• 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 IV
Combinatorial theory of orthogonal polynomials and continued fractions

The Institute of Mathematical Sciences, Chennai, India  (January-March 2019)

Lectures related to the course

Laguerre heaps of segments for the PASEP
algebra and combinatorics seminar IISc, Bangalore, March 8, 2019
slides (pdf, 29 Mo)
abstract:
The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems strongly related to the moments of some classical orthogonal polynomials (Hermite, Laguerre, Askey-Wilson). The partition function has been interpreted with various combinatorial objects such as permutations, alternative and tree-like tableaux, etc. We introduce a new one called "Laguerre heaps of segments", which seems to play a central role in the network of bijections relating all these interpretations.
A survey of the combinatorial theory of orthogonal polynomials and continued fractions.
colloquium IISc, Bangalore, March 7, 2019
slides (pdf, 23 Mo)
abstract: 
The theory of orthogonal polynomials started with analytic continued fractions going back to Euler, Gauss, Jacobi, Stieljes ... The combinatorial interpretations started in the late 70's and is a an active research domain. I will give the basis of the theory interpreting the moments of general (formal) orthogonal polynomials, Jacobi continued fractions and Hankel determinants with some families of weighted paths. In a second part I will give some examples of interpretations of classical orthogonal polynomials and of their moments (Hermite, Laguerre, Jacobi, ...) with their connection to theoretical physics.
Proofs without words: the example of the Ramanujan continued fraction
 colloquium IMSc, Chennai, February 21, 2019
slides (pdf, 28 Mo)   
video  link to Ekalavya  (IMSc Media Center)
video  link to YouTube  (from IMSc Matsciencechanel Playlist)
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. 

Combinatorial aspects of continued frations and applications

Colloque PFAC, "Philippe Flajolet and analytic combinatorics", Paris, 15 December 2011
slides  (pdf, 20 Mo)
abstract
In his 1980 seminal paper, Philippe Flajolet stated a fundamental theorem interpreting any analytic continued fractions in terms of certain weighted lattice paths (the so-called Motzkin paths). This theory is equivalent to give an interpretation of the moments of any family of orthogonal polynomials in term of these weighted paths. Combining this general statement with some specific bijections between classical combinatorial objects and the so-called "histories" related to weighted Motzkin paths, Flajolet deduce several combinatorial proofs for many continued fraction expansions of well known power series. In particular the classical "Françon-Viennot bijection" between permutations and "Laguerre histories" play a key role and give rise to a combinatorial theory of the Sheffer class of orthogonal polynomials (i.e. Hermite, Laguerre, Charlier, Meixner I and II). 
Using Françon's concept of "data histories", a spectacular application of this combinatorial theory of continued fraction has been made by P. Flajolet (with J. Françon and J. Vuillemin) for the evaluation of the integrated cost of data structures subject to arbitrary sequences of insert, delete and search operations. Each classical data structures is related to a classical family of Sheffer type polynomials.
Extensions of the theory has been made by E. Roblet for Padé approximants and T-fractions. Further combinatorial developments have been made more recently by P. Flajolet with E. van Fossen Conrad and R. Bacher for continued fractions expansions of some elliptic functions. Some recent applications are also related to physics with some discrete integrable systems and positivity in cluster algebras involving continued fraction rearrangements (P. Di Francesco and R. Kedem) or the work of J. Bouttier and E. Guitter relating random planar maps and continued fractions. One of the last paper of Philippe (with P. Blasiak) connects continued fractions with the normal ordering of creation-annihilation operators in quantum physics.