Cyril Banderier

Generating functions for generating trees

Generating trees describe conveniently certain families of combinatorial objects: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) while providing efficient random generation algorithms. In this paper, we investigate the relationship between structural properties of the rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating functions.

Basic analytic combinatorics of directed lattice paths

This paper develops a unified enumerative and asymptotic theory of directed two-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are specified by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially one-dimensional objects.) The theory relies on a specific “kernel method” that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.

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 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.

Smoothed Analysis of Three Combinatorial Problems

Smoothed analysis combines elements over worst-case and average case analysis. For an instance x, the smoothed complexity is the average complexity of an instance obtained from x by a perturbation. The smoothed complexity of a problem is the worst smoothed complexity of any instance. Spielman and Teng introduced this notion for continuous problems. We apply the concept to combinatorial problems and study the smoothed complexity of three classical discrete problems: quicksort, left-to-right maxima counting, and shortest paths.

Symplectic transformations and quantum tomography in finite quantum systems

There is difference between coefficients of thermal expansion of the first member and the second member. A mismatch occurs between the walls of the first member and the second member when the temperature is changed significantly. Stress generated by this mismatch is distributed among and received by respective fitting sections of respective concave portions and respective convex portions. Because these fitting sections exist above and below the joining member, stress generated by the mismatch is distributed and received by the fitting sections before the stress reaches the joining member. Therefore, only a little stress acts on the joining member.

Enumeration and asymptotics of restricted compositions having the same number of parts

We study pairs and m-tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of parts. Under the uniform probability model, we obtain the asymptotics for the probability that two or, more generally, m randomly and independently chosen compositions of n have the same number of parts. For a large class of compositions, we show how a nice interplay between complex analysis and probability theory allows to get full asymptotics for this probability. Our results extend an earlier work of Bona and Knopfmacher. While we restrict our attention to compositions, our approach is also of interest for tuples of other combinatorial structures having the same number of parts.

Formulae and Asymptotics for Coefficients of Algebraic Functions

We study the coefficients of algebraic functions ∑ n ≥0 f n z n . First, we recall the too-little-known fact that these coefficients f n always admit a closed form. Then we study their asymptotics, known to be of the type f n ~ CA n n α . When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values for A . We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar ( i.e. , their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).

A Generalized Cover Time for Random Walks on Graphs

Given a random walk on a graph, the cover time is the first time (number of steps) that every vertex has been hit (covered) by the walk. Define the marking time for the walk as follows. When the walk reaches vertex vi, a coin is flipped and with probability pi the vertex is marked (or colored). We study the time that every vertex is marked. (When all the pi’s are equal to 1, this gives the usual cover time problem.) General formulas are given for the marking time of a graph. Connections are made with the generalized coupon collector’s problem. Asymptotics for small p i ’s are given. Techniques used include combinatorics of random walks, theory of determinants, analysis and probabilistic considerations.

Analysis of an Exhaustive Search Algorithm in Random Graphs and the

We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual

n^{c\log n}

\mathscr{G}_{n,p}