Combinatoire des fonctions de parking : espèces, énumération d’automates et algèbres de Hopf

This thesis comes within the scope of algebraic, bijective and enumerative combinatorics. It deals with the study of generalized parking functions following those axes. In the first part, we are interested in generalized parking as a species of combinatorial structures. We define this species from a functional equation involving the species of set sequences. We lift the cycle index serie to the non-commutative symmetric functions, express in several bases. By specialization, we obtain new enumeration formula of generalized parking and their isomorphism types. In the functional equation, the species of sets can be replaced by some other species. This defines new structures: the χ-parking tables. In particular cases with χ : m ↦ a + b(m-1), we define a bijection between the χ-parking tables and new tree structures. This defines a generalization of the C. H. Yan bijection. In the second part, we are interested in the enumeration of automata. Firstly, we construct a simple bijection between (non-initial) automata and sequences of sets. From this bijection we extract a subfamily of quasi-distinguished automata. We obtain a better upper bound of the number of minimal automata than the one of M. Domaratzki. Then we construct a new bijection between 2m^k-parking functions and (non-initial) acyclic automata over an alphabet of k symbols. From this bijection we extract, from parking function, informations about automata structures. We deduce an enumeration formula of the minimal acyclic automata.In a third part, we formalize the common technique of polynomial realization of Hopf algebras: FQSym, WQSym, PQSym, etc.. We define a notion of type of alphabet and partitioning map. We highlight some operation which stabilizes these notions. Based on this, we define two constructions of dual combinatorial Hopf algebra; and we show that they are automatically endowed of dendriform coalgebra, and #-product.As an application, we define, for every family of χ-parking functions, a generalization of the parkization. We show that this is a partitionning map if and only if χ : m ↦ 1 + b(m-1).Cette thèse se situe dans les domaines de la combinatoire algébrique, bijective et énumérative.Elle s'intéresse à l'étude des fonctions de parking généralisées suivant ces trois axes.medskip. Dans une première partie, on s'intéresse aux fonctions de parking généralisées en tant qu'espèce de structures combinatoires (théorie introduite par A.Joyal et développée F. Bergeron, G. Labelle et P.Leroux). On définit cette espèce à partir d'une équation fonctionnelle faisant intervenir l'espèce des séquences d'ensembles.On obtient un relèvement non-commutatif de la série indicatrice de cycles dans les fonctions symétriques non-commutatives, exprimé dans différentes bases.Par spécialisation, on obtient de nouvelles formules d'énumérations des fonctions de parking généralisées et de leurs types d'isomorphismes.En remplaçant l'espèce des ensembles par d'autres espèces dans l'équation fonctionnelle, on définit de nouvelles structures: les seqPF-tables de parking. Dans les cas particuliers où seqPF : m ↦ a + b(m-1), on établit une bijection entre les seqPF-tables de parking et de nouvelles structures arborescentes, généralisant la bijection de C. H.Yan entre les seqPF-fonctions de parking et les séquences de a-forêts de b-arbres. Dans une seconde partie, on s'intéresse à l'énumération d'automates. On commence par construire une bijection simple entre les automates(non-initiaux) et les séquences d'ensembles. À partir de cette bijection, on extrait la sous-famille des automates quasi-distingués (c'est-à-dire les automates pour lesquels les couples status de terminaison et fonction de transition des états sont distincts). L'énumération de ces automates quasi-distingués fournit une meilleure borne supérieure pour le nombre d'automates minimaux que celle obtenue par M.Domaratzki & textit{al}. Ensuite, on construit une nouvelle bijection entre les 2m^k-fonctions de parking et les automates acycliques (non-initiaux) sur un alphabet à k symboles. De cette dernière, on extrait, directement sur les fonctions parking, denombreuses informations de structure sur les automates, en particulier des informations liées à la minimalité. À partir de ces informations, on déduit une formule d'énumération des automates acycliques minimaux. Dans une troisième partie, on formalise la technique commune de réalisation polynomiale des algèbres de Hopf: fqsym, wqsym, pqsym, etc. Pour ceci, ondéfinit la notion de type d'alphabet et d'application partitionnante. La notion d'application partitionnante formalise les bonnes propriétés de la standardisation, le tassement, la parkisation, etc associées à ces précédentes algèbres de Hopf. On montre que certaines opérations, produit cartésien, coloration, union ouencore intersection, stabilisent ces notions. À partir de celles-ci, on définit deux constructions d'algèbres de Hopf combinatoire en dualité; et l'on montre qu'elles sont automatiquement munies de structures d'algèbres dendriformes et du produit #. En guise d'applications, on définit, pour toute famille de seqPF-fonctions deparking, une application généralisant la parkisation. On montre que cette dernière est une application partitionnante si et seulement si seqPF : n ↦ 1 + m(n-1). Ceci permet de retrouver les algèbres de Hopf sur les m-fonctions de parking généralisées de J.-C. Novelli et J-.Y.Thibon

Uncertainty in Multi-Commodity Routing Networks: When does it help?

We study the equilibrium behavior in a multi-commodity selfish routing game with many types of uncertain users where each user over- or under-estimates their congestion costs by a multiplicative factor. Surprisingly, we find that uncertainties in different directions have qualitatively distinct impacts on equilibria. Namely, contrary to the usual notion that uncertainty increases inefficiencies, network congestion actually decreases when users over-estimate their costs. On the other hand, under-estimation of costs leads to increased congestion. We apply these results to urban transportation networks, where drivers have different estimates about the cost of congestion. In light of the dynamic pricing policies aimed at tackling congestion, our results indicate that users' perception of these prices can significantly impact the policy's efficacy, and "caution in the face of uncertainty" leads to favorable network conditions.Comment: Currently under revie

Parking Spaces

Let $W$ be a Weyl group with root lattice $Q$ and Coxeter number $h$. The elements of the finite torus $Q/(h+1)Q$ are called the $W$-{\sf parking functions}, and we call the permutation representation of $W$ on the set of $W$-parking functions the (standard) $W$-{\sf 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 {\sf noncrossing parking space} and the {\sf algebraic parking space}, with the following features: 1) They are defined more generally for real reflection groups. 2) They carry not just $W$-actions, but $W\times C$-actions, where $C$ is the cyclic subgroup of $W$ generated by a Coxeter element. 3) In the crystallographic case, both are isomorphic to the standard $W$-parking space. Our Main Conjecture is that the two new parking spaces are isomorphic to each other as permutation representations of $W\times C$. This conjecture ties together several threads in the Catalan combinatorics of finite reflection groups. We provide evidence for the conjecture, proofs of some special cases, and suggest further directions for the theory.Comment: 49 pages, 10 figures, Version to appear in Advances in Mathematic

Optimisation of policies for transport integration in metropolitan areas: report on work packages 30 and 40

The overall objectives of Project OPTIMA are:- (i) to identify optimal urban transport strategies for a range of urban areas within the EU; (ii) to compare the strategies which are specified as optimal in different cities, and to assess the reasons for these differences; (iii) to assess the acceptability and feasibility of implementation of these strategies both in the case study cities and more widely in the EU, and (iv) to use the results to provide more general guidance on urban transport policy within the EU. There is a wide range of objectives of transport policy in urban areas, but most can be grouped under the broad headings of economic efficiency, including economic development, on the one hand, and sustainability, including environment, safety, equity and quality of life, on the other. It is now generally accepted that the overall strategy for achieving these objectives must include an element of reduction of private car use and transfer of travel to other modes. The policy instruments for achieving these objectives can include infrastructure provision, management measures to enhance other modes and to restrict car use, and pricing measures to make public transport more attractive and to increase the marginal cost of car use. It is now widely accepted that the most appropriate strategy will involve several of these measures, combined in an integrated way which emphasises the synergy between them. The most appropriate strategy for a city will depend on its size, the current built form, topography, transport infrastructure and patterns of use; levels of car ownership, congestion and projected growth in travel; transport policy instruments already in use; and the acceptability of other measures in political and legislative terms. These will differ from city to city. Policy advice cannot therefore be generalised, but must be developed for a range of different types of city. This is the approach adopted in this study, in which nine different cities in five countries (Edinburgh, Merseyside, Vienna, Eisenstadt, Trams@, Oslo, Helsinki, Torino and Salerno) have been studied in detail, using a common study methodology. This report summarises the output of two work packages in OPTIMA: WP30: Test Combinations of Policy Instruments WP40: Identify Optim

A vision-guided parallel parking system for a mobile robot using approximate policy iteration

Reinforcement Learning (RL) methods enable autonomous robots to learn skills from scratch by interacting with the environment. However, reinforcement learning can be very time consuming. This paper focuses on accelerating the reinforcement learning process on a mobile robot in an unknown environment. The presented algorithm is based on approximate policy iteration with a continuous state space and a fixed number of actions. The action-value function is represented by a weighted combination of basis functions. Furthermore, a complexity analysis is provided to show that the implemented approach is guaranteed to converge on an optimal policy with less computational time. A parallel parking task is selected for testing purposes. In the experiments, the efficiency of the proposed approach is demonstrated and analyzed through a set of simulated and real robot experiments, with comparison drawn from two well known algorithms (Dyna-Q and Q-learning)