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