• XAVIER VIENNOT
  • Foreword
    • Preface
    • Introduction
    • Acknowledgements
    • Lectures for wide audience
  • PART I
    • Preface
    • Abstract
    • Contents
    • Ch0 Introduction to the course
    • Ch1 Ordinary generating functions
    • Ch2 The Catalan garden
    • Ch3 Exponential structures and genarating functions
    • Ch4 The n! garden
    • Ch5 Tilings, determinants and non-intersecting paths
    • Lectures related to the course
    • List of bijections
    • Index
  • PART II
    • Preface
    • Abstract
    • Contents
    • Ch1 Commutations and heaps of pieces: basic definitions
    • Ch2 Generating functions of heaps of pieces
    • Ch3 Heaps and paths, flows and rearrangements monoids
    • Ch4 Linear algebra revisited with heaps of pieces
    • Ch5 Heaps and algebraic graph theory
    • Ch6 Heaps and Coxeter groups
    • Ch7 Heaps in statistical mechanics
    • Lectures related to the course
  • PART III
    • Preface
    • Abstract
    • Contents
    • Ch0 overview of the course
    • Ch1 RSK The Robinson-Schensted-Knuth correspondence
    • Ch2 Quadratic algebra, Q-tableaux and planar automata
    • Ch3 Tableaux for the PASEP quadratic algebra
    • Ch4 Trees and tableaux
    • Ch5 Tableaux and orthogonal polynomials
    • Ch6 Extensions: tableaux for the 2-PASEP quadratic algebra
    • Lectures related to the course
    • References, comments and historical notes
  • PART IV
    • Preface
    • Introduction
    • Contents
    • Ch0 Overview of the course
    • Ch1 Paths and moments
    • Ch2 Moments and histories
    • Ch3 Continued fractions
    • Ch4 Computation of the coefficients b(k) lambda(k)
    • Ch5 Orthogonality and exponential structures
    • Ch6 q-analogues
    • Lectures related to the course
    • Complements
    • References
  • Epilogue

The Art of Bijective Combinatorics   Part I
An introduction to enumerative, algebraic and bijective combinatorics

The Institute of Mathematical Sciences, Chennai, India  (January-March 2016)

Some lectures for a wide audience as an introduction to the Art of Bijective Combinatorics 

lecture in french
D'une lettre oubliée d'Euler (1707-1783) à la combinatoire et la physique contemporaine
(from a forgotten letter of Euler (1707-1783) to modern combinatorics and combinatorial physics)
Conférence à la BNF (bibliothèque Nationale de France), Paris,   14 Mars 2007
avec Mariette Freudenthiel et Gérard Duchamp (violons) et Marcia Pig Lagos (textes)
organisée par la SMF (Société Mathématique de France) dans le cadre de la série de conférences
"Un texte, un mathématicien"
avec le concours de la BNF, France Culture, Animath et la revue "Tangente"
 résumé, slides et vidéo avec une description détaillée de la vidéo sur la page Euler

(abstract, slides and video with a detailed description of the video on the page Euler)

Trees in various science
Institute colloquium, IIT Bombay, 29 January 2013
slides (pdf, 55Mo)     video(?) 
(similar lecture in french: trees in the stars, trees in particles of light, Nancy, 30 May 2013)

lecture in french
Des arbres dans les étoiles, des arbres dans les grains de lumière
 avec Gérard DUCHAMP (violon)  et la conteuse Marcia PIG LAGOS
Collection: Sciences et société,  Université de Nancy, Jeudi 30 Mai 2013
slides (pdf, 59 Mo)   video
(conférence analogue en anglais, IIT Bombay, 29 Janvier 2013)
abstract
 Les arbres apparaissent dans toutes les parties de l'informatique. Plus généralement des structures arborescentes sont présentes dans diverses sciences : réseaux fluviaux en hydrogéologie, molécules d'ARN, structures fractales en physique...  
Des paramètres sur les arbres venant de considérations d'optimisation en informatique se retrouvent dans ces structures fluviales ou biologiques.
 Ces "mathématiques des arbres" apparaissent dans des recherches récentes en physique théorique,
aussi bien vers "l'infini grand" avec la structure de notre "espace temps", que vers "l'infini petit" dans notre compréhension des particules élémentaires et de la lumière. C'est le moment des contes pour pouvoir imaginer cette science en mouvement . Le titre de la conférence est sa conclusion.

The birth of a new domain: combinatorial physics

colloquium IMSc, Chennai, India, 12 february 2015
slides (pdf, 42 Mo )   video
abstract
The interaction between Combinatorics and Physics is not new: the classical combinatorial solution of the Ising model for ferromagnetism goes back to the 60's. In the last 30 years, there has been a renaissance of combinatorics, especially what is called enumerative, algebraic and bijective combinatorics. Powerful combinatorial tools have been discovered, in relation with other domains of pure mathematics, and such tools appear to be useful for theoretical physics.
With some examples I will illustrate this fruitful interaction between combinatorics and physics, giving rise to a domain which can be called "Combinatorial Physics''. A new journal is born "Combinatorics, Physics and their Interactions'' in the prestigious series of the Annales of Poincare Institute in Paris. On the front page one can read "The unfolding of new ideas in physics is often tied to the development of new combinatorial methods, and conversely some problems in combinatorics have been successfully attacked using methods inspired by statistical physics or quantum field theory".

From  a  letter  of  Leonhard  Euler to  modern  researches  at  the  crossroad  of algebra, geometry,  combinatorics  and  physics
K.Madhava  Sarma  Memorial  Distinguished  Lecture, CMI (Chennai Mathematical Institute, Chennai, India), 24 February 2016
slides (pdf, 42 Mo)
abstract:
In a letter to  Goldbach in September 1751, Euler introduced the notion of triangulation of a convex polygon. These  objects are enumerated by the ubiquitous and very classical Catalan numbers, which also enumerate trees, Dyck paths and many other combinatorial objects. I will show how some considerations on these triangulations have been the starting point of works on various related topics which are nowadays hot subjects of researches . This lecture is accessible to everyone.

Proofs without words: the example of the Ramanujan continued fraction
 colloquium IMSc, Chennai, February 21, 2019 
slides (pdf, 28 Mo)   
video link to YouTube          video link to bilibili
abstract:
Visual proofs of identities were common at the Greek time, such as the Pythagoras theorem. In the same spirit, with the renaissance of combinatorics, visual proofs of much deeper identities become possible. Some identities can be interpreted at the combinatorial level, and the identity is a consequence of the construction a weight preserving bijection between the objects interpreting both sides of the identity.
 In this lecture, I will give an example involving the famous and classical Ramanujan continued fraction. The construction is based on the concept of "heaps of pieces",
which gives a spatial interpretation of the commutation monoids introduced by Cartier and Foata in 1969.