Lattice paths of slope 2/5

We analyze some enumerative and asymptotic properties of Dyck paths under a line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014.Our approach relies on the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths. A key ingredient in the proof is the generalization of an old trick of Knuth himself (for enumerating permutations sortable by a stack),promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic,and they offer some new combinatorial identities, which can be also tackled in the A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar results for other slopes than 2/5, an interesting case being e.g. Dyck paths below the slope 2/3, which corresponds to the so called Duchon's club model.Comment: Robert Sedgewick and Mark Daniel Ward. Analytic Algorithmics and Combinatorics (ANALCO)2015, Jan 2015, San Diego, United States. SIAM, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), eISBN 978-1-61197-376-1, pp.105-113, 2015, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO

Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)

In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and Algebraic Combinatorics (Vancouver, 2004), electronic proceeding

Du Bartas et l’ars memorativa

Malgré les efforts entrepris depuis plusieurs décennies, l’oeuvre de Guillaume Salluste Du Bartas (1544-1590) est loin d’avoir livré tous ses secrets. De manière générale, son importance semble encore sous-évaluée, alors que cet auteur connut, de son vivant et jusqu’à quarante ans après sa mort, une gloire et un rayonnement exceptionnels. Afin de mieux éclairer les raisons de ce succès, cet article étudie La Sepmaine comme ressortissant à la tradition de l’ars memorativa. Le succès de ce poème à la fois didactique, apologétique et encyclopédique correspond, en effet, à la période de l’histoire occidentale où la mnémotechnie, héritée de l’Antiquité grecque, suscite un vif regain d’intérêt. Aussi l’auteur propose-t-il d’analyser tout particulièrement la place et la fonction de la mnémotechnique dans ce texte, ainsi que les divers moyens rhétoriques dont il se sert.Despite efforts undertaken in many recent decades, the work of Guillaume Salluste Du Bartas (1544-1590) has by no means revealed all its secrets. Its importance, generally speaking, still appears under-estimated, whereas there were outstanding recognition and popularity for this author during his lifetime and up to forty years after his death. To shed light on the reason for this success, this article studies the Sepmaine as issuing from the tradition of the ars memorativa. The success of this poem at once didactic, apologetic and encyclopaedic corresponds, in fact, to that period of Western history when the mnemotechnics inherited from Greek antiquity experienced a lively renewal of interest. Our purpose is therefore to analyze, very particularly, the position and function of mnemotechnics in this text, as well as the various rhetorical methods it employs

Bounded discrete walks

International audienceThis article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work

On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution

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FORMULAE AND ASYMPTOTICS FOR COEFFICIENTS OF ALGEBRAIC FUNCTIONS

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