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 IV 

Combinatorial theory of orthogonal polynomials and continued fractions

The Institute of Mathematical Sciences, Chennai, India  (January-March 2019) 

Chapter 6   q-analogues of some orthogonal polynomials

 

Chapter 6a

 March 4 , 2019

 

slides of Ch6a   (pdf 20 Mo )                

video Ch6a  link to YouTube  (from IMSc Matsciencechanel Playlist)

 

q-analogue, n! and binomials coefficients   3

scheme of basic hypergeometric orthogonal polynomials   13

Continuous q-Hermite polynomials   14

continuous q-Hermite polynomials (Hermite I)   15

recalling Hermite histories   16-37

q-analogue of Hermite histories   38

crossing number   42

q-analogue of Hermite histories wit nestings   45

moments of continous q-Hermite   60

formula of these moments   62

the philosophie of histoires and its q-analogues   63

interpretation of the coefficients of continous Hermite   71

Discrete q-Hermite  (Hermite II)   86

moments of discrete q-Hermite   88

number of "inversion" Inv(I) of a chord diagram I   95

relation relating Inv(I) and number of crossings and nestings   97

Charlier polynomials   104

discrete q-Charlier (Charlier II) (de Médicis, Stanton, White)   108

formula expressing discrete q-Charlier polynomials   111

interpretation of the discrete q-Charlier polynomials   112

formula for the moments of discrete q-Charlier polynomials with q-striling numbers   113

restricted growth functions for set partitions  

the parameter rs for restricted growth functions (Wachs, White)   116

0-1 tableaux (Leroux)   119

interpretation of the moments of discre q-Charlier with the parameter rs   120

classical q-Charlier polynomials (Zeng)   122-123

Continuouos q-Charlier  (Charlier I) (Kim, Stanton, Zeng)   124

formula for expressing the q-Charlier polynomials   126

combinatorial interpretation of q-Chalier polynomials (with Simion-Stanton)   128

formula for the moments mu_n(a;q)  129

interpretation of the moments of continuous Charlier   130

restricted crossing and restricted nesting of a partition   131-132

in conclusion   134

The end   140

   

Chapter 6b

 March 11 , 2019

 

slides of Ch6b   (pdf 29Mo)                

video Ch6b  link to YouTube  (from IMSc Matsciencechanel Playlist)

 

Reminding Ch 6a   3

Basic hypergeometric series   14

Continuous q-Laguerre polynomials (q-Laguerre I)   18

Al-Salam - Chihara polynomials   21

formula for the q-Laguerre polynomials

Moments of the continous q-Laguerre polynomials   24

weighted q-Laguerre histories   30

interpretation with the patterns 31-2   31 and 35

formula for the moments of the continous q-Laguerre polynomials (Corteel, Josuat-Vergès, Prellberg, Rubey)   36

Continous q-Laguerre polynomials with parameter beta   38

Subdivided Laguerre histories (and its q-analogues)   43

Bijection subdivided Laguerre histories  H -- restricted Laguerre histories  h   (Ch 3b, 43-72)  51

parameter q with crossings of the associated pairs of Hermite histories   66

relation with the number of crossings of a permutation (Corteel)   67-68

the commutative diagram: permutations sigma  - pairs of Hermite histories - subdived Laguerre histories H   - restricted Laguerre histories  h   71

Interpretation with Laguerre heaps of segments

bijection restricted Laguerre histories  h - Laguerre heaps of segments  E   74

number of crossings of a Laguerre heap of segments   87

intepretation of the moments of the continuous q-Laguerre with parameter beta with Laguerre heaps   88

Bijection restricted Laguerre histories h -- permutations tau  (Ch 3b, 127-129)  90

commutative diagram h, E, tau, inverse of tau

q-analogue of Euler continued fraction with parameter beta 

discrete q-Laguerre polynomials (Laguerre II)   101

definition of discrete q-Laguerre polynomials with the 3 terms recurrence relation   102

relation between the number of inversions of a permutation and

the number of crossing and nesting of the associated pair of Hsermite histories   106

Bijective proof for the Askey-Wilson integral   109

PASEP and orthogonal polynomials   118

The end 126