**The Art of Bijective Combinatorics**

The Institute of Mathematical Sciences, Chennai, India (January-March 2016-2019)

**Preface**

Some lectures for a wide audience as an introduction to Bijective Combinatorics

**The Art of Bijective Combinatorics **(for short "**ABjC**") is not a book, it is not an e-book, it is not reduced to a collection of videos, it is a kind of "video-book"

**This "video-book" "ABjC" has 3 components:**

- a set of 77 lectures video-recorded at the Insitute of Mathematical Science (IMSc), Chennai, India, as a series of four different courses on "bijective combinatorics" during four consecutive years (2016-2019) at the invitation of Amritanshu Prasad (Amri),

- the correponding set of slides of each lecture,

- a website which enable one to navigate inside the videos, in the same way you turn around the pages of a book. For example if you click on the time given just after the slide number corresponding to one of the videos, you will get, up to one second, to exact position in the video.

an introduction to the video-book ABjC is given at the beginning of the video of the Epilogue (from 2' 02" to 8' 42")

**This video-book on bijective combinatorics is in four parts** (or "volumes"), each part corresponds to a combinatorial course given at IMSc.

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)

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

**A series of courses at two different levels**

Each course can be followed independently. There are many overlaps between the differents courses. I gave these courses at two different levels at the same time. Most of the courses are supposed to be followed by good undergraduate students (at the Master level) and graduate students. 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*». In this video-book their are some new results and also some new ways to present classical theory. These courses were 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.

**Book or video-book ?**

The discussion is open if it is better to write a book (here in fact 4 books !) or "to speak" a "video-book". Nowadays many lectures and courses are recorded in videos.

One of the main disadvantage of videos is that it is difficult to find or listen just a part or a particular point in which one is interested. I tried to overcome this difficulty by giving a detailed description of each video, using the corresponding slides, with some direct links going inside the video. Also nowadays, most people give together with a research paper, a set of slides giving a survey of the paper. I found very convenient to look first at the set of slides coming from a talk, and then to the paper in order to go inside the proofs ot the details. Here in this video-book, I tried to imitate this attitude: first the reader can look at the set of slides, and for more details go to the corresponding portion of the video with the links given in this website.

The emphasis of the course being on the bijective point of view in combinatorics, there are many figures, visual mathematics and some animations.

Here is the advantage of a video-book, keeping the beauty of bijections and bijective proofs. 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. This website is in progress for completing the biography, references, historical notes, index, adding complementary lectures and videos, etc ... The videos and slides will remain unchanged, even if I find some mistakes or errors which will be added to this website.

**How to quote this video-book in the literature**

The videos, slides and the address of this video-book ABjC will remain unchanged. It is possible to quote this video-book in the literature in the same way as quoting a book. If you give only the direct link to a set of slides, or to the YouTube address of a video, the reader will miss the corresponding page giving the precise decription of the slides, video and direct links in order to navigate inside the video. Also think of our Chinese colleagues, for which the link to the video is only through the popular Chinese chain "bilibili".

I suggest the following:

- if you want to quote one of the four part (volume) of ABjC, I suggest to quote in an analogous way to a book, with the editor (here IMSc), the place of edition (here Chennai) and the year, for example for the second volume:

*The Art of Bijective Combinatorics*, Part II,* Commutations and heaps of pieces*, IMSc, Chennai, 2017, http://www.viennot.org/abjc2.html

the suffix /abjcn.html refers to Part "N" of ABjC and the pointer goes to the web page "Preface" of Part "N"

- if you want to quote a precise point inside the video-book, as for example a definition, theorem or bijection, I suggest to refer to the number(s) of the corresponding slide(s). Example: the "Tamil bijection" is introduced in

*The Art of Bijective Combinatorics*, Part III, *The cellular ansatz: bijective combinatorics and quadratic algebra*, Chapter 4c, IMSc, Chennai, 2019, p 178, http://www.viennot.org/abjc3-ch4.html

the suffix abjc3-ch4.html refers to the web page of Chapter 4 of Part "III", this page containing the different slides and videos of the 3 lectures Ch4a, Ch4b, Ch4c corresponding to this Chapter 4.

- if you want to quote the whole collection of the 4 volumes then:

*The Art of Bijective Combinatorics*, IMSc, Chennai, 2016-2019, http://www.viennot.org/abjc.html

the suffix /abjc.html refers to the web page of the general preface of ABjC (this page)

**Mirror image of this video-book**

A mirror image of this website is here, kindly supported by The Institute of Mathematical Science (IMSc), Chennai, India.

After March 16, 2019, I may not be able to update this mirror image. Please go to the original website www.viennot.org

**Special thanks**

to The Institute of Mathematical Sciences (IMSc), Chennai, to Amritanshu Prasad and the video department

**part I**: The playlist from matsciencechannel of the 17 videos of this course is here** **

**part II:** The playlist from matsciencechannel of the 19 videos of this course is here

**part III:** The playlist from matsciencechannel of the 22 videos of this course is here** **

**part IV:** The playlist from matsciencechannel of the 19 videos of this course is here** **

**last update: **february 8, 2021

Xavier Viennot

email: first name (at) name (dot) org

Some lectures for a wide audience** as an introduction to the Art of Bijective Combinatorics **