• 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)

January 10, 2019

Chapter 0     Overview of the course

slides of Ch0   (pdf  31 Mo)                
video Ch0  link to Ekalavya  (IMSc Media Centerr)
video2 Ch0  link to YouTube.  (Video link has been changed) (from IMSc Matsciencechanel Playlist)
video Ch0  link to bilibili
About the video-book "The Art of Bijective Combinatorics"
Introduction to the classical theory of orthogonal polynomials, relation with continued fractions,
the birth of the combinatorial approach  p.5    03:28
the origin:continued fractions   10    06:59
An example with the combinatorial interpretation of the Hermite polynomials   19   15:46
combinatorial interpretation of Laguerre polynomials   32   22:13
the Askey scheme of hypergeometric orthogonal polynomials   37   25:29
Formal orthogonal polynomials   40   28:02
definition of a sequence of (formal) orthogonal polynomials   43   29:52
Combinatorial theory of orthogonal polynomials   46   35:48
Favard's theorem   47   37:35
weighted Motzkin paths   49-50   39:05
linearization coefficients   54   44:20
The notion of histories   55   44:43
Hermite history, definition   61-62   48:05
bijection Hermite histories -- chord diagrams   63-79   48:56
Laguerre histories   80   51:06
the FV bijection between Laguerre histories and permutations   82   51:59
Sheffer polynomials   83   52:28
the five Sheffer orthogonal polynomials: Hermite, Laguerre, Charlier, Meixner and Meixner-Pollaczek   84, 86   53:25
Duality   88   56:34
Analytic continued fractions   1:00:52
Stieljes and Jacobi continued fractions   98, 99   1:00:57
the classical equivalence between orthogonal polynomials and Jacobi continued fractions   100   1:01:02
Flajolet fundamental Lemma   104   1:02:12
back to Euler continued fractions related to Hermite polynomials   107   1:01:31
Contraction of continued fractions   109-113   1:04:39
The quotient-difference (qd- ) algorithm   113-116   1:06:19
Hankel determinants   117   1:08:30
Ramanujan's algorithm  121   1:10:21
The same "essence" of five bijective proofs using sign reversing involutions  122-128   1:11:05
Some q-analogs of orthogonal polynomials   129   1:15:18
the scheme of basic hypergeometric orthogonal polynomials   139   1:18:31
Furthers chapters ... 141   1:19:50
Chapter 7 Linerarization coefficients   143    1:20:14
Chapter 8 Operators, quadratic algebra and orthogonal polynomials   147   1:21:04
The PASEP in physics (partialy asymmetric exclusion process)   150   1:21:36
Chapter 9 Applications and interactions   156   1:22:48
Data structures and integrated cost in computer science   157   1:23:20
Chapter 10 extensions   165   1:25:09
Padé approximants   166   1:25:19
L- fractions, T-fractions, ... 172   1:26:34
The end   177   1:29:15

The  playlist from matsciencechannel of the videos of this course is here