION PROJECT MANAGEMENT OF CITY PARK

City people need a quality city park. It makes the city park development project anticipate the needs of the community and the environment. Currently, city parks in Jakarta aren’t qualified, because the gaps in the management of city park construction projects. The research aim is to provide a set of variables and indicators that influence the pre-construction stage of managing a sustainable City Park construction project to measurement city park project cost performance and quality, as a reference for increasing the achievement of city park functions. This study applies a combination of Soft System Methodology (SSM) and Hard System Methodology (HSM) with Focus Group Discussion (FGD) and questionnaires. This research is consultant's point of view pilot project. The research results are a set of variables and indicators for performance measurement as a basis for developing a model of sustainable city park construction projects management in future research.

Asymptotic behaviour of the first positions of uniform parking functions

AbstractIn this paper we study the asymptotic behaviour of a random uniform parking function $\pi_n$ of size n. We show that the first $k_n$ places $\pi_n(1),\ldots,\pi_n(k_n)$ of $\pi_n$ are asymptotically independent and identically distributed (i.i.d.) and uniform on $\{1,2,\ldots,n\}$ , for the total variation distance when $k_n = {\rm{o}}(\sqrt{n})$ , and for the Kolmogorov distance when $k_n={\rm{o}}(n)$ , improving results of Diaconis and Hicks. Moreover, we give bounds for the rate of convergence, as well as limit theorems for certain statistics such as the sum or the maximum of the first $k_n$ parking places. The main tool is a reformulation using conditioned random walks.

A symmetry on parking functions via Dyck paths

This paper proves a symmetric joint distribution on parking functions via constructing two involutions on Dyck paths. The second involution is built with the aid of plane trees, leading to the discovery of two natural symmetric pairs on rooted labeled trees.