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)
Monday and Thursday, 11h30-13h, video room, first class: 4th January 2018
Contents
Chapter 0 overview of the course
Chapter 1 RSK The Robinson-Schensted-Knuth correspondence
Ch 1a The RS correspondence: Schensted insertions, geometric version with shadow lines
Ch1b The RS correspondence: Fomin growth diagrams, edge local rules,
the bilateral RSK planar automaton, dual of a tableau
Ch1c the cellular ansatz: planarisation of the rewriting rules, Q-tableaux,
commutations diagrams and growth diagrams, rook placements
Ch1d Schützenberger jeu de taquin, Knuth transpositions, Fomin (vertex) local rules for jeu de taquin,
edge local rules for jeu de taquin, nil-Temperley-Lieb algebra
Ch1e Greene theorem, from RS to RSK, edge local rules for RSK and dual RSK, bijections for rook placements
Chapter 2 Quadratic algebra, Q-tableaux and planar automata
Ch 2a The philosophy of the cellular ansatz: Q-tableaux and complete Q-tableaux
the PASEP algebra, alternative tableaux, a quadratic algebra for ASM
Ch 2b Planar automata, The RSK planar automaton, reverse Q-tableaux and reverse quadratic algebra,
tree-like tableaux,duplication of equations and RSK
Ch 2c Duplication of equations in the PASEP algebra, the XYZ algebra
XYZ tableaux: rhombus tilings, Aztec tilings, ASM, 6 and 8-vertex model
Ch 2d The LGV Lemma, binomial determinants,
XYZ-tableaux: rhombus tilings, plane partitions and non-intersecting paths
XYZ-tableaux: ASM, osculating paths and FPL
Complements: the beautiful garden of some jewels of combinatoric: ASM, TSSCPP, DPP, FPL, RS,....
Chapter 3 Tableaux for the PASEP quadratic algebra
Ch 3a The 5-parameters PASEP model, the matrix ansatz, PASEP and alternative tableaux
Catalan alternative tableaux, representation with 4 operators
Ch 3b Laguerre histories, the exchange-fusion algorithm, commutations diagrams
from the representation of the PASEP algebra, equivalence commutations diagrams and exchange-delete algorithm
Ch 3c interpretation of the parameters of the 3-PASEP, Genocchi sequence of a permutation, permutation tableaux,
bijection inversion tables - tree-like tableaux with the insertion algorithm
Chapter 4 Trees and Tableaux
Ch 4a Trees and Tableaux: binary trees and Catalan alternative tableaux
Ch 4b Trees and tableaux: the Loday-Ronco algebra of binary trees
Ch 4c Trees and tableaux: alternative binary trees, non-ambiguous trees and beyond
Chapter 5 Tableaux and orthogonal polynomials
Ch 5a Moments of orthogonal polynomials with weighted Motzkin paths, continued fractions and the Flajolet Lemma,
Laguerre, Hermite and Charlier histories
Ch 5b permutation and inversion table, q-Laguerre polynomials I and II, q-Hermite polynomials I,
the bijection permutations subdivided Laguerre histories, Dyck tableaux, Laguerre heaps
Ch 5c the parameter "q". Interpretation of the 3-parameters distribution of PASEP tableaux (alternative and tree-like) with
Laguerre histories, subdivided Laguerre histories, Dyck tableaux, Laguerre heaps of segments and permutations
the 5 parameters PASEP intrepretation with staircase tableaux, moments of Askey-Wilson polynomials
Chapter 6 Extensions: tableaux for the 2-PASEP quadratic algebra
The playlist from matsciencechannel of the 22 videos of this course is here
last update: 25 April 2018
A mirror image of this website is here at IMSc , the Institute of Maths Science at Chennai, India (last update 22 April 201