• 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 III
The cellular ansatz:  bijective combinatorics and quadratic algebra

The Institute of Mathematical Sciences, Chennai, India  (January-March 2018)

January 4, 2018

Chapter  0     Overview of the course

slides of Ch0   (pdf  17 Mo, version2)                
video Ch0:  link to Ekalavya  (IMSc Media Centerr)
video Ch0:  link to YouTube
video Ch0:  link to bilibili
recalling Part I of the bijective combinatorics course:  enumerative combinatorics:   01:27
Catalan numbers and n!   p6   01:40
number of Young tableaux  10   02:13
A beautiful identity 15   04:07
algebraic combinatorics: an example 21   05:40
bijective combinatorics: RSK  24   07:36
the idea of "bijective tools" 30   10:22
Part III of the bijective course: the cellular ansatz 38   12:20
First step of the cellular ansatz: quadratic algebra Q and associated Q-tableau 41   13:18
with the example Q defined by the relation  UD = DU + Id   13:41
normal ordering   14:34
permuations and normal ordering  16:43
why the name "cellular ansatz": planarization of the rewriting rules   17:29
planarization of the rewriting rules  17:43
permutations as complete Q-tableaux 45    20:53
permutations as Q-tableaux   21:59
Rothe diagrams as Q-tableaux  22:19
Planar automata and Q-tableaux 80   23:20
finite automaton  23:42
alternating sign matrices recognized by a planar automaton  24:57
the idea of planar automata   25:54
the quadratic algebra associated to alternating sign matrices   27:33
First step of the cellular ansatz (sumary)   27:58
Second step of the cellular ansatz   28:23
The Robinson-Schensted-Knuth (RSK) correspondence 89   28:51
Fomin's local rules (growth diagrams)  29:50
combinatorial representation of the algebra UD = DU +Id  31:02
Second step of the cellular ansatz (sumary) 91  32:25
Combinatorial physics   33:16
Combinatorial physics: example with the PASEP 95   33:38
The PASEP algebra  37:15
"normal ordering" for the PASEP algebra   37:53
alternative tableaux: definition 100   38:20
planarization of the rewriting rules of the PASEP algebra   40:30
alternative tableaux as Q-tableau of the PASEP algebra 105  42:19
Catalan alternative tableaux and the TASEP (q=0) 133   43:27
Enumeration of alternating tableaux   44:19
Representation of the PASEP algebra and the bijection EXF ("exchange-fusion") 138  44:35
RSK, EXF, tilings, paths, ASM, 8-vertex model, ... and much more, under the same roof 140   47:20
The njumber of alternating matrices  49:11
A summary of the course in a single tableau   49:58
demultiplication of the equations in a quadratic algebra   51:11
the end 150   0:53:58

The  playlist from matsciencechannel of the 22 videos of this course is here