The Art of Bijective Combinatorics Part I

** An introduction to enumerative, algebraic and bijective combinatorics**

The Institute of Mathematical Sciences, Chennai, India (January-March 2016)

**Preface**

This website is devoted to the combinatorial course I gave at IMSc (Institute of Mathematical Science), Chennai, India, in January-March 2016. The whole slides and videos of the course are available (2024 slides and 17 videos for 17 lectures, for a total of almost 24 hours). The course was followed by undergraduate and graduate students, together with some professors, in mathematics, physics and computer science, from IMSc and the neighbourless institutes in Chennai: CMI (Chennai Mathematical Institute) and IIT (Indian Institute of Technology) Madras.

I gave the course at two different levels at the same time. Most of the course is supposed to be followed by undergraduate students (at the Master level). I also gave some more advanced topics, opening some «windows» without proof, for graduate students, researchers and professors, in connection with active researches in combinatorics, and sometimes with connection in theoretical physics and computer science. Such sections are listed under the name «Complements».

The emphasis of the course being on the bijective point of view in combinatorics, there are many figures and visual mathematics. Although transparencies and overhead projectors had been replaced by video-projectors, I tried to keep the spirit of the so-called «Viennotique» with handmade transparencies in color which are incorporated in the modern techniques.

Warm thanks to colleagues and friends: Amritanshu Prasand (Amri) who invited me to give this combinatorics course at IMSc, Suresh Govindarajan, Arul Lakshminarayan, V.S.Sunders, Meena Mahajan and N. Narayanan, for their support, enthusiasm, and interest in the course. Many thanks to the students from IMSc, CMI and IIT for their endurance and interest for the course and to the technicians of IMSc for making the videos. Special thanks to Sridhar, Varsha, Jinu for writing the forthcoming notes from the course.

Xavier Viennot

**Contents of the course**

Chapter 0 Introduction to the course (1 lecture)

Chapter 1 Ordinary generating functions (4 lectures)

Chapter 2 The Catalan garden (4 lectures)

Chapter 3 Exponential structures and generating functions (2 lectures)

Chapter 4 The n! garden (4 lectures)

Chapter 5 Tilings, determinants and non-intersecting paths (2 lectures)

A detailed content of each chapter is given in the corresponding pages, with references to the page number for slides, and to the corresponding time for the video.