• 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)

List of bijections
Trees

bijection binary trees — complete binary trees, Ch2a  26    video
bijection binary trees — (forests of) planar trees, Ch2a  36    video
bijection binary trees — 2-colored Motzkin paths, Ch2a  90   video
bijection (complete) binary trees — Dyck paths, Ch2a  65  video  and (with violin) 70  video
bijection (complete) binary trees — triangulations  Ch2b 31  video
bijection planar trees — Dyck paths, Ch2a  92   video 
bijection planar trees — Lukasiewicz paths, Ch2a  98  video

Paths 

bijection Dyck paths — (complete) binary trees , Ch2a  65  video  and  72   video
bijection Dyck paths — planar trees, Ch2a  92  video
bijection Dyck paths — 2-colored Motzkin paths, Ch2a  55  video
bijection 2-colored Motzkin paths — binary trees Ch2a  90   video
bijection Dyck paths — semi-pyramids of dimers, Ch2c  71 (with violin) video
bijection Lukasiewicz paths — Dyck paths , Ch2a  60  video 
bijection Lukasiewicz paths — planar trees, Ch2a  98  video

Staircase polygons

bijection staircase polygons — 2-colored Motzkin paths, Ch2a  105  video
bijection staircase polygons — Dyck paths, Ch2a  110    video
bijection staircase polygons — basis of TL_n , Ch2b-complements-TLn, 23  video
bijection staircase polygons — (321)- avoiding permutations, Ch2b-complements-TLn,  34 (no video) 

Non-crossing partitions

bijection non-crossing partitions — Dyck paths, Ch2b  8  video 
bijection non-crossing partitions  — Catalan permutations , Ch2b  53  video

Planar maps

bijection planar maps — planar quartic maps, Ch2d  18  video
bijection rooted planar maps — rooted planar quartic maps, Ch2d  20  video
                reverse bijection, Ch2d  22  video   more explanations in this  video
bijection rooted planar quartic maps — well balanced blossoming trees,
                (Schaeffer bijection) Ch2d  25-44   video
bijection triangulations — (complete) binary trees, Ch2b  23 (with violin)  video

Temperley-Lieb algebra  TL_n

bijection basis of TL_n — some heaps of dimers, Ch2b-complements-TL_n, 18  video
bijection basis of TL_n — Dyck paths,  Ch2b-complements-TL_n, 19  video
bijection basis of TL_n — staircase polygons,  Ch2b-complements-TL_n, 23  video
bijection basis of TL_n — (321)- avoiding permutations, Ch2b-complements-TL_n, 31  video

Permutations

bijection permutations (cycles notation) — permutations (word), Ch4a  7-8  video and 20 video
bijection permutations — assemblées of increasing arborescences, Ch4a  96 video
bijection permutations — increasing binary trees, Ch4a  75-76   video 
bijection permutations — inversion table,  Ch4a  29-30  video
                reverse  bijection  33-44  video
bijection permutations — inversion table (with the Maj index), Ch4a  52-72  video
bijection permutations — pair of Young tableaux same shape (Robinson-Schensted, RS)
                RS with Schensted insertions, Ch4c  24-46  video
                RS with light and shadows, Ch4c  53-73   video
                RS with growth diagrams  Ch4d  4-26   video
                RSK matrices — pairs of semi-standard Young tableaux, Ch4d  63  video
bijection involutions (no fixed points) — oscillating tableaux, Ch4d  52-55  video
bijection involutions (no fixed points) — Hermite histories, Ch4b 68-77  video
bijection Catalan permutations — non-crossing partitions, Ch2b  53  video
bijection (321)- avoiding permutations — basis of the TL_n,  Ch2b-complements-TLn, 31  video

Histories

bijection Hermite histories — chord diagrams, Ch4b  68-77   video
bijection Laguerre histories — permutations (with binary trees), Ch4b  11-22  video
                (with words), Ch4b  23-33   video
                reciprocal bijection Ch4b  34-40   video 

Rook placements

bijection rook placements — oscillating tableaux, Ch4d 52  video
bijection rook placements — Hermite histories, Ch4d 53  video
bijection rook placements — involutions with 2-colored fixed points, Ch4d 56  video
bijection staircase rook placements — Partitions, Ch4d  58  video

Non-intersecting configurations of paths

bijection plane partitions — non-intersecting configurations of paths, Ch5a  97-105  video
bijection aztec tilings — non-intersecting configurations of Schröder paths, Ch5b  94  video
bijection semi-standard Young tableaux — non-intersecting configurations of paths, 
             Ch5a  69-70  (by rows)   video 
             Ch5b  19-22  (by rows)  video  (by columns)   video

Bijective proof of identities 

visual proof of Pythagorus theorem,  Ch1a  84   video
bijective proof  of an identity with q-series,  Ch1a  113   video
bijective proof for the number of aztec tilings, Ch5b  94-113  video 
bijective proof for the number of Baxter permutations, 
            Ch4b  108-122,    video    (bijection  Baxter permutations triple of paths with Laguerre histories) 
            and   Ch5a  45-48    video     (triple of paths as binomial determinant)  
            and  Ch5a   81-86  video    (with the formula  contents / hook length) 
bijective proofs for the formula giving the Catalan number: 
            the philosophy of bijective proofs for various  Catalan formulae, Ch1a  127  video
            bijective proof of the multiplicative recursion for Catalan numbers,    Ch2b  60   video
            bijective proof  related to the cyclic lemma,  Ch2c 11  video
            bijective proof with binary trees, Ch2c  22  video ,   with  Dyck paths,  Ch2d  80    video 
            bijective proof with the reflexion principle, 44  video
bijective proof of Lagrange inversion formula, Ch2c  28-39   video
bijective proof of Mehler identity, Ch3b  27-34    video
bijective proof of Touchard identity, Ch2b  73   video