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)

Some lectures related to the course

**Introduction to the theory of heaps of pieces with applications to statistical mechanics and quantum gravity**

Isaac Newton Institute for Mathematical Scinces, 07 April 2008

workshop Combinatorial Identities and their Applications in Statistical Mechanics

*a lecture giving a survey of Part II of the ABjC course, lecture dedicated to my dear friend *Pierre Leroux (1942-2008)

slides (pdf 21 Mo)

video (1h 5 mn) (producer: Steve Greenham)

abstract:

The notion of "heaps of pieces" has been introduced by the author in 1985, as a "geometrization" of the algebraic notion of commutation monoids defined by Cartier and Foata. The theory has been developed by the Bordeaux group of combinatorics, with strong interaction with theoretical physics.

We state three basic lemmas of the theory: an "inversion lemma" giving generating functions of heaps as the quotient of two alternating generating functions of "trivial" heaps, the "logarithmic lemma", and the "path lemma" saying that any path can be put in bijection with a heap. Many results and explicit formulae or identities in various papers scattered in the combinatorics and physics literature can be unified and viewed as consequence of these three basic lemmas, once the translation of the problem into heaps methodology has been made.

Applications and interactions in statistical mechanics will be given with the now classical directed animal models and gas models with hard core interaction, such as Baxter's hard hexagons model; the appearance of some q-Bessel functions in two lattice models: the staircase polygons and the Solid-on-Solid model; and some 2D Lorentzian quantum gravity models introduced by Ambjorn, Loll, Di Francesco, Guitter and Kristjansen.

Finally, I shall present a heap bijective proof of a new identity of Bauer about loop-erased walks, in relation with recent work of Brydges and Abdesselam about loop ensembles and Mayer expansion.

**Proofs without words: the example of the Ramanujan continued fraction**

colloquium IMSc, Chennai, February 21, 2019 (lecture also related to Part IV of the ABjC course)

slides (pdf, 28 Mo)

video link to Ekalavya (IMSc Media Center)

video link to YouTube (from IMSc Matsciencechanel Playlist)

video link to bilibili

abstract:

Visual proofs of identities were common at the Greek time, such as the Pythagoras theorem. In the same spirit, with the renaissance of combinatorics, visual proofs of much deeper identities become possible. Some identities can be interpreted at the combinatorial level, and the identity is a consequence of the construction a weight preserving bijection between the objects interpreting both sides of the identity.

In this lecture, I will give an example involving the famous and classical Ramanujan continued fraction. The construction is based on the concept of "heaps of pieces",

which gives a spatial interpretation of the commutation monoids introduced by Cartier and Foata in 1969.

the same conference has also been given in the following Institutions:

Ramanujan Institute, Chennai, India, 10 January 2017

Twenty Second Srinivasa Rajan Memorial Lecture

Amrita Vishwa Vidyapeetham, Amrita University,

Coimbatore, 7 March 2017

Indian Institute of Sciences, Bangalore, 9 March 2017

**Zeta function on graphs revisited with the theory of heaps of pieces**

ICGTA19, International Conference on Graph Theory and Applications, Amrita Vishwa Vidyapeetham Coimbatore, India, 5th January 2019

slides (second version after the talk, pdf, 39Mo)

abstract

An identity of Euler expresses the classical Rieman zeta function as a product involving prime numbers. Following Ihara, Selberg, Hashimoto, Sunada, Bass and many others, this function has been extended to arbitrary graphs by defining a certain notion of « prime » for a graph, as some non-backtracking prime circuits. Some formulae has been given expressing the zeta function of a graph. We will revisit these expressions with the theory of heaps of pieces, initiated by the speaker, as a geometric interpretation of the so-called commutation monoids introduced by Cartier and Foata. Three basic lemma on heaps will be used: inversion lemma, logarithmic lemma and the lemma expressing paths on a graphs as a heap. A simpler version of the zeta function of a graph proposed recently by Giscard and Rochet can be interpreted as heaps of elementary circuits, in relation with MacMahon's Master theorem.

** «How to color a map with (-1) color»**

Amrita Vishwa Vidyapeetham, Amrita University,

Coimbatore, 8 March 201

slides first part (pdf 21 Mo)

slides second part (pdf 15 Mo)