Why Delannoy numbers?

This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems. These numbers appear in probabilistic game theory, alignments of DNA sequences, tiling problems, temporal representation models, analysis of algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of Statistical Planning and Inference

Uniform random sampling of planar graphs in linear time

This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Gim\'enez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $O(n^7)$.Comment: 55 page

Combinatorics and Stochasticity for Chemical Reaction Networks

Stochastic chemical reaction networks (SCRNs) are a mathematical model which serves as a first approximation to ensembles of interacting molecules. SCRNs approximate such mixtures as always being well-mixed and consisting of a finite number of molecules, and describe their probabilistic evolution according to the law of mass-action. In this thesis, we attempt to develop a mathematical formalism based on formal power series for defining and analyzing SCRNs that was inspired by two different questions. The first question relates to the equilibrium states of systems of polymerization. Formal power series methods in this case allow us to tame the combinatorial complexity of polymer configurations as well as the infinite state space of possible mixture states. Chapter 1 presents an application of these methods to a model of polymerizing scaffolds. The second question relates to the expressive power of SCRNs as generators of stochasticity. In Chapter 2, we show that SCRNs are universal approximators of discrete distributions, even when only allowing for systems with detailed-balance. We further show that SCRNs can exactly simulate Boltzmann machines. In Chapter 3, we develop a formalism for defining the semantics of SCRNs in terms of formal power series which grew as a result of work included in the previous chapters. We use that formulation to derive expressions for the dynamics and stationary states of SCRNs. Finally, we focus on systems that satisfy complex balance and conservation of mass and derive a general expressions for their factorial moments using generating function methods

an introduction to the history of European computer science and technology

(essays and documents

Combinatoire des cartes et polynome de Tutte

Les cartes sont les plongements, sans intersection d'arêtes, des graphes dans des surfaces. Les cartes constituent une discrétisation naturelle des surfaces et apparaissent aussi bien en informatique (codage d'informations visuelles) quén physique (surfaces aléatoires de la physique statistique et quantique). Nous établissons des résultats énumératifs pour de nouvelles familles de cartes. En outre, nous définissons des bijections entre les cartes et des classes combinatoires plus simples (chemins planaires, couples d'arbres). Ces bijections révèlent des propriétés structurelles importantes des cartes et permettent leur comptage, leur codage et leur génération aléatoire. Enfin, nous caractérisons un invariant fondamental de la théorie des graphes, le polynôme de Tutte, en nous appuyant sur les cartes. Cette caractérisation permet d'établir des bijections entre plusieurs structures (arbres cou- vrant, suites de degrés, configurations du tas de sable) comptées par le polynôme de Tutte.A map is a graph together with a particular (proper) embedding in a surface. Maps are a natural way of representing discrete surfaces and as such they appear both in computer science (encoding of visual data) and in physics (random lattices of statistical physics and quantum gravity). We establish enumerative results for new classes of maps. Moreover, we define several bijections between maps and simpler combinatorial classes (planar walks, pairs of trees). These bijections highlight some important structural properties and allows one to count, sample randomly and encode maps efficiently. Lastly, we give a new characterization of an important graph invariant, the Tutte polynomial, by making use of maps. This characterization allows us to establish bijections between several structures (spanning trees, sandpile configurations, outdegree sequences) counted by the Tutte polynomial

Coalescent Theory and Yule Trees in time and space

Mathematically, Coalescent Theory describes genealogies within a population in the form of (binary) trees. The original Coalescent Model is based on population models that are evolving neutrally. With respect to graph isomorphy, the tree-structures it provides can be equivalently described in a discrete setting by the Yule Process. As a population evolves (in time), the genealogy of the population is subject to change, and so is the tree structure associated with it. A similar statement holds true if the population is assumed to be recombining; then, in space, i.e. along the genome, the genealogy of a sample may be subject to change in a similar way. The two main focuses of this thesis are the description of the processes that shape the genealogy in time and in space, making use of the relation between Coalescent and Yule Process. As for the process in time, the presented approach differs from existing ones mainly in that the population considered is strictly finite. The results we obtain are of mainly theoretical nature. In case of the process along the genome, we focus on mathematical properties of Linkage Disequilibrium, a quantity that is relevant in the analysis of population-genetical data. Similarities and differences between the two are discussed, and a possibility of performing similar analyses when the assumption of neutrality is abandoned is pointed out