Parking spaces

Let W be a Weyl group with root lattice Q and Coxeter number h  . The elements of the finite torus Q/(h+1)Q$Q/(h+1)Q$ are called the W-parking functions, and we call the permutation representation of W on the set of W-parking functions the (standard) W-parking space. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new W-parking spaces, called the noncrossing parking space and the algebraic parking space, with the following features:

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They are defined more generally for real reflection groups.     

Parking with density

We study parkings with n places, where m(n) cars are placed according to a nonuniform probability. The aim of this paper is to show a threshold function (depending on the distribution) for the emergence of a giant component. The size of the largest blocks of consecutive occupied places and the total displacement of the cars are also studied. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 364–380, 2001.     

多目的地多停车场系统动态停车分配模型

在一个多目的地多停车场系统中,每个目的地附近有多个停车场可供选择,每个停车场也可供多个目的地的停车需求停车。每个目的地的停车需求在各停车场的停车量即为停车分配问题. 本文定义了停车量分配均衡原则:各目的地的驾驶员总是首选最低费用(包括在停车场内的停车费用以及停车时间、步行时间转换后的货币成本,其中停车费用及在停车场内的车位找寻时间都正相关于该停车场当前时刻停车饱和度)的停车场停车;由于停车场具有容量限制,当费用最低的停车场已满则再选择费用次低的停车场,以此类推;所有没被使用的停车场或者比被使用的停车场具有更高的费用或者停车位已满。考虑停车需求和各停车场内空余泊位数的实时动态特性,提出了与该均衡原则等价的数学规划模型,证明了其解的唯一性并设计了求解算法。通过两个算例对模型进行了验证,并再现了目的地附近各停车场内车位占用变化规律,从而为动态停车收费、停车选址规划等提供依据。     

Parking garages with optimal dynamics

We construct generalized polygons (??parking garages??) in which the billiard flow satisfies the Veech dichotomy, although the associated translation surface obtained from the Zemlyakov?CKatok unfolding is not a lattice surface. We also explain the difficulties in constructing a genuine polygon with these properties.     

ON THE OPTIMAL PARKING PROBLEM

In this paper,the concept of dual predictable projection is used,for the optimal parkingproblem.A strictly rigorous,simpler treatment is introduced and the optimal stopping rule is alsogiven explicitly.     

Parking Functions and Descent Algebras

We show that the notion of parkization of a word, a variant of the classical standardization, allows us to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and contains the descent algebra as a left ideal. Received March 1, 2005     

Parking functions and vertex operators

We introduce several associative algebras and families of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we define a family of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct families of vector spaces whose dimensions are Catalan numbers and Fuss–Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras.      

Expected Sums of General Parking Functions

A (u₁, u₂, . . . )-parking function of length n is a sequence (x₁, x₂, . . . , x_n) whose order statistics (the sequence (x₍₁₎, x₍₂₎, . . . , x_(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x_(i) u_(i). The Gonarov polynomials g _n (x; a₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals (a _i) Dⁱ, where (a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gonarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u_i+1 - u_i. Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35.     

Permutation Factorizations and Prime Parking Functions

Permutation factorizations and parking functions have some parallel properties. Kim and Seo exploited these parallel properties to count the number of ordered, minimal factorizations of permutations of cycle type (n) and (1, n − 1). In this paper, we use parking functions, new tree enumerations and other necessary tools, to extend the techniques of Kim and Seo to the cases (2, n − 2) and (3, n − 3). Received October 13, 2004     

Rational Parking Functions and Catalan Numbers

The “classical” parking functions, counted by the Cayley number (n+1) ^n?1, carry a natural permutation representation of the symmetric group S _n in which the number of orbits is the Catalan number \({\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}\). In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by b ^a?1, carry a permutation representation of S _a in which the number of orbits is the “rational” Catalan number \({\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}\). First, we compute the Frobenius characteristic of the S _a -module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers \({\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}\) and for the q-binomial coefficients \({{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}\). We give a bijective explanation of the division by [a+b] _q that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.     

停车设施选址的双层规划模型

为了解决城市停车难问题,针对目前日益紧张的土地资源和快速增长的交通流量,考虑停车设施对路网上交通流的影响,提出一种在土地利用规划基础上的,对路网影响最小,停车场使用者步行时间最省的停车设施选址的双层规划模型,借助现有的交通规划软件(Emme)进行求解,并应用于上海市江湾-五角场副中心地区停车设施规划研究中.     

Computable analogs of cardinal characteristics: Prediction and rearrangement

There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles. Each property yields a highness notion in the Turing degrees. In this paper we study the highness notions that result from the translation of the evasion number and its dual, the prediction number, as well as two versions of the rearrangement number. When translated appropriately, these yield four new highness notions. We will define these new notions, show some of their basic properties and place them in the computability-theoretic version of Cichoń's diagram.     

Cycle Lemma, Parking Functions and Related Multigraphs

For positive integers a and b, an ${(a, \overline{b})}$ -parking function of length n is a sequence (p ₁, . . . , p _n ) of nonnegative integers whose weakly increasing order q ₁ ≤ . . . ≤ q _n satisfies the condition q _i  < a + (i ? 1)b. In this paper, we give a new proof of the enumeration formula for ${(a, \overline{b})}$ -parking functions by using of the cycle lemma for words, which leads to some enumerative results for the ${(a, \overline{b})}$ -parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between ${(a, \overline{b})}$ -parking functions and rooted forests, we enumerate combinatorially the ${(a, \overline{b})}$ -parking functions with identical initial terms and symmetric ${(a, \overline{b})}$ -parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to ${(a, \overline{b})}$ -parking functions.     

Parking Capacity and Pricing in Park'n Ride Trips: A Continuous Equilibrium Network Design Problem

In this paper we consider the problem of designing parking facilities for park'n ride trips. We present a new continuous equilibrium network design problem to decide the capacity and fare of these parking lots at a tactical level. We assume that the parking facilities have already been located and other topological decisions have already been taken.The modeling approach proposed is mathematical programming with equilibrium constraints. In the outer optimization problem, a central Authority evaluates the performance of the transport network for each network design decision. In the inner problem a multimodal traffic assignment with combined modes, formulated as a variational inequality problem, generates the share demand for modes of transportation, and for parking facilities as a function of the design variables of the parking lots. The objective is to make optimal parking investment and pricing decisions in order to minimize the total travel cost in a subnetwork of the multimodal transportation system.We present a new development in model formulation based on the use of generalized parking link cost as a design variable.The bilevel model is solved by a simulated annealing algorithm applied to the continuous and non-negative design decision variables. Numerical tests are reported in order to illustrate the use of the model, and the ability of the approach to solve applications of moderate size.     

Parking buses in a depot with stochastic arrival times

Given buses of different types arriving at a depot during the evening, the bus parking problem consists of assigning these buses to parking slots in such a way that they can be dispatched adequately to the next morning routes without moving them between their arrivals and departures. In practice, the bus arrival times deviate stochastically from the planned schedule. In this paper, we introduce for this problem two solution approaches that produce solutions which are robust to variations in the arrival times. The first approach considers that each arrival can deviate from its planned arrival order (sooner or later) by at most k positions, where k is a predefined parameter. In the second approach, the objective aims at minimizing the expectation of a function positively correlated with the number of buses that make the planned solution infeasible because they arrive too late or too early. In both approaches, the problem is modeled as an integer linear program that can be solved by a commercial mip solver. Computational results obtained on instances derived from a real-world dataset are reported.     

顾客延时离开的共享车位预订控制策略研究

基于车位共享用户未按约定时间离开而导致车位使用时间冲突的问题,构建了随机动态规划模型。通过模型,研究了平台的预订控制策略,并针对平台能否拒绝延时需求的情形,设计两种算法对模型进行求解,最后通过数值仿真验证了算法的有效性。研究结果表明：(1)随着预订周期的增加,平台的边际收益会先增加后减少,因此存在最优预订周期;(2)若平台可以决策延时需求是否被接受,则随着延时概率的增加,机会成本将减少,而平台服务率和车位利用率将增加;(3)若平台只能被动接受延时需求,则随着延时概率的增加,平台服务率将减少,而机会成本和车位利用率将增加。