On the number of lattice convex chains

On the number of lattice convex chains, Discrete Analysis 2016:19, 15pp. A _lattice convex polygon_ is a convex polygon whose vertices have integer coordinates. This simple definition leads quickly to some very interesting questions that have been studied by several authors. For example, one can ask for an asymptotic formula for the number of lattice convex polygons with vertices in the set $\{1,2,\dots,n\}^2$ when $n$ is large. One can also ask what the typical shape of such a polygon is: in particular, is there a limiting shape that a random lattice convex polygon in an $n\times n$ box approaches with high probability? The answer to this last question was shown to be yes by Barany, Vershik, and Sinai (independently): a typical lattice convex polygon in an $n\times n$ box is (approximately) made out of four parabolic arcs, each one tangent to two adjacent sides of the box at their midpoints. Given the form of this answer, it is not surprising that what really matters for this analysis is the shape of a piecewise linear convex function $f:[0,n]\to [0,n]$ such that $f(0)=0$, $f(n)=n$, and the only points at which the gradient changes have integer coordinates. Equivalently, one is interested in sequences $(x_0,y_0), (x_1,y_1),\dots,(x_n,y_n)$ of points in $\{0,1,\dots,n\}^2$ with $(x_0,y_0)=(0,0)$, $(x_n,y_n)=(n,n)$, and $(y_i-y_{i-1})/(x_i-x_{i-1})$ strictly increasing with $i$. These are called _lattice convex chains_. Barany, Vershik and Sinai proved that the number of lattice convex chains is $\exp(3\kappa^{1/3}n^{2/3}(1+o(1)))$, where $\kappa=\zeta(3)/\zeta(2)$. Here $\zeta$ is the Riemann zeta function. This paper obtains a much more precise version of the above formula, which is too complicated to give here, which gives the correct result up to a factor $1+o(1)$. (This is of course much stronger than having the $1+o(1)$ inside the exponential.) The authors achieve this by using a statistical-mechanical model developed by Sinai and analysing its partition function with the help of an integral representation that they have discovered. However, their formula involves a sum over zeros of the zeta function, and it is not easy to estimate its magnitude. Assuming the Riemann hypothesis, the rough order of this term can be given. The authors also show the converse: that is, a suitable estimate for the magnitude of this term would imply the Riemann hypothesis

Computational aspects of tomographic and neuroscientific problems

4siopenopenBrunetti, S.; Dulio, P.; Frosini, A.; Rozenberg, G.Brunetti, S.; Dulio, P.; Frosini, A.; Rozenberg, G

Constructions par greffe, combinatoire analytique et génération analytique

Analytic combinatorics is a field which consist in applying methods from complex ana- lysis to combinatorial classes in order to obtain results on their asymptotic properties. We use for that specifications, which are a way to formalise the (often recursive) structure of the objects. In this thesis, we mainly devote ourselves to find new specifications for some combinatorial classes, in order to then apply more effective enumerative or random sampling methods. Indeed, for one combinatorial class several different specifications, based on different decompositions, may exist, making the classical methods - of asymptotic enu- meration or random sampling - more or less adapted. The first set of presented results focuses on Rémy’s algorithm and its underlying holonomic specification, based on a grafting operator. We develop a new and more efficient random sampler of binary trees and a random sampler of Motzkin trees based on the same principle. We then address some question relative to the study of subclasses of λ-terms. Finally, we present two other sets of results, on automatic specification of trees where occurrences of a given pattern are marked and on the asymptotic behaviour and the random sampling of digitally convex polyominoes. In every case, the new specifications give access to methods which could not be applied previously and lead to numerous new results.La combinatoire analytique est un domaine qui consiste à appliquer des méthodes issues de l’analyse complexe à des classes combinatoires afin d’obtenir des résultats sur leurs propriétés asymptotiques. On utilise pour cela des spécifications, qui sont une manière de formaliser la structure (souvent récursive) des objets. Dans cette thèse, nous nous attachons principalement à trouver des nouvelles spécifications pour certaines classes combinatoires, afin de pouvoir ensuite y appliquer des méthodes efficaces d’énumération ou de génération aléatoire. En effet, pour une même classe combinatoire il peut exister différentes spécifications, basées sur des décompositions différentes, rendant les méthodes classiques d’énumération asymptotique et de génération aléatoire plus ou moins adaptées. Le premier volet de résultats présentés concerne l’algorithme de Rémy et la spécification holonome qui y est sous-jacente, basée sur un opérateur de greffe. On y développe un nouvel algorithme, plus efficace, de génération aléatoire d’arbres binaires et un générateur aléatoire d’arbres de Motzkin basé sur le même principe. Nous abordons ensuite des questions relatives à l’étude de sous-classes de λ-termes. Enfin, nous présentons deux autres ensembles de résultats, sur la spécification automatique d’arbres où les occurrences d’un motif donné sont marquées et sur le comportement asymptotique et la génération aléatoire de polyominos digitalement convexes. Dans tous les cas, les nouvelles spécifications obtenues donnent accès à des méthodes qui ne pouvaient pas être utilisées jusque là et nous permettent d’obtenir de nombreux nouveaux résultats

Space programs summary no. 37-49, volume 3 for the period December 1, 1967 to January 30, 1968. Supporting research and advanced development

Space program research projects on systems analysis and engineering, telecommunications, guidance and control, propulsion, and data system

Asymptotic Analysis and Random Sampling of Digitally Convex Polyominoes

Recent work of Brlek et al. gives a characterization of digitally convex polyominoes using combinatorics on words. From this work, we derive a combinatorial symbolic description of digitally convex polyominoes and use it to analyze their limit properties and build a uniform sampler. Experimentally, our sampler shows a limit shape for large digitally convex polyominoes