• 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)

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

March 15, 2018
slides of Ch 6   (pdf,   35Mo )                 
video Ch6: link to Ekalavya  (IMSc Media Center)
video Ch6: link to YouTube
video Ch6: link to bilibili
Reminding the essential of the cellular ansatz   3     0:29
The 2-species PASEP   12     7:23
Matrix ansatz for the 2-species PASEP   14     12:26
Rhombic alternative tableaux (RAT)   17     15:44
the diagram  Gamma(X) associated to a word X   20     16:33
West-  and-  North strips associated to a tiling T  of  Gamma(X)   22     18:09
definition of a rhombic alternative tableau (RAT) (associated to a tiling T)   23     18:40
the weight of a RAT   25     20:33
proposition: the weight generating function does not depend of the tiling   28     25:24
a flip in a tiling   29     26:58
weight preserving bijection between two RAT's associated to two different tilings  T  and  T'   30-31     27:14
Combinatorial interpretation of the stationary probabilities of the 2-PASEP   32     29:10
Some remarks   34     31:06
The 2-PASEP quadratic algebra   43     34:38
Planarization of the rewriting rules   49     38:10
An example   52     39:45
Combinatorial interpretation of the stationary probabilities (proof)   81     43:39
Enumeration of rhombic alternative tableaux   84     46:27
assemblées of permutations enumerated by Lah numbers   85     46:51
formula for the partition function with parameters alpha and beta   86     51:08
Assemblées and species  (remindng BJC I, Ch3)   87     52:46
Proof of the formula for Lah numbers   95-98     56:58
From assemblées of permutations to rhombic alternative tableaux   99     58:05
the exchange-fusion algorithm for assemblées of permutations, defintion: 106,107,11     1:03:27
an example   111     1:06:09
interpretation of the parameters alpha and beta   114-115     1:06:42
visualizatinon of the algorithm with network of blue, green and red threads   117-119     1:07:25
The inverse algorithm: from rhombic alternative tableaux to assemblées of permutations   120     1:08:03
Bijection "assemblées of permutations" -- (subset of r elements among n) X (r-truncated subexceedant functions)   133     1:09:13
interpretation of the formula for the partition function with two parameter alpha and beta   142-143     1:12:52
Further enumerative results   144     1:13:27
Tree-like rhombic tableaux   146     1:13:41
Relation with Koorwinder-Macdonald polynomials   155     1:14:15
rhombic alternative tableaux with staircase shape     158     1:17:12
bijection rhombic alternative tableaux -- rhombic alternative tableaux with staircase shape   159     1:17:32
the 2-PASEP model with 5 parameters, interpretation with rhombic staircase tableaux   165     1:18:40
Koorwinder moments   167     1:20:00
Expression of the partition function of the 5 parameters 2-PASEP model with Koorwinder moments   168     1:20:34
(from Corteel, Mandelshtam, Williams)
analogue of Jacobi-Trudi identities (Schur functions) for Koorwinder polynomials and its moments   169     1:22:19
The end of the bijective course III   170      1:23:04
The godess Saraswathi  173     1:25:56