• 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)

Some possible complementary chapters

Some possible complementary chapters:
Chapter 7    Linearization coefficients

Linearization coefficients of the five Scheffer orthogonal polynlomials, combinatorial interpretation
Linearization coefficiens of q-Hermite, q-Charlier and q-Laguerre polynomials

Chapter 8    Operators, quadratic algebra and orthogonal polynomials

q-Hermite polynomials and the (Weyl-Heisenberg) quadratic algebra defined by UD = qDU + Id
Rook placements, q-Hermite, operators U,D and Al-Salam--Chihara polynomials
q-Laguerre polynomials and the (PASEP) quadratic algebra defined by DE = qED + E + D

Chapter 9    Applications and interactions

The birth and death process in probability theory revisited with the combinatorics of orthogonal polynomials
Computing integrated cost for data structures in computer science:
stack,  priority queue, dictionnary, linear list, symbol table
The PASEP model in physics and its matrix ansatz
Orthogonal polynomials and Smith normal form

Chapter 10    Extensions

Biorthogonality
L-fractions, extension of the matrix inversion theorem between orthogonal polynomials and inverse (= vertical) polynomials.
multicontinued fractions, T-fractions,  tree-like continued fractions, some examples
Combinatorial theory of Padé approximants and P-fractions