You are on the first page of X.V. new website www.viennot.org

During the reconstruction the "old" website www.xavierviennot.org is still active,

together with the sub-websites (Part I and II) dedicated to the "The Art of Bijective Combinatorics" , course I am giving at IMSc, Chennai, India.

Part I: An introduction to enumerative, algebraic and bijective combinatorics (January-March 2016)

Part II: Commutations and heaps of pieces with interactions in physics, mathematics and computer science (January-March 2017)

Part III: The cellular ansatz: bijective combinatorics and quadratic algebra (January-March 2018)

*Robinson-Schensted-Knuth, Asymmetric Exclusion Process, Tilings, Alternating Sign Matrices ... under the same roof*

Part IV: A combinatorial theory of orthogonal polynomials and continued fractions (January-March 2019)

A mirror image of this website is here at IMSc , the Institute of Maths Science at Chennai, India (last update 22 April 2018.

NEW: **last update: 3 January 2019**

**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 (pdf, 38Mo)

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.

Slides and video related to this talk can be found in the video-book « The Art of Bijective Combinatorics », Part II, Chapter 5b.

**Empilements de Laguerre pour le PASEP**

Groupe de travail Combinatoire, LaBRI, Bordeaux, 24 Septembre 2018

slides (in english, pdf 21 Mo, v2 some corrections after the talk)

résumé

Le PASEP ("partially asymmetric exclusion process) est un modèle très connu en physique des systèmes dynamiques ayant une combinatoire sous-jacente très riche et particulièrement étudiée par l'équipe bordelaise. Josuat-Vergès a donné une belle interprétation combinatoire de la fonction de partition du modèle à 3 paramètres en termes de permutations, en utilisant une combinaison de 4 bijections. Nous en donnons une autre preuve en introduisant un nouvel objet dans le jardin combinatoire du PASEP: les empilements de Laguerre, c'est-à-dire certains empilements de segments pointés, en bijection avec les permutations. Nous montrons le lien avec les tableaux de Dyck, les "histoires de Laguerre" et la structure de donnée "dictionnaire" en informatique.

**The essence of bijctions: from growth diagrams to Laguerre heaps of segments for the PASEP**

KrattenthalerFest, 81th SLC (Séminaire Lotharingien de Combinatoire) 11 September 2018

concert Christian

slides (pdf, 34Mo)

abstract

The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems with a very rich underlying combinatorics in relation with orthogonal polynomials culminating in the combinatorics of the moments of the Askey-Wilson polynomials. I will begin with a tour of the PASEP combinatorial garden with many objects such as alternative, tree-like and Dyck tableaux, Laguerre and subdivided Laguerre histories, all of them enumerated by n!. Using several bijections relating these objects, Josuat-Vergès gave the most simple interpretation of the partition function of the 3 parameters PASEP in terms of permutations related to the moments of the Al-Salam-Chihara polynomials.

This beautiful interpretation can be "explained" by introducing a new object called "Laguerre heaps of segments" having a central position among the several bijections of the PASEP garden. I will discuss some relations between these bijections and extract what can be called the "essence" of these bijections, some of them having the same "essence" as the Robinson-Schensted correspondence expressed with Fomin growth diagrams, dear to Christian.

**Growth diagrams and edge local rules**

GASCom 2018, Athens, Greece, 18 June 2018

slides

slides preliminary version: (pdf, 33Mo) GT LaBRI, Bordeaux, June 1st 2018

paper (extended abstract, 10p. in Proceedings GASCom 2018)

abstract

Fomin growth diagrams for the Robinson-Schensted correspondence and for the *jeu de taquin* are some rules allowing the construction of the 4th Ferrers diagram once 3 Ferrers diagrams around an elementary cell are known.We propose here to replace these rules by some local rules on edges instead of local rules on vertices.

Keywords

Fomin growth diagrams, edge local rules, Schützenberger jeu de taquin, RSK product of two words, duality of Young tableaux.

**Growth diagrams, local rules and beyond**

21st Ramanujan Symposium: National Conference on algebra and its applications,

Ramanujan Institute, University of Madras, Chennai, India, 28 February 2018

slides part I (pdf, 17 Mo)

slides part II (pdf, 15 Mo)

abstract

Robinson-Schensted-Knuth correspondence (RSK), Schützenberger "jeu de taquin", Littlewood-Richarson coefficients are very classical objects in the combinatorics of Young tableaux, Schur functions and representation theory. Description has been given by Fomin in terms of "growth diagrams" and operators satisfying the commutation rules of the Weyl-Heisenberg algebra. After a survey of Fomin's approach, I will make a slight shift by introducing an equivalent description with "local rules on edges", and show how such point of view can be extended to some other quadratic algebras.

79th SLC (Séminaire Lotharingien de Combinatoire), Bertinoro, Italy, 11 September 2017

**Maule: tilings, Young and Tamari lattices under the same roof **

slides part I** **(pdf, 25 Mo) tilings, Young and Tamari lattices as maules

slides part II (pdf, 26 Mo)** ** Tamari(v) as a maule + comments and references added after the talk** **

Same lectures is given at IMSc, Chennai, India in two video-recorded Maths seminars 19 and 26 February 2018

(slightly augmented) set of slides and video links are available on this page.