Courses X.V.
Courses Talca, Chile 2010/2011
Cours Universidad de Talca
(December 2010 - January 2011)
Combinatorics and interactions (with physics)
«The Cellular Ansatz» (24 h)
some references:
- A paper( in french) introducing enumerative, algebraic, bijective ... combinatorics from elementary to recent researches, LMA3.Viennot.v4.pdf
“Enumérons ! (De la combinatoire énumérative classique aux nouvelles combinatoires : bijective, algébrique, expérimentale, quantique et … magique)”, article dans “Leçons de mathématiques d’aujourd’hui ”, vol 3, éds. Eric Charpentier et Nicolas Nikolski, Cassini, Paris, 2007, pp 165-238.
-
-A complete text (in french) with more informations, covering Ch1 and Ch2 of the Talca course. (160p), see my first website at the page «cours», the lecture Notes 1988 «séries génératrices énumératives»
-
- A survey (in 1984) of the Robinson-Schensted correspondence:
Chain and antichain families, grids and Young tableaux, Annals of Discrete Maths., 23 (1984) 409-464.
- A survey (in french) about combinatorial interpretations of Euler and Genocchi numbers, with some introduction to basic bijections for combinatorics:
Interprétations combinatoires des nombres d’Euler et de Genocchi (pdf, 9,2 Mo) Séminaire de Théorie des nombres de Bordeaux, Publi. de l’Université Bordeaux I, 1982-83, 94p.
1 december 2010 (pdf 4,1 Mo)
Ch1a Ordinary generating functions, the Catalan garden
1 december 2010 (pdf 6 Mo, 54 p.)
Binary trees p.2
ordinary generating functions, formal power series p.11
formal power series algebra, formalisation p.25
operations on combinatorial objects, example: binary trees p.37
bijective combinatorics, example: Catalan numbers p.44
Dyck paths p.52
Ch1b Ordinary generating functions, the Catalan garden
7 december 2010 (pdf 10,8 Mo, 81p.)
Leonhard Euler and Catalan numbers p.4
from triangulations to binary trees p.10
from binary trees to Dyck paths p.17
binary trees, complete binary trees p.21
chord diagrams p.24
parallelogram polyominoes (staircase polygons), 2-colored Motzkin paths p.27
operations on combinatorial objects, example: integers partitions, q-series p.33
operations on combinatorial objects: formalisation p.49
bijective proof of an identity, the "bijective paradigm" p.56
algebricity, decomposable structures, example: planar maps p.67
directed animals p.73
Ch1c The Catalan garden, algebraic complements in relation with physics
7 december 2010 (pdf 2,8 Mo, 30 p.)
the associahedron p.2
the Temperley-Lieb algebra p.7
Ch 2a Exponential generating functions, permutations
22 december 2010 (pdf 12,2 Mo, 40 p.)
permutations: classical combinatorics p.2
inversion table, q-analogue p.10
increasing binary trees p.17
exponential generating functions, species and (exponential) structures p. 29
operations on species, exponential structures p.37
Ch 2b Exponential generating functions, permutations
4 january 2011 (pdf 8,2 Mo, 63 p.)
operations on species: exp, log, substitution p.3
Striling numbers and binomial type polynomials p.9
differential equations p.12
weighted species p.26
examples: orthogonal polynomials p.30
Mehler identity for Hermite polynomials p. 45
complements p.58
Ch 2c Permutations and Laguerre histories
4 january 2011 (pdf 2,5 Mo, 33 p.)
weighted paths p.2
Laguerre histories p.7
bijection Laguerre histories - permutations p.12
Ch 3a Permutations and Young tableaux:
the Robinson-Schensted correspondence
6 january 2011 (pdf 13,5 Mo, 117 p.)
Young tableaux p.2
Hook length formula p.7
Introduction to RSK p.16
RSK with Schensted's insertions p.22
A geomeric version of RSK with "light" and "shadow lines" p.48
proof of the equivalence insertions - geometric construction p.716
"jeu de taquin" p.78
increasing and decreasing subsequences p.107
Ch 3b The Robinson-Schensted correspondence with operators
6, 11 january 2011 (pdf 6,2 Mo, 121 p.)
"local" algorithms on a grid or "growth diagrams" p.2
Heisenberg operators UD = qDU + I, normal ordering p.9
representation of the operators U and D p.41
RSK with Fomin's "local rules" p.55
local RSK and geometric RSK p.64
alternating sign matrices (ASM) and a quadratic algebra p.84
Ch 4a Alternative tableaux and the PASEP
(partially asymmetric exclusion process)
13 january 2011 (pdf 6 Mo, 98 p.)
The PASEP p.2
The matrix Ansatz of Derrida, Evans, Hakim, Pasquier p.19
The PASEP algebra DE = qED + E + D, p.26
alternative tableaux p.50
stationary probabilities for the PASEP p.61
permutation tableaux p.64
PASEP and orthogonal polynomials p.89
references p.92
Ch 4b Alternative tableaux and the PASEP
the bijection alternative tableaux -- permutations
13, 18 january 2011 (pdf 9,8 Mo, 102 p.)
the "exchange-fusion" algorithm p.2
the inverse "exchange-fusion" algorithm p.21
Genocchi sequence of a permutation p.36
some parameters p.43
"q-analogue" of Laguerre histories p.53
the cellular Ansatz: from the PASEP algebra DE = ED + E +D to bijections p.61
representation of the operators E and D with DE = ED + E +D p.63
the cellular Ansatz: bijection permutations - alternative tableaux p.74
the inverse bijection permutations - alternative tableaux p.90
two bijections, one theorem p.98
Ch 4c Alternative tableaux and the PASEP
complements 18 january 2011 (pdf 2,2 Mo, 81 p.)
the "exchange-deletion" algorithm p.2
an alternative "jeu de taquin" p.20
the same example denoted with the inverse permutation p.37
proof of the main theorem p.52
Postnikov's bijection permutation tableaux - permutations p.76
The seminal paper of Derrida, Evans, Hakim and Pasquier on the Matrix Ansatz for the PASEP
A «video-preprint» by XGV introducing Alternative tableaux
(Isaac Newton Institute, 23 April 2008)
Ch 5 Combinatorial theory of orthogonal polynomials
20 january 2011 (pdf 11,7 Mo, 110 p.)
some trigonometry p.2
combinatorial interpretations of some orthogonal polynomials p.9
formal orthogonal polynomials: definition, moments p.15
Favard theorem p.25
3-terms recurrence relation and pavages p.27
moments and weigthed Motzkin paths p.32
Laguerre histories p.38
data structures histories p.48
quadratic algebra operators and orthogonal polynomials p.56
q-analogues of Hermite and Laguerre polynomials p.78
PASEP and orthogonal polynomials p.84
combinatorial proof of identities p.89
the Askey-Wilson integral p.91
continued fractions p.97
Ch 6a Jeu de taquin for binary trees, Catalan tableaux and the TASEP
24 january 2011 (pdf 7,1 Mo, 98 p.)
Catalan alternative tableaux p.2
bijection Catalan alternative tableaux - binary trees p.11
TASEP and Catalan alternative tableaux p.29
Catalan permutation tableaux p.42
caracterisation of Catalan alternative tableaux p.47
TASEP with Catalan tableaux, realtion with Shapiro-Zeilberger interpretation p.60
other description of the bijection binary trees - pair of paths p.80
Ch 6b Tableaux and increasing / alternative binary trees
24 january 2011 (pdf 6,3 Mo, 111p.)
increasing binary trees: canopy and up-down sequence p.2
"jeu de taquin" for increasing binary trees p.13
from "jeu de taquin" for increasing binary trees to Catalan alternative tableaux p.30
alternative binary trees p.50
"jeu de taquin" for alternative binary trees p.57
the twisted symmetric order p.82
complements p.86
the "exchange-fusion" algorithm in the Catalan case p.87
Bernardi permutations p.103
Ch 6c Catalan tableaux and the Loday-Ronco algebra
24 january 2011 (pdf 2,1 Mo, 21 p.)
Hopf algebras: Malvenuto-Reutenauer, Loday-Ronco, descents algebra p.2
Loday-Ronco product of binary trees with Catalan alternative tableaux p.4
proof of the equivalence of the two products p.10
the # product for binary trees p.18
complements:
2 talks by J.-C. Aval on Hopf algebra, alternative Catalan tableaux and Loday-Ronco algebra
Ch 7a The cellular Ansatz
25 january 2011 (pdf 8,7 Mo, 117 p.)
Q-tableaux p.2
Q-tableaux: exemples p.11
planar automata p.32
"figures" accepted by planar automata ? p.38
the 8-vertex algebra (or Z-algebra, or XYZ-algebra) p.43
rhombus tiling p.46
Aztec diamond p.61
geometric interpretation of Z-tableaux p.68
non-intersecting paths p.72
bijection plane partitions -- non-intersecting paths p.75
osculating paths p.83
fully packed loops (FPL) p.90
configurations B.A.BA p.105
Z-tableaux and configurations B.A.BA p.113
Ch 7b The cellular Ansatz
complements 25 january 2011 (pdf 4,7 Mo, 49 p.)
symmetries for plane partitions p.3
totaly symmetric sef-complementary plane partitions (TSSCPP) p.12
Razumov-Stroganov (ex)-conjecture p.20
8-vertex model , analog of Razumov-Stroganov conjecture ? p.35
some bijections p.36
