• 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 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 Ekalavya  (IMSc Media Center)
video Ch6a  link to YouTube  (from IMSc Matsciencechanel Playlist)
video Ch6a link to bilibili
q-analogue, n! and binomials coefficients   3     0:31
6 q-analogues of orthogonal polynomials   11     9:12
continuous and discrete Hermite, Charlier and Laguerre polynomials
scheme of basic hypergeometric orthogonal polynomials   13     12:55
discussion: what makes a "good" q-analogue     14:05
Continuous q-Hermite polynomials   14     16:48
continuous q-Hermite polynomials (Hermite I)   15   
recalling Hermite histories   16-37     17:06
q-analogue of Hermite histories   38     21:20
crossing number   42     22:19
proposition: interpretation of the moments of continuous q-Hermite I with crossings   44     23:23
q-analogue of Hermite histories with nestings   45     24:07
moments of continous q-Hermite   60     27:16
formula of these moments   62     27:55
the philosophy of histoires and its q-analogues   63     28:57
exercise: the double distribution (crossings, nestings)   70     33:25
q-analogue of continous Hermite polynomials   71     33:48
proposition: interpretation of the coefficients of continous Hermite   83     38:48
Discrete q-Hermite  (Hermite II)   86     42:25
proposition: formula for the moments of discrete q-Hermite   88     44:13
number of "inversions" Inv(I) of a chord diagram I   95     45:25
relation relating Inv(I) and number of crossings and nestings   97     46:16
proof of the formula for the moments of discrete q-Hermite   98-102        47:50
a funny relation between continuous and discrete q-Hermite   103     50:36
q-Charlier polynomials   104     51:42
Discrete q-Charlier (Charlier II) (de Médicis, Stanton, White)   108     54:42
formula expressing discrete q-Charlier polynomials   111     56:27
interpretation of the discrete q-Charlier polynomials   112     58:01
formula for the moments of discrete q-Charlier polynomials with q-Stirling numbers   113     58:48
restricted growth functions for set partitions   114     59:47
the parameter  rs  for restricted growth functions (Wachs, White)   116     1:02:05
0-1 tableaux (Leroux)   119     1:04:35
proposition: interpretation of the moments of discre q-Charlier with the parameter rs   120     1:05:26
classical q-Charlier polynomials (Zeng)   122-123     1:08:26
Continuous q-Charlier  (Charlier I) (Kim, Stanton, Zeng)   124     1:09:40
formula for expressing the q-Charlier polynomials   126     1:10:42
definition of the index w(k) for a permutation (Simion, Stanton) 127-128     1:11:20
proposition: combinatorial interpretation of the continuous q-Chalier polynomials (with Simion-Stanton)   128     1:12:34

formula for the moments mu_n(a;q)  129     1:13:38

moments of continuous Charlier   130     1:14:27
restricted crossing and restricted nesting of a partition   131-132     1:14:34
proposition: interpretation of the moments of continuous q-Charlier   133     1:18:30
in conclusion   134     1:19:04
The end   140     1:24:44

Chapter 6b

 March 11 , 2019
slides of Ch6b   (v2, pdf 29Mo)                
video Ch6b  link to Ekalavya  (IMSc Media Center)
video Ch6b  link to YouTube  (from IMSc Matsciencechanel Playlist)
video Ch6b link to bilibili
Reminding Ch 6a   3     0:24
Basic hypergeometric series   14     6:58
Continuous q-Laguerre polynomials (q-Laguerre I)   18     12:13
Al-Salam - Chihara polynomials   21     14:08
formula for the q-Laguerre polynomials (Kasraoui, Stanton, Zeng ; Simion, Stanton)   22-23     15:06
Moments of the continous q-Laguerre polynomials   24     17:01
weighted q-Laguerre histories   30     22:30
lemma: relation with the patterns 31-2   31     23:27
proposition: interpretation of the moments with the patterns 31-2   35     31:29
formula for the moments of the continous q-Laguerre polynomials (Corteel, Josuat-Vergès, Prellberg, Rubey)   36     32:22
Continous q-Laguerre polynomials with parameter beta   38     33:37
proposition: interpretation of the moments of the continuous q-Laguerre with parameter beta   42     38:58
Subdivided Laguerre histories (and its q-analogues)   43     39:57
q-analogue of Euler's (Stieljes) continued fraction for n!   48-49     41:33
Bijection subdivided Laguerre histories  H -- restricted Laguerre histories  h   (Ch 3b, 82-91)  51     44:19
Bijection permutations sigma -- subdivided Laguerre histories  H    (Ch 3b, 43-72)   58     47:35
parameter q  and nestings of the associated pairs of Hermite histories   64     51:58
parameter q  and crossings of the associated pairs of Hermite histories   65     53:08
relation with the number of crossings of a permutation (Corteel)   66-67     54:10
corollary: interpretation of the moments of continuous q-Laguerre polynomials with numbers of crossing of permutations   68     55:52
the commutative diagram: permutations sigma  - pairs of Hermite histories - subdived Laguerre histories H   - restricted Laguerre histories  h   70    56:18
q-moments with the eulerian parameter  y   71     58:02
Interpretation with Laguerre heaps of segments
bijection restricted Laguerre histories  h - Laguerre heaps of segments  E  (Ch2c, p95-104, p113-133)  72     58:48
number of crossings of a Laguerre heap of segments   85     1:02:36
intepretation with Laguerre heaps of the moments of the continuous q-Laguerre polynomials with parameter beta    86     1:03:34
Bijection restricted Laguerre histories h -- permutations tau  (Ch 3b, 127-129)  88     1:07:09
commutative diagram h, E, tau, inverse of tau   91     1:07:58
q-analogue of Euler continued fraction with parameter beta   93     1:09:50
interpretation of the beta-q  parameters on  the (A,K) chord diagrams related to the subdivided Laguerre history   98     1:11:50
discrete q-Laguerre polynomials (Laguerre II)   99     1:13:13
definition of discrete q-Laguerre polynomials with the 3 terms recurrence relation   100     1:13:18
relation between the number of inversions of a permutation and
the number of crossing and nesting of the associated pair of Hermite histories   103     1:14:28
proposition: expresssion for the moments of discrete q-Laguerre polynomials   (Heine, Biane)   105     1:17:12
proposition: expresssion for the moments of discrete beta-q-Laguerre polynomials   106     1:18:22
Bijective proof for the Askey-Wilson integral (Ismail, Stanton, X.V.)   108     1:18:53
Askey-Wilson polynomials (definition)   110     1:19:07
orthogonality of Askey-Wilson polynomials   111     1:19:30
the Askey-Wilson integral   112     1:19:54
bijective proof: the Askey-Wilson integral as a product of four continuous q-Hermite polynomials   115      1:21:05
PASEP and orthogonal polynomials   117     1:22:50
The end 126     1:26:15