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)

Monday and Thursday, 11h30-13h, video room, first class: 4th January 2018

Contents

 

Chapter 0   overview of the course

 

Chapter 1    RSK   The Robinson-Schensted-Knuth  correspondence

Ch 1a     The RS correspondence: Schensted insertions, geometric version with shadow lines

Ch1b     The RS correspondence: Fomin growth diagrams, edge local rules, 

the bilateral RSK planar automaton, dual of a tableau

Ch1c     the cellular ansatz: planarisation of the rewriting rules, Q-tableaux,

commutations diagrams and growth diagrams, rook placements

Ch1d     Schützenberger jeu de taquin, Knuth transpositions, Fomin (vertex) local rules for jeu de taquin,

edge local rules for jeu de taquin, nil-Temperley-Lieb algebra

Ch1e     Greene theorem, from RS to RSK, edge local rules for RSK and dual RSK, bijections for rook placements 

 

Chapter 2   Quadratic algebra, Q-tableaux and planar automata

Ch 2a     The philosophy of the cellular ansatz: Q-tableaux and complete Q-tableaux

 the PASEP algebra, alternative tableaux, a quadratic algebra for ASM

Ch 2b     Planar automata, The RSK planar automaton, reverse Q-tableaux and reverse quadratic algebra,

tree-like tableaux,duplication of equations and RSK

Ch 2c     Duplication of equations in the PASEP algebra, the XYZ algebra

XYZ tableaux: rhombus tilings, Aztec tilings, ASM, 6 and 8-vertex model

Ch 2d     The LGV Lemma, binomial determinants,

XYZ-tableaux: rhombus tilings, plane partitions and non-intersecting paths 

XYZ-tableaux: ASM, osculating paths and FPL

Complements: the beautiful garden of some jewels of combinatoric: ASM, TSSCPP, DPP, FPL, RS,....

  

Chapter 3   Tableaux for the PASEP quadratic algebra

Ch 3a     The 5-parameters PASEP model, the matrix ansatz, PASEP and alternative tableaux

Catalan alternative tableaux, representation with 4 operators

Ch 3b     Laguerre histories, the exchange-fusion algorithm, commutations diagrams 

from the representation of the PASEP algebra, equivalence commutations diagrams and exchange-delete algorithm

 Ch 3c     interpretation of the parameters of the 3-PASEP, Genocchi sequence of a permutation, permutation tableaux,

bijection inversion tables - tree-like tableaux with the insertion algorithm

 

Chapter 4   Trees and Tableaux 

Ch 4a  Trees and Tableaux: binary trees and Catalan alternative tableaux

Ch 4b   Trees and tableaux: the Loday-Ronco algebra of binary trees

Ch 4c   Trees and tableaux: alternative binary trees, non-ambiguous trees and beyond

 

Chapter 5    Tableaux and orthogonal polynomials

Ch 5a     Moments of orthogonal polynomials with weighted Motzkin paths, continued fractions and the Flajolet Lemma, 

Laguerre, Hermite and Charlier histories

Ch 5b   permutation and inversion table, q-Laguerre polynomials I and II, q-Hermite polynomials I,  

the bijection permutations subdivided Laguerre histories, Dyck tableaux, Laguerre heaps

Ch 5c  the parameter "q".  Interpretation of the 3-parameters distribution of PASEP tableaux (alternative and tree-like) with

Laguerre histories, subdivided Laguerre histories, Dyck tableaux, Laguerre heaps of segments and permutations

 the 5 parameters PASEP intrepretation with staircase tableaux, moments of Askey-Wilson polynomials

 

Chapter 6    Extensions: tableaux for the 2-PASEP quadratic algebra

 

 

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

 

 

last update: 25 April 2018

 

A mirror image of this website is here at IMSc , the Institute of Maths Science at Chennai, India (last update 22 April 201