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)

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

 

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