The Bijective Combinatorics Course 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.

**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