Q\mathbb{Q}-bonacci words and numbers

We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer we rediscover classical multi-step Fibonacci numbers: Fibonacci, Tribonacci, Tetranacci, etc. When $q$ is not an integer, obtained recurrence relations are connected to certain restricted integer compositions. We also discuss Gray codes for these words, and a possibly novel generalization of the golden ratio.Comment: 10 pages, 2 tables, 3 figure

Asymptotics of strongly overlapping permutations

In this work, we introduce the concept of strongly (non-)overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a permutation can be represented as a sequence of non-overlapping ones. The above structural decomposition allows us to obtain equations for the corresponding generating functions, as well as the complete asymptotic expansions for the probability that a large random permutation is strongly (non-)overlapping. In particular, we show that almost all permutations are strongly non-overlapping, and that the corresponding asymptotic expansion has the self-reference property: the involved coefficients count strongly non-overlapping permutations once again. We also discuss the similarities of the introduced concept to already existing permutation building blocks, such as indecomposable and simple permutations, as well as their associated asymptotics.Comment: 9 pages, 5 figure

The complexity of deciding whether a graph admits an orientation with fixed weak diameter

International audienceAn oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems

A lattice on Dyck paths close to the Tamari lattice

We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate generating function counting the number of Dyck paths with respect to the semilength, the numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of coverings, meet and join irreducible elements. As a byproduct, we present a new involution on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges into its reverse. Finally, we give a generating function for the number of intervals, and we compare this number with the number of intervals in the Tamari lattice

Grand Dyck paths with air pockets

Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths with air pockets by allowing them to go below the $x$-axis. We present enumerative results on GDAP (or their prefixes) subject to various restrictions such as maximal/minimal height, ordinate of the last point and particular first return decomposition. In some special cases we give bijections with other known combinatorial classes.Comment: 20 pages, 4 figure

Emperical study and modelling of the internet topology dynamics

De nombreux travaux ont étudié la topologie de l’Internet, mais peu d’entre eux se sont intéressés à comment elle évolue. Nous étudions la dynamique de la topologie de routage au niveau IP et proposons une première étape vers une modélisation réaliste de cette dynamique. Nous étudions les mesures périodiques des arbres de routage à partir d’un moniteur vers un ensemble de destinations et nous observons certaines propriétés invariantes de la dynamique de leur topologie. Ensuite nous proposons un modèle simple qui simule la dynamique d’une topologie de réseau réel. En étudiant les résultats de la simulation, nous montrons que ce modèle captures les invariantes observés. De plus, l’analyse des résultats de simulations de différents types de réseaux nous permet de trouver des caractéristiques structurelles qui ont le plus grand impact sur ​la dynamique de la topologie. Nous étudions également comment la fréquence des mesures affecte la dynamique observée. Nous sommes intéressés par les processus sous-Jacents qui causent les dynamiques observées. Nous introduisons une méthode non-Classique de l'estimation des paramètres de un processus stochastique et nous appliquons cette méthode pour les mesures modélisées et réelles afin de caractériser le taux de l'évolution de la topologie. Nous montrons aussi que la dynamique de réseau est une dynamique non-Uniforme: les parties différentes du réseau peuvent avoir différentes vitesses d'évolution.Many works have studied the Internet topology, but few have investigated the question of how it evolves over time. This thesis focuses on the Internet routing IP-Level topology dynamics and offer a first step towards a realistic modeling of these dynamics. For this end we study data from periodic measurements of routing trees from a single monitor to a fixed destination set. Next we propose a simple model that simulates the dynamics of a topology real network. By studying the results of the simulation, we show this model catches some observed invariant properties of the real-World data. In addition, analysing the simulation results of different types of networks, we found several structural features that have great impact on the dynamics of the topology. We study also how the frequency of measurement affects the observed dynamics. We are interested in the underlying process causing the observed dynamics. We introduce a method non-Classical parameter estimation of a stochastic process apply this method to the real-World and modelled measures in order to characterise the rate of the topology evolution. We also show that the network have non-Uniform dynamics: different parts of the network can have different rates of change

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Q-bonacci words and numbers

International audienceWe present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer we rediscover classical multi-step Fibonacci numbers: Fibonacci, Tribonacci, Tetranacci, etc. When $q$ is not an integer, obtained recurrence relations are connected to certain restricted integer compositions. We also discuss Gray codes for these words, and a possibly novel generalization of the golden ratio