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 II

Commutations and heaps of pieces with interactions in physics, mathematics and computer science

 

The Institute of Mathematical Sciences, Chennai, India  (January-March 2017)

Preface

 

To be completed

Contents

 

-introduction to the combinatorial theory of heaps: commutation monoids, basic definitions about heaps, equivalence commutation monoids and heaps monoids, graphs, posets and linear extension of a poset

-reminding formal power series and generating functions

-the 3 basic lemma: inversion formula and generating function for heaps, the logarithmic formula, equivalence between paths and heaps of cycles

-combinatorial proof with heaps of classical theorems in linear algebra, MacMahon master theorem

-heaps and algebraic graph theory: zeros of matching polynomials, acyclic orientations, chromatic polynomial

-heaps for a combinatorial theory of formal orthogonal polynomials and continued  fractions

-interpretation of the reciprocal of the Rogers-Ramanujan identities with heaps of dimers

-fully commutative elements in Coxeter groups and Temperley-Lieb algebra

-applications to statistical physics: directed and multidirected animals, parallelogram polyominoes and Bessel functions, SOS models, hard gas models, Baxter hard hexagons model,

-application to 2D Lorentzian quantum gravity: causal triangulations.

 

 The  playlist from matsciencechannel of the 19 videos of this course is here  

 

complementary topics

- zeta function on graph and number theory

 -minuscule representations of Lie algebra with operators on heaps

 -basis for free partially commutative Lie algebra

 -Ising model revisited with heaps of pieces

 -interactions with string theory in physics

 -the SAT problem in computer science revisited with heaps (from D. Knuth)

 -heaps in computer science: Petri nets, aynchronous automata and Zielonka theorem