XAVIER VIENNOT
  • PART I
    • 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
    • list of bijections
    • index
  • PART II
    • 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
    • Some lectures related to the course
  • PART III
    • 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
    • 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

 

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