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