numerative ombinatorics

A pplications Enumerative Combinatorics and Applications ECA 2:1 (2022) Interview #S3I4 ecajournal.haifa.ac.il

Interview with Xavier Viennot Toufik Mansour

Xavier Viennot completed his studies at Ecole Normale Sup´erieure Ulm in 1969. He obtained a PhD at the Uni- versity of Paris in 1971, under the direction of Marcel-Paul Sch¨uzenberger. Since 1969 he has been working as Researcher at CNRS (National Centre for Scientific Research, France). His present position is Emeritus Research Director at CNRS, mem- ber of the combinatorial group at Laboratoire Bordelais de Recherche en Informatique (LaBRI), Universit´ede Bordeaux. Photo by Nicolas Viennot Professor Viennot held visiting positions at 20 different univer- sities or research centres in Europe, North and South America, India, China, and Australia. He has received several awards, including A. Chˆateletmedal (for his work in algebra) in 1974 and Silver Medal at CNRS (for his work in combinatorics and control theory) in 1992. He has given numerous communications and colloquia and has been invited speaker in some major conferences in pure mathematics, combinatorics, computer science and physics. The main research domains of Xavier Viennot is enumerative, algebraic and bijective combinatorics.

Mansour:a Professor Viennot, first of all, we torics (just think of the number of simple finite would like to thank you for accepting this in- groups!). Nowadays combinatorics, especially terview. Would you tell us broadly what com- bijective combinatorics, is rather an attitude binatorics is? that is transversal to all mathematics (and also Viennot: First, I want to thank you for your physics, computer science, theoretical biology, invitation to this interview with no limit in the etc) where classical (and non-classical) theo- length of my answers and anecdotes. There ries are studied with a “combinatorial” point are many possible definitions, and many dif- of view. This denomination “attitude” is due ferent kinds of combinatorics, such as enu- to Volker Strehl, one of the creators in 1980 merative, algebraic, bijective, analytic, “ex- of the “S´eminaire Lotharingien de Combina- istentialist”, extremal, geometric, probabilis- toire”1. tic, “integrable” (i.e. combinatorial physics), I believe that the world, at a very small scale, and even magic. Let us say that combina- such as Planck length is discrete. Michel torics is the study of finite mathematical ob- Mend`es-France, a number theorist in Bor- jects having a poor underlying structure. In deaux, was talking about the fractal character other words, some objects where a kid can give of time and that we are becoming older by in- examples once you explain to him the defi- finitesimal little jumps of time. We can think nition (permutations, Young tableaux, move- of space-time particles. In this sense, our en- ments of a tower on a chessboard, tilings the tire world is combinatorics and the continuum same chessboard with dimers, etc). Enumerat- is just an illusion in the same way when you ing finite structures is not necessarily combina- look at a computer screen with a high resolu-

The authors: Released under the CC BY-ND license (International 4.0), Published: April 23, 2021 Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is [email protected] 1See http://www.mat.univie.ac.at/~slc/. Interview with Xavier Viennot tion or a movie. G.C. Rota said that math- natorics, trying to put some order in the jungle ematicians were first studying continuum as a of various bijections, to develop some aspects first step before going to the study of the finite. of these new theories or methodologies, or re- With this (broad) point of view, the beauty of proved in a better way some known fact or for- Nature is a reflection of the underlying beauty mulae. I like to find the ultimate elegant bijec- of combinatorics. Even if it is a belief, I love tion which will explain at the same time differ- this point of view. ent formulae, even if bijective proofs already Mansour: What do you think about the de- exist for these formulae. Of course, I have also velopment of the relations between combina- been happy to work on specific open problems, torics and the rest of mathematics? find a formula or a bijection for a specific enu- Viennot: As I just said, I believe that combi- merative problem, such as the enumeration of natorics is an attitude transversal to all math- polyominoes and directed animals. An exam- ematics. The relation with the rest of mathe- ple was to find a bijection to prove the amazing matics is deep, fruitful, amazing, spectacular. formula2 3n for the number of compact source Many parts of mathematics, especially algebra, directed animals of size (n + 1)m. analysis, etc, can be rewritten with a combina- In particular, I have been developing the torial point of view. Classical theories have a following methodologies: commutations and new birth among the garden of combinatorial heaps of pieces, combinatorial theory of or- objects (for example, the theory of symmetric thogonal polynomials, and continued fractions, functions, representation of groups and Lie al- combinatorial theory of differential equations gebra, or in analysis, continued fractions and (with Pierre Leroux) and what I call the “cel- orthogonal polynomials). Even some very dif- lular ansatz” (i.e. the relation between bijec- ficult problems can be solved using combina- tive combinatorics on a grid and quadratic al- torics therapy. gebra). The starting point is bijections such I have been very lucky to belong to the are RSK (the Robinson-Schensted-Knuth cor- generation of students where this relation was respondence3,4,5) viewed with Fomin’s “local starting in all directions. Like the birth of a rules”6 or bijections coming from the PASEP star from different pieces of matter scattered model in physicsd relating permutations and in the space, the birth of modern combina- some tableaux7. torics is developing from many individual facts In the last ten years I concentrated my ef- and formulae. We are in a golden age of this forts to write (euh sorry ... to speak), a book part of mathematics. Everything was waiting (in fact a “video book”) on bijective combina- to be developed, from the foundations of the torics. This video-book is in four “Volumes” house to the bathroom of the seventh floor un- (called “Parts”), each Part corresponds to a til some sophisticated details of the balcony on course given at the IMSc (The Institute of the tenth floor. Mathematical Science, Chennai, India). This Mansour:b What have been some of the main video-book called “The Art of Bijective Com- goals of your research? binatorics”8 (ABjC for short) is the achieve- Viennot: Some people like to solve some spe- ment of the 1985 seminal paper on the the- cific problems. My preferences are to try to ory of heaps of pieces9 (Part II), the 1983 unify different facts or formulae, try to under- monograph on orthogonal polynomials10 (Part stand them, to develop a theory or a methodol- IV) and the development of the methodology ogy from them, in the spirit of bijective combi- I called the “cellular ansatz”11 (Part III). I am 2D. Gouyou-Beauchamps and X. Viennot, Equivalence of the two-dimensional directed animals problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), 334–357. 3D.E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific Journal of Mathematics 34 (1970), 709–727. 4G. de B. Robinson, On the representations of the symmetric group, American Journal of Mathematics, 60:3 (1938), 745–760. 5C. Schensted, Longest increasing and decreasing subsequences, Canadian Journal of Mathematics 13 (1961), 179–191. 6S. Fomin, Generalised Robinson-Schensted-Knuth correspondence, Journal of Soviet Mathematics, 41 (1988), 979–991. Trans- lation from Zapiski Nauchn Sem. LOMI 155 (1986), 156–175. 7E. Steingrımsson and L. Williams, Permutation tableaux and permutation patterns, J. Combin. Theory Ser. A 114:2 (2007), 211–234. 8See http://www.viennot.org/abjc.html. 9See http://www.viennot.org/abjc2.html. 10See http://www.viennot.org/abjc4.html. 11See http://www.viennot.org/abjc3-abstract.html.

ECA 2:1 (2022) Interview #S3I4 2 Interview with Xavier Viennot very extremely grateful to IMSc and my col- courses at the Ecole given by famous profes- league and friend Amritanshu Prasad (Amri) sors. Bruhat was the director for mathematics to have made possible this video book. studies, after 25 years of direction by Henri Mansour: We would like to ask you about Cartan and his famous seminar. The studies your formative years. What were your early are four years long. The ambiance was very experiences with mathematics? Did that hap- stimulating, science and letters students are pen under the influence of your family or some mixed and living together in the hot ambiance other people? of the Latin quarter in the middle of Paris. Viennot: At school, I loved physics and math- Freedom was the rule. I became friend with ematics and was a good student, nothing more. Alain Connes, who entered the school one year My father was an engineer and he wanted me after me. He get the Fields medal, respect- to be the same and pushed me to prepare for ing the tradition that the 10 Fields medal- the exams of the “great schools” as they are ists in France (as today), all of them come called in France, such as the prestigious Ecole from ENS Ulm (!). There was also the fa- Polytechnique. I succeeded at the “Ecole Nor- mous Physics Laboratory of ENS. I remem- male Sup´erieure”,rue d’Ulm, where instead of ber the special events when Alfred Kastler re- an engineering school and Ecole Polytechnique, ceived the Nobel Price in Physics. Former stu- I was attracted by the perfume of freedom with dents became Prime Minister (such as Laurent the passion for science and research. Fabius, promotion 1966) or President of France Mansour: Were there specific problems that such as Georges Pompidou. I remember a visit made you first interested in combinatorics? with some friends from ENS at Hotel Matignon Viennot: There were no specific problems when he was Prime Minister of President De that made my interest in combinatorics. In Gaulle. It was the time when candid students general, I just loved discrete and concrete were joking with a Prime Minister, few months mathematics. After my ”th`ese d’Etat”, I before the May 1968 “revolution”. The school learnt about the Robinson-Schensted corre- was created during the French revolution of spondence between permutations and pairs 1789 and was initially intended for the forma- of Young tableaux and wrote my first com- tion of college professors. At the historical en- binatorial paperj. Then I read Foata- trance one can read “D´ecretde la Convention, Sch¨utzenberger12 Lecture Note in Mathemat- 9 Brumaire, An III”. ics on Eulerian polynomials and was surprised We also had the privilege to have some spe- by the interpretations of secant and tangent cial lectures. This was the way I met Marco numbers with alternating permutations and Sch¨utzenberger for the first time. He gave a D´esir´eAndr´e’spermutations13. Since then, I lecture on a topic in theoretical computer sci- started my bijective exploration in the fasci- ence. When you meet such an original person, nating world with the combinatorial interpre- it is like falling in love. This one-hour lecture tations of the Genocchi numbers14. decided for me the rest of my scientific life. Mansour: What was the reason you chose the Mansour:c What was the problem you worked University of Paris for your Ph.D. and your ad- on in your thesis? visor Marcel-Paul Sch¨utzenberger? Viennot: In France, at that time, the process Viennot: As I mentioned above, I succeeded was the following: a “th`esede 3`emecycle”, a for Ecole Normale Sup´erieure.In France, after kind of “light” Ph.D., where you are supposed secondary schools, there are two options: uni- to spend 2-3 years, now replaced by “thesis” (= versity or the so-called “great schools”. But Ph.D.) and several years later, the “big thesis” paradoxically for Ecole Normale Sup´erieure, called “Th`esed’Etat”, now equivalent to what we follow courses at university (mainly the is called “Habilitation”. I am not going to de- University of Paris, very close to “rue d’Ulm” scribe my “th`esede 3`eme cycle” which contains in the Latin quarter) and have only a few some notions related to automata theory and 12D. Foata and M.-P. Sch¨utzenberger, Th´eorieg´eom´etriquedes polynˆomeseul´eriens, Lecture Notes in Mathematics, 138, Berlin, Springer-Verlag, 1970. Electronic reedition in the section “book” of SLC, see 1. 13D. Foata and M.-P. Sch¨utzenberger, Nombres d’Euler et permutations alternantes, A survey of Combinatorial Theory, J.N. Sri- vastava et al. eds., 173–187, Amsterdam, North-Holland, 1973. Available http://irma.math.unistra.fr/~foata/paper/pub18.pdf. 14D. Dumont, Interpr´etationscombinatoires des nombres de Genocchi, Duke Math. J. 41 (1974), 305–318.

ECA 2:1 (2022) Interview #S3I4 3 Interview with Xavier Viennot the premise of my “Th`esed’Etat”, but will go pretations, and in a second step lift all these directly to this last one. constructions in a more general framework of Sch¨utzenbeger15 introduced the notion of general theory. Maybe in a third step use such “factorizations” of free monoids. A particu- methodology for the construction of new bijec- lar example is the so-called “Lyndon words” tions. which are in bijection with a basis of the free Mansour: When you are working on a prob- Lie algebra. The subject of the “Th`esed’Etat” lem, do you feel that something is true even was to make a general theory of the relation before you have the proof? between various classes of “complete” factor- Viennot: When I work hard on a specific re- izations and the constructions of the various search problem or in the construction of a more basis of free Lie algebra, including all known general bijective methodology, I have some- basis. For general factorizations, the relation times the feeling to be connected to a wonder- is with the construction of a decomposition of ful word underlying the so-called “real” word the free Lie algebra into a direct sum of (free) that we can see. It can be a brief flash or a de- Lie subalgebra, together with the construction licious sensation of floating on a cloud (a few of a “basic family” freely generating each sub- hours or even several days). I have a strong algebra. impression that something is true before hav- Such topic has a flavor of theoretical com- ing the proof. In the beginning, it cannot be puter science with the study and construc- put in terms of words, but I feel that it is true. tions on words and free monoids, together Maybe it will take weeks or months to give the with combinatorial algebra (groups and Lie proof and write it. Sometimes it happens to algebra defined by relations and generators, me that long after, I found a counter-example, Michel Lazard’s commutators calculus in free but with a slight modification of the definitions groups16, etc). involved, the initial intuitive idea became true. Pierre Cartier was the referee of my “Th`ese d’Etat” and for publication as a Lecture Notes Mansour:d What three results do you con- in Maths he asked me to restrict myself to sider the most influential in combinatorics dur- “complete” factorizations, equivalent to the ing the last thirty years? basis of free Lie algebra, and thus to rewrite Viennot: Of course I will talk only about my completely the manuscript (!). Luckily, at that favorite and familiar world of enumerative with time, when you enter at CNRS as a trainee, my bijective combinatorics glasses. Instead of not only you have an adviser but also a “God- selecting three particular results, I think it is father”, in the person of Dominique Foata (in better to quote three hot and influential areas French “parrain”, there is no mention of God). of research where there is a package of major I am extremely grateful to him for the help results. to write these Lecture Notes. With his many First of all, I will quote the long-standing corrections, I learned how to write. conjecture (Mills, Robins, and Rumsey17) for Mansour: What would guide you in your re- the enumeration of alternating sign matrices search? A general theoretical question or a (ASM). Such objects are very simple to define specific problem? and can be viewed as an extension of permuta- Viennot: In the first step, I like to start from a tions. It has been first solved by Zeilberger18 specific identity, find a combinatorial interpre- in 1992, in association with the proof by An- tation of both sides, and then find a bijection drews19 for the number of the so-called totally that will give proof of the identity. This can symmetric self-complementary plane partitions be very difficult, maybe impossible. You need (TSSCPP), which are enumerated by the same some imagination. Then I will try to relate number. A second shorter proof, using the six- this bijection with other bijections and inter- vertex model in statistical physics was given 15M.-P. Sch¨utzenbeger, On a factorization of free monoids, Proc. Amer. Math. Soc. 16 (1965), 21–24. 16M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Annales Sci. ENS, 3, 71 (1954), 101–190. 17W. H. Mills, D. P. Robbins, and H. C. Rumsey Jr. Proof of the Macdonald conjecture, Invent. Math. 66:1 (1982), 73–87. 18D. Zeilberger, Proof of the alternating sign matrix conjecture, Electron. J. Combin. 3:2 (1996), Article R13. 19G. E. Andrews, Plane partitions V: The TSSCPP conjecture, J. Combin. Theory Ser. A 66:1 (1994), 28–39. 20G. Kuperberg, Another proof of the alternating sign matrix conjecture, Int. Math. Res. Not. 3 (1996), 139–150.

ECA 2:1 (2022) Interview #S3I4 4 Interview with Xavier Viennot by Kuperberg20 in 1996. Nobody could surpass Richard Askey, who was Other combinatorial objects are enumer- called “the fastest chalk in West”! Then in the ated by the same numbers as ASM: descend- 80’s appear transparencies, overhead projec- ing plane partitions (DPP) introduced by An- tors, and came slowly the combinatorial climb- drews21 in 1979, alternating sign triangles ing of the Askey tableau with interpretations (AST) introduced by Ayyer, Behrend and Fis- of the coefficients or generating function for cher22 in 2016. At the end of the 90’s a book Laguerre, Jacobi, Hahn, ... polynomials (by 23 Bergeron, Leroux, Foata, Strehl, Kreweras, La- on these topics has been written by Bressoud , 30 just before another conjecture was discovered belle, Yeh , ...). The proof of Mehler’s formula by Razumov and Stroganov24: these numbers is now an exercise in a standard course on bi- appear in Physics with quantum spins chains, jective combinatorics. conjecture solved by physicists Cantini and The interpretation of the moments of Sportiello25 in 2010 with bijective techniques. Askey-Wilson polynomials is in fact the cul- Finding bijections between these classes is a mination of a series of works in physics and challenge for more than 30 years. Very recently in combinatorics. In physics, it is the famous a bijective construction has been given by Fis- PASEP (also called ASEP) model (partially cher and Konvalinka26. But many mysteries asymmetric exclusion process), a toy model in are still open. I do not see an interest to select the physics of dynamic systems far from equi- which one is the most influential between these librium and the computation of the stationary remarkable results that solve hard conjectures. probabilities. The most general case (with five parameters q, α, β, γ, and δ) has been solved in A second fascinating result (to me) is the 31 first combinatorial interpretation of the mo- 2003 by Uchiyama, Sasamoto, and Wadati . ments of the Askey-Wilson polynomials by The Askey-Wilson polynomials are central in Corteel and Williams27 in 2009 with the so- this resolution. In combinatorics, it is the in- terpretation of these probabilities in terms of called staircase tableaux. These polynomials 32 are at the top level classification of orthogo- alternative tableaux (in bijection with permu- nal polynomials (the famous Askey tableau28). tations in the case of 3 parameters q, α, and When I start my life as a combinatorist, it was β) coming naturally from a quadratic algebra time for interpretation of Hermite polynomi- related to this PASEP, in the same way, the als, at the bottom level of the Askey tableau. famous RSK correspondence between Young tableaux and permutations is related to the At the end of the 70’s, I remember a talk at f Oberwolfach given by Foata29 about his proof Heisenberg quadratic algebra . of Mehler’s formula for Hermite polynomials. A third fascinating subject is diagonal har- It was the time of blackboards and chalks (but monics and the resolution of the so-called with colors chalks, a must for combinatorics!). (n + 1)n−1 conjecture33, n! conjecture34, delta 21G. E. Andrews, Plane partitions (III): The weak Macdonald conjecture, Invent. Math. 53 (1979), 193–225. 22A. Ayyer, R. E. Behrend, and I. Fischer, Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order, Adv. Math. 367 (2020), Article 107125. 23D. Bressoud, Proofs and confirmations. The story of the alternating sign matrix conjecture, MAA Spectrum, Mathematical Association of America and Cambridge University Press, Washington, DC and Cambridge, 1999. 24A. V. Razumov and Yu. G. Stroganov, Spin chains and combinatorics, J.Phys. A34 (2001), Article 3185. 25L. Cantini and A. Sportiello, Proof of the Razumov–Stroganov conjecture, J. Combin. Theory, Ser. A 118:5 (2011), 1549–1574. 26I. Fischer and M. Konvalinka, The first bijective proof of the refined ASM, S´em.Lothar. Combin. 84B (2020), Article #18. 27S. Corteel and L. K. Williams, Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials, PNAS 107:15 (2010), 6726–6730. 28R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54:319 (1985), 1–55. 29D. Foata, A combinatorial proof of the Mehler formula, J. Combin. Theory, Ser. A 24:3 (1978), 367–376. 30X. Viennot, The Art of Bijective Combinatorics, Part IV, A combinatorial theory of orthogonal polynomials and continued fractions, Chapter 5a and 5b, IMSc, Chennai, 2019. See http://www.viennot.org/abjc4-ch5.html. 31M. Uchiyama, T. Sasamoto, and M. Wadati, Asymmetric simple exclusion process with open boundaries and Askey–Wilson polynomials, J. Phys. A: Math. Gen. 37:18 (2004), Article 4985. 32X. Viennot, Alternative tableaux, permutations and partially asymmetric exclusion process, talk in the workshop “Statistical Mechanics and Quantum Field Theory Methods in Combinatorial Enumeration”, Isaac Newton Institute for Mathematical Sciences, 23 April 2008. See https://sms.cam.ac.uk/media/1004. 33A. M. Garsia and M. Haiman; A graded representation model for the Macdonald polynomials, Proc. Nat. Acad. Sci. 90:8 (1993), Article 36073610. 34M. Haiman, Hilberts schemes, polygraphs, and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14:4 (2001), 941–1006.

ECA 2:1 (2022) Interview #S3I4 5 Interview with Xavier Viennot conjecture35, shuffle conjecture36,37,(q, t)- given in 42. Jousuat-Verg`es43 gave an inter- Catalan38, etc. We are in deep algebraic pretation with the alternating tableaux of the combinatorics. Many people are working PASEP11. You may ask what is the relation hard for thirty years, in particular Garsia with the symmetry problem for Askey-Wilson the pioneer in this story, Bergeron, Haiman, polynomials? Gessel and Zeng showed that the Haglund, Loehr, Armstrong, and many oth- 3-variables polynomials expressing the ternary ers. But many mysteries are waiting to be ex- symmetry of Genocchi numbers are moments plained. I am not qualified to talk on that of some orthogonal polynomials known as con- subject, I just follow from far away this sub- tinuous dual Hahn polynomials, an important ject, talking with my friends Adriano Gar- sequence in the Askey-Wilson hierarchy. I sus- sia and Fran¸coisBergeron, except that I have pect the existence of a super combinatorial just been involved in some related work with object above the staircase tableaux interpret- Louis-Fran¸coisPr´eville-Ratelle39, see my An- ing the moments of Askey-Wilson polynomials swerf about Tamari and diagonal harmonics. where the 4-parameters symmetry appear nat- Mansour:e What are the top three open ques- urally, together with the 3-variables related to tions in your list? Hahn polynomials and Genocchi polynomials, Viennot: In the previous question, I men- and why not going through the interpretation tioned 3 fascinating areas of research with re- of Koorwinder polynomials discussed in f and markable achievements. In each of these ar- going down to the famous problem of finding eas, the story is far from the end. I will a natural explanation to the symmetry of the mention only open questions which bother me (q, t)-Catalan numbers discussed just below. It for a long time. First, coming back to the is possible to dream. interpretation of the Askey-Wilson polynomi- In the same philosophy about combinatorial als with some tableaux. The interpretation interpretations where you can “see” in a “nat- of the moments of Askey-Wilson polynomials ural” way the symmetry, I have always been has been improved with Corteel, Stanley, Stan- puzzled by the different interpretations of the ton, and Williams40. These polynomials are (q, t)-Catalan polynomials38 (Dyck paths, area, q-polynomials with 4 parameters α, β, γ, and bounce parameter, ...) mentioned above, but δ having natural symmetries. One cannot see where the symmetry of the two parameters q these symmetries on the tableaux. An ultimate and t is hidden. I remember Adriano Garsia interpretation explaining these symmetries is introducing me to this problem 30 years ago missing. when we were in the magnificent environment Maybe this symmetry problem is related to at Mittag-Leffler Institute. another analog problem. Dumont and Foata41 I would have many other top open questions discovered a remarkable ternary symmetry on about bijective proofs and combinatorial inter- Genochi numbers. To my knowledge, to- pretations. Let’s mention a question that re- day, there is no combinatorial interpretation mains in a corner of my mind for 40 years. In of Genocchi numbers where the ternary sym- statistical physics, there is the so-called hard metry appears ”naturally”. A discussion is hexagon gas model. This model was solved 35J. Haglund, J. B. Remmel, and A. T. Wilson, The delta conjecture, Discrete Math. Theor. Comput. Sci. proc. BC (2016), 611–622. 36J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov. A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J. 126 (2005), 195–232. 37E. Carlsson and A. Mellit, A proof of the shuffle conjecture, J. Amer. Math. Soc. 31 (2018), 661–697. 38A. Garsia and J. Haglund, A proof of the q, t-Catalan positivity conjecture, Adv. Math. 175 (2003), 319–334. 39L.-F. Pr´eville-Ratelleand X. Viennot, The enumeration of generalized Tamari intervals, Trans. Amer. Math. Soc. 369 (2017), 5219–5239. The Journal gave a wrong title, it should be “An extension of Tamari lattice”. 40S. Corteel, R. Stanley, D. Stanton, and L. Williams, Formulae for Askey-Wilson moments and enumeration of staircase tableaux, Tran. Amer. Math. Soc. 364:11 (2012), 6009–6037. 41D. Dumont and D. Foata, Une propri´et´ede sym´etriedes nombres de Genocchi, Bull. Soc. Math. France 104 (1976), 433–451. 42X. Viennot, Interpr´etationcombinatoire des nombres d’Euler et de Genocchi, S´eminairede Th´eoriedes nombres de Bordeaux, Publi de l’Universit´ede Bordeaux I, 1982–83, §5;4, 54–60. See http://www.xavierviennot.org/xavier/articles_files/Euler_ Genocchi81.pdf. 43M. Jousat-Verg`es, Generalized Dumont-Foata polynomials and alternative tableaux, S´em.Lothar. Combin. 64 (2010), Article B64b. 44R. J. Baxter, Hard hexagons: exact solution, J. Phys. A: Mathematical and General 13:3 (1980), L61–L70. 45G. E. Andrews, The hard-hexagon model and the Rogers-Ramanujan type identities, Proc. Natl. Acad. Sci. 78 (1981), 5290–5292.

ECA 2:1 (2022) Interview #S3I4 6 Interview with Xavier Viennot by Baxter44 using Roger-Ramanujan identities see that I am more interested in develop- and 14 analogous identities45. “Solved” means ing general bijective methodologies (starting giving an expression (in fact a system of equa- from particular facts, formulae, or bijections) tions) for the partition function, or equiva- than solving particular problems. Three main lently the so-called density of the gas. G.S. methodologies emerge from my work: heaps Joyce proved that this density is a formal of pieces, combinatorial theory of orthogo- power series that satisfies an algebraic equa- nal polynomials with weighted paths, and a tion (of degree 12). The open question is to third methodology I propose to call “cellular “explain” combinatorially this algebricity. ansatz”. These 3 methodologies are Parts II, When I was at Tata Institute in Bombay III, and IV of ABjCb. (Mumbai), Deepak Dhar gave me a system of Heaps of pieces (Part II of ABjC)49 are algebraic equations (given by some colleagues related to various topics such as orthogo- working on modular forms). I was surprised. nal polynomials, algebraic graph theory (chro- This system involves some simple algebraic matic, spanning trees, zeros of matching poly- equations having a flavor of Catalan numbers nomials), representation theory of Lie alge- and some similarity with the system of equa- bras, fully commutative elements in Coxeter tions for the number of planar maps given a groups, Petri nets in computer science, Ising long time ago by Tutte, where various combi- model and gas model in statistical mechanics, natorial explanations have been given starting Lorentzian triangulations in quantum gravity. with the first bijection of Cori and Vauquelin46 Much beautiful works on the relation between (as a difference of two algebraic languages), heaps and representation of Lie algebra has then the “direct” bijection of Schaeffer47, and been done by Richard Green50 and its students also many others, in particular in relation with at Boulder University. I wish the continuation physics48. Using the theory of heaps of pieces, of such fruitful developments. the density of the gas becomes the generat- ing function of pyramids of hexagons on a tri- I am somewhat surprised to see that re- angular lattice. From my education in Paris cently several papers have appeared in the with Sch¨utzenberger and being since 40 years last six months on arXiv where heaps method- at LaBRI, a computer science laboratory, I ology plays a key role, while the basic pa- have a background in theoretical computer sci- per on heaps appears in 1985 and the Lec- ence. Algebricity is sometimes related to al- ture Notes (in French) on orthogonal poly- gebraic languages which can be recognized by nomials in 1983. Here are some: Garsia automata with one stack. With two stacks and Ganzberger (heaps and orthogonal poly- you lose the algebricity. From a naive point of nomials)51; Tamm, Pospelov, and Nechaev view, trying to recognize pyramids of hexagons (heaps in statistical mechanics)52; Giscard would need an automaton with many (infinite!) (heaps, paths and self-avoiding polygons)53; stacks. Bagno, Biagioli, Jouhet, and Roichman (heaps Mansour:f What kind of mathematics would and fully commutative elements in Coxeter you like to see in the next ten-to-twenty years groups)54; Cigler and Krattenthaler (heaps as the continuation of your work? and orthogonal polynomials)55; Fredes and Viennot: From my previous answers you can Marckert (heaps and spanning tree in prob- 46R. Cori and B. Vauquelin, Planar maps are well labeled trees, Canad. J. Math. 33 (1981), 1023–1042. 47G. Schaeffer, Conjugaison d’arbres et cartes combinatoires al´eatoires, Ph.D. thesis, Universit´eBordeaux I, 1998, (1999 SPECIF price). See http://www.lix.polytechnique.fr/~schaeffe/Biblio/PhD-Schaeffer.pdf. 48J. Bouttier, P. Di Francesco, and E. Guitter, Planar maps as labeled mobiles, Electron. J. Combin. 11:1 (2004), Article #R69. 49See http://www.viennot.org/abjc2.html. 50R. M. Green, Combinatorics of minuscule representations, Cambridge University Press, 2013, Cambridge Tracts in Maths 199. 51A. M. Garsia and G. Ganzberger, Fibonacci polynomials, https://arxiv.org/pdf/2009.10213.pdf. 52M. Tamm and N. Pospelov and S. Nechaev, Growth rate of 3D heaps of pieces, arXiv: 2009.12540v2 [cond-math.stat-mech], 2020. 53P.-L. Giscard, Counting walks by their last erased self-avoiding polygons using sieves, Discrete Math. 344:4 (2021), Article 112305. 54E. Bagno, R. Biagioli, F. Jouhet, and Y. Roichman, Block number, descents and Schur positivity of fully commutative elements in Bn, https://arxiv.org/abs/2012.06412. 55J. Cigler and C. Krattenthaler, Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity, https://arxiv.org/abs/2012.03878.

ECA 2:1 (2022) Interview #S3I4 7 Interview with Xavier Viennot abilities)56; Rani and Arunkumar (heaps and from RSK defined from a representation of the Borcherds-Kac-Moody Lie superalgebras, with Heisenberg algebra, as explained by Fomin6 some flavor of Lalonde’s Lyndon heaps)57. with “local rules”. These local rules are usu- With Lyndon heaps, we are in the flavor of ally defined by attaching 3 Ferrers diagrams Lyndon basis of free Lie algebrac. I have no to 3 adjacent vertices of an elementary cell doubts that heaps will continue to have a fruit- on the square lattice. This process can also ful and exciting life. be applied for the PASEP algebra and leads About orthogonal polynomials (part IV of to the known bijections between permutations ABjC)10, my first dream would be to see an and alternating tableaux, corresponding to the ultimate, complete, and unified combinato- PASEP with 3 parameters. Another quadratic rial “understanding” of the classical orthogonal algebra related to the PASEP with two types of polynomials, following two paths in the Askey- particle leads, applying the philosophy of the Wilson tableau. An ascending path starting “cellular ansatz” to the notion of rhombic al- from the known interpretations of the coeffi- ternating tableaux (work with Mandelshtam60, cients (or generating function) of the polyno- FPSAC 2016) and bijections with assembl´ees mials (Hermite, Laguerre, Jacobi, Hahn, ...) of permutations. and reaching the top with the Askey-Wilson The process of extending alternative polynomials. A descending path starting from tableaux (PASEP with 3 parameters) to stair- an interpretation of the moments of the Askey- case tableaux (PASEP with 5 parameters) Wilson polynomials where the symmetries of has an analog with this rhombic alternative the four parameters can be seene, and then tableaux and leads to an interpretation of the going down through Hahn and Jacobi poly- Koorwinder polynomials with rhombic stair- nomials and reaching the known interpreta- case tableaux (work of Corteel, Mandelshtam, tions of moments for Laguerre, Tchebycheff, and Williams61). Now I can see as a spectator Hermite polynomials. Inspiration maybe can the development of this new active field where come from the “bridge” between Hermite and the tower going from Hermite polynomials to Askey-Wilson with our (semi)-bijective proof Koorwinder polynomials via Askey-Wilson is (with Ismail and Stanton58) for the famous joining the tower going from Young tableaux, Askey-Wilson integral expressed as an inte- symmetric function, MacDonald polynomials. gral of a product of four q-Hermite polynomi- Maybe an extension of the cellular ansatz phi- als. An exciting fact is a relation between or- losophy is possible and will be useful? thogonal polynomials and the combinatorics of Going back to the beginning of the motiva- ASM (alternating sign matrices) discussed in d. tion of the cellular ansatz with RSK, represen- Colomo and Pronko59 showed that some enu- tation of the Heisenberg algebra and Fomins’s meration problems of ASM can be solved using local rules, I think it is better to write these some Hankel determinants related to Continu- local rules by attaching labels on the edges, as ous Hahn, Meixner-Pollaczek, and continuous shown in the paper62 11th GASCom, Athens, dual Hahn polynomials. 2018. Then RSK can also be defined as the re- Finally, the “cellular ansatz” (Part III of sult of another process I call “demultiplication ABjC)11 contains some general tools for con- of equations” in a quadratic algebra. In gen- structing new bijections. Let me explain eral, we can define the dual of this quadratic briefly the philosophy of this “cellular ansatz”, algebra. In the case of RSK and the Heisen- which is not written anywhere. We start berg algebra is self-dual and no new bijection 56L. Fredes and J.-F. Marckert, Aldous–Broder theorem: extension to the non reversible case and new combinatorial proof, https://arxiv.org/pdf/2102.08639.pdf. 57S. Rani and G. Arunkumar, A study on free roots of Borcherds-Kac-Moody Lie Superalgebras, https://arxiv.org/abs/2103. 12332. 58M. E. H. Ismail, D. Stanton, and X. Viennot, The Combinatorics of q-Hermite polynomials and the Askey—Wilson Integral, Europ. J. Combin. 8:4 (1987), 379–392. 59F. Colomo and A.G. Pronko, Square ice, alternating sign matrices and classical orthogonal polynomials, arXiv:math- ph/0411076v2, 2004. 60O. Mandelshtam and X. Viennot, Rhombic alternative tableaux and assembl´eesof permutations, Europ. J. Combin. 73 (2018), 1–19. 61S. Corteel, O. Mandelshtam, and L. Williams, Combinatorics of the two-species ASEP and Koornwinder moments, Adv. Math. 321 (2017), 160–204. 62See http://ceur-ws.org/Vol-2113/paper22.pdf.

ECA 2:1 (2022) Interview #S3I4 8 Interview with Xavier Viennot is deduced. For the PASEP algebra, the dual to me. Around a pisco sour, he explained to algebra is different. The tableaux become the me his work65 in the context of diagonals har- well-known tree-like tableaux63 associated with monic, the problem of extending the Tamari the PASEP (with 3 parameters), and the ana- lattice for the so-called rational Catalan com- log principle gives rise to new bijections (such binatorics, corresponding to define an analog of as the “Tamil bijection” for binary trees and Tamari lattice on paths (with North and East the “Adela bijection” for permutations). steps) located below a line of rational slope. These are powerful tools that should be de- His adventurous idea was to search for a much veloped. In particular for alternating sign ma- more general Tamari lattice for the set of paths trices (ASM) and fully packed loops (FPL), that are below any arbitrary path ν with North which can also be defined as some tableaux as- and East steps. He had a possible definition sociated with a quadratic algebra with 4 gen- with 3 beautiful conjectures. He has a strong erators. feeling that this was related to the talk I just Another topic I like and which should con- gave. The next day we realized39 that the ν- tinue its development is around Tamari lat- Tamari he was looking for was in bijection with tices. On the set of binary trees enumerated by the interval of binary trees with a given canopy Catalan numbers, inspired by the associativity ν. property for well parenthesis expressions, Dov People were looking for an extension of Tamari has defined a lattice in his thesis, Uni- Tamari lattice to rational Tamari, Louis- versit´ede Paris, 1951. A classical geometric Fran¸coisfor a much more general extension, representation is a so-called associahedron. and this extension was there, inside the ordi- Let us tell another story. In November nary Tamari! We presented this work at FP- 2013, I was invited to give a talk at the work- SAC 2015 in South Korea and published it in shop in Madrid “Recent trends in Algebraic the Transactions of the AMS39 amazingly the and Geometric Combinatorics” where many title given in the Transactions is wrong (!), “Tamarists” would be there. I should present the correct title is “An extension of Tamari something related to Tamari but had nothing lattice”, see https://arxiv.org/pdf/1406. in my combinatorial suitcase. In the past I 3787.pdf. used the notion of “canopy” of a binary tree in Immediately, this was the starting point of relation with some bijections solving the enu- many beautiful works by Fang and Pr´eville- meration of convex polyominoesl. The canopy Ratelle66, Caballos, Padrol and Sarmiento67,68 of a binary tree is a word in two letters which and many others, in particular by Defant69.I has been defined (without giving a name) by see that we are going from ν-Tamari to the Loday and Ronco64 in the context of Hopf al- combinatorics of PASEP described above. gebras, analog to up-down sequences for per- Mansour: Do you think that there are core or mutations. For the Madrid workshop, I study mainstream areas in mathematics? Are some the behavior of the canopy in the Tamari lat- topics more important than others? tice and made a modest contribution showing Viennot: Mainstream areas or important sub- that binary trees with a given canopy form an jects will vary from time to time and is sub- interval of the Tamari lattice. ject to one personal’s taste. One quality for Then at the invitation of Luc Lapointe, we a domain is to be active, with plenty of facts, move to Talca, Chile, for a few months, and I formulae, problems that are all connected to start to repeat the talk given in Madrid. At the each other by a network of links, correspon- end of the talk, a young qu´ebecois post-doc, dence. Moreover, another important criterium Louis-Fran¸coisPr´eville-Ratellewanted to talk is when the domain has many interactions with 63J.-C. Aval, A. Boussicault, and P. Nadeau, Tree-like tableaux, Elec. J. Comb. 20 (2013), Article P34. 64J.-L. Loday and M. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139:2 (1998), 293–309. 65F. Bergeron and L.-F. Pr´eville-Ratelle, Higher trivariate diagonal harmonics via generalized Tamari posets, J. Combin. 3:3 (2012), 317–341. 66W. Fang and L.-F. Pr´eville-Ratelle, The enumeration of generalized Tamari intervals, European J. Combin. 61 (2017), 69–84. 67C. Ceballos, A. Padrol, and C. Sarmiento, Geometry of ν-Tamari lattices in types A and B, Trans. Amer. Math. Soc. 371 (2019), 2575–2622. 68C. Ceballos, A. Padrol, and C. Sarmiento, The ν-Tamari lattice ν-trees, ν-bracket vectors, and subword complexes, Electron. J. combin. 27:1 (2020), Article #P1.141. 69C. Defant, Meeting covered elements in ν-Tamari lattices, arXiv: 2104.03890v1 [math.CO], 2021.

ECA 2:1 (2022) Interview #S3I4 9 Interview with Xavier Viennot other domains and other fields. Nowadays ev- tains are the more important? Himalaya or erything goes so fast. New domains are just Andes? Alpes or Pyr´en´ees?I went to the four born and a few years later appear to be “impor- of them and my preference is for Pyr´en´ees, tant”, such as the beautiful new notion of clus- even if the highest mountain is Pic d’Aneto ter algebra of Fomin and Zelevinski70. Even in with only 3404 meters high. a few years, it became a new domain classified Mansour:g What do you think about the dis- in Maths Review 13F60. Although it is clas- tinction between pure and applied mathemat- sified as a subdomain of commutative algebra, ics that some people focus on? Is it mean- it contains a lot of combinatorics. ingful at all in your case? How do you see the In 1977, in “Panomara des math´ematiques relationship between so-called “pure” and “ap- pures, le choix bourbachique”, Dieudonn´e71 plied” mathematics? gave a classification of mathematical theories Viennot: You can go by a continuous and into six classes. The best seems to be in fruitful path from the purest mathematics to class IV “Les probl`emesqui s’ordonnent au- the most applied mathematics. It is not a lin- tour d’une th´eorieg´en´erale, f´econde et vivante, ear path, it a network of fruitful connections in avec l’apport ininterrompu de probl`emesnou- all directions. In my case I have been naturally veaux”. In this classification Combinatorics is connected via “pure” notions of combinatorics in class II “les probl`emessans post´erit´es”. In to various topics such as the shape of rivers in more than 40 years what a change! I think Hydrogeology, mRNA (now well known with “Combinatorics” can be classified in class IV the vaccine for Covid-19), radiology of lungs, and that the classification of Maths Review is control theory, etc. I spent maybe one year somewhat obsolete. But how to classify the of my life applying pure beautiful concepts re- subdomains of a domain which is transverse lated to trees and Catalan numbers to com- to all mathematics, with incursion in physics, puter graphics72. Let me talk a little about computer science and theoretical bsiology? this adventure. Maybe the best criterium is “beauty”. The idea comes from a parameter on bi- Mathematics is a form of art, beauty is fun- nary trees, called Strahler number, introduced damental. The paradox is that at first, one by hydrogeologists Horton and Strahler in the can believe that he or she is creating a piece study of rivers networks, which also appears of this art, a nice bijection, a nice space, but in computer science as the minimum number after some more exploration, one will realize of registers needed to compute an arithmeti- that all this was already there, no piece of art cal expression. The generating function has has been created, we are just walking in a mu- very beautiful properties. Flajolet and co- seum, but not a closed museum with past dirty authors73 gave asymptotic properties related collections, but a living world, like exploring to Delange function in number theory and mountains. also Riemann zeta function. Fran¸con74 gave Some mountains like the mountains near a highly recursive bijective proof with Dyck Chamonix, Mont-Blanc have been explored in paths. Surprisingly another parameter (com- every corner of each face and peaks and it is plexity of the secondary structure of RNA) to- impossible to inaugurate a new route some- tally different, appears to have the same dis- where. Probably all main peaks in Himalaya tribution (and it is really difficult to prove it). and Andes have been climbed (except an im- This is the Ph.D. thesis of Mireille Vauchaus- portant one for Tibetan, Kailash Mount, and sade de Chaumont75. With Didier Arqu`es should never be climbed). Comparing explo- (from Besan¸con), and our students Georges ration of maths and mountains, which moun- Eyrolles and Nicolas Janey72, we start from 70S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15:2 (2002), 497–529. 71J. Dieudonn´e, Panorama des math´ematiques pures. Le choix bourbachique, Gauthier-Villars, Paris, France, 1977. 72D. Arqu`es,G. Eyrolles, N. Janey, and X. Viennot, Combinatorial analysis of ramified patterns and computer imagery of trees, Proc. SIGGRAPH’89, Computer Graphics 23 (1989), 31–40. 73P. Flajolet, J. C. Raoult, and J. Vuillemin, The number of registers required for evaluating arithmetic expressions, Theoret. Comput. Sci. 9 (1979), 99–105. 74J. Fran¸con, Sur le nombre de registres n´ecessaires `al’´evaluationd’une expression arithm´etique, RAIRO Inform. Th´eor. 18 (1984), 355–364. 75M. V. de Chaumont and X. Viennot, Enumeration of RNAs by complexity, Proc. Intern. Conf. of Medicine and Biology, Bari, Italie, 1983. Lecture Notes in Biomathematics, 1985.

ECA 2:1 (2022) Interview #S3I4 10 Interview with Xavier Viennot this Strahler analysis for binary trees to do the end of our stay, he found a very nice “di- some synthetic images of natural trees, we rect” bijection78 replacing the one of Fran¸con, work very hard to get some beautiful images, a style of bijection not familiar in combina- submit a paper to the big annual meeting SIG- torics where the reverse bijection is defined in GRAPH in USA (20 thousands participants, the same way. But there was a big problem. 5000 for the scientific part) and we were ac- His bijection was using an intermediate cepted. We get a glimpse of another world. structure between paths and binary trees, a You have to give a talk in front of thousands very classical data structure in computer sci- of people (Boston 1989) in a huge room used ence, called “heaps”. Nothing to do with heaps for meetings for presidential elections, with 3 “`ala Xavier”. Don wanted me to change the big screens (two for your slides, one for video name “heap” throughout my works. I told or for your face which is lighted with very him that this was not possible, there were al- strong projectors, so that you do not see any- ready many papers in various domains using body! I cannot use my usual transparencies the name “heaps”. Even worse: in our con- with an overhead projector and do the usual versations during this Mittag-Leffler month, I so-called “viennotique”. Here, there are no was inspired to introduce a fourth combinato- transparencies, just slides in the old way (on rial interpretation of Strahler numbers, with a a photographic plate). I followed the advice new interpretation of Catalan numbers as se- of my colleague Claude Puech, and after 3 or quences of heaps of dimers on cylinders of size 4 slides, without seeing anybody, I hear the 2, 4, 8, 16, ... and Don was going to put it in his crowd laughing. I know it was going to go report with his new bijection using “computer well. After that, we have been invited speak- science heaps”. Finally, I proposed to call this ers at IMAGINA’90, the International forum interpretation “Kepler towers”79 for different of new images of Monte-Carlo, inauguration reasons: this is a reminder of Kepler model by the Prince Albert de Monaco, luxury hotel for solar system and heaps can also be used in and helicopter to go to Nice airport, an invita- quantum gravity, see k. Don prefered to make tion to give a course at EUROGRAPHICS, etc. a projection on a plane of these sequences of Then, having a glimpse of such world, having cylinder heaps, looking like a CD Rom, see the pushed the Strahler idea at the maximum, it cover of Stanley’s book on Catalan numbers. was good to stop and go back to our world of But back home, another problem rises. Don bijective combinatorics. wanted to use heaps (`ala Xavier) in his Volume Let’s tell another personal story. Sev- 4 of TAOCP, Pre-fascicle 6A, section 7.2.2.2 eral years after this incursion into the world about satisfiability, starting with Lov´asz’slocal of Computer Graphics, I have the chance of lemma, and of course, he cannot use the term spending one month at Mittag-Leffler Institute “heaps” in Computer Science books. What with Donal Knuth. He loves the Strahler pa- to do? After long exchanges of emails, Don rameter with the 3 interpretations on binary agreed to use in his book the French name for trees, Dyck paths, and planar trees (coming heaps, that is “empilements”. from the mathematics of mRNA), and the 3 Mansour: Do you think that research and the bijections between them: between binary trees field of teaching mathematics can benefit from and Dyck paths (Fran¸con74), between binary the notion of video lectures? trees and planar trees (Zeilberger76 who won Viennot: About video/lecture, there is a long the price of 10 bottles of “Domaine des Mat- discussion in Igor Pak’s blog80 “You should tes” 1981) and a bijection between planar trees watch combinatorics videos!”. (May 2, 2015), and Dyck paths77. Don had worked hard to im- and a collection of 400 videos (prior to 2015) on prove these bijections. The night just before his website. There are more and more videos 76D. Zeilberger, A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number, Discrete Math. 82 (1990), 89–92. 77X. Viennot, A Strahler bijection between Dyck paths and planar trees, Discrete Math. 246:1-3 (2002), 317–329. 78D. Knuth, Program to read: Zeilberger, Fran¸con, and Viennot. See https://www-cs-faculty.stanford.edu/~knuth/ programs/francon.w. 79X. Viennot, Kepler towers, Catalan numbers and Strahler distribution, FPSAC’05, special session dedicated to Adriano Garsia, June 2005, Taormina, Italy. http://www.xavierviennot.org/xavier/articles_files/Kepler-Towers2.pdf. 80See https://igorpak.wordpress.com/page/3/.

ECA 2:1 (2022) Interview #S3I4 11 Interview with Xavier Viennot related to combinatorics. It can be a series can be spoiled when bijections are described in of lectures for a full course, or a lecture in a a written way. For each conference or course, seminar, workshop, or congress. You will find there are downloadable slides, a full page de- pieces of advice on how to watch videos and voted to a detailed map of the talk, with sec- also how to give a talk which is going to be tions, subsections, key facts, definitions, or recorded and available for the public to watch. propositions. In front of each line, there is the Probably in the future, with the improve- page number of the slides with a link giving ment of this kind of communication, which the time. If you click on this link, the reader sometimes can be an “art”, there will be a li- goes directly into the video at the correspond- brary of videos (satisfying a certain standard ing place up to one second. Thus you can nav- for publication) freely available, analogue to igate better in the video, in the same way, you ArXiv for papers. I have practiced this kind navigate in a paper, with 3 screens: the video, of activity with many videos recorded and can the slides, and the website. give some pieces of advice from my own expe- In the field of teaching mathematics, there rience of giving and watching videos. is the idea of “video book”. There are several If it is a video recorded in real-life, in front possible definitions. A possibility is to start of an audience, it happens very often, that to from a written book and then every chapter make a good video, you need a good profes- and subchapter are put into videos as if you sional recording the video. I have seen some were giving a course from the written book. A videos of some conferences where the camera nice example is with the book of Philippe Fla- insists on the speaker and not on the slides, jolet and Robert Sedgewick “Analytic combi- or sometimes you even do not see some slides. natorics”81. On the left part of the screen, one In that case, you should watch the videos with can navigate inside the contents of the book, two screens, one for the video, the other for the for each portion selected, you get the corre- slides. An important detail is that when you sponding video. The speaker (here Sedgewick) put your laser pointer on the screen, you will explains with slides as if he was in a course not see it on the video. It is better to have a for students. On the right part of the screen long stick. Another way, which is now widely you can see in written words what the speaker used since Covid-19 is to record the speaker says, this text is following exactly the video, with a fixed camera, the slides related to the with some marks giving the time. This kind of talk appear on a full screen, the speaker ap- video book is for students who follow a course. pears in a little window on one of the corners There are many other such courses, mainly for of the screen and the arrow of the computer computer science and engineering. It is avail- replaces the laser pointer. able on the platform CUvids82 (nothing to do Usually, people do not look at videos in to- with Covid!) from Princeton University. tality, just a few portions or even a few min- Another kind of video book is based on the utes. Looking at a video instead of reading a reverse process. It may be something trying to paper, is like reading the paper where you can propose a set of videos replacing a book, where see only one line at the same time. The main you can navigate in the same way you turn the problem is to be able to go inside a video in pages of a book. I try to develop this idea with the same way you can go back and forth inside the “video book” ABjC. It has 3 components, a a paper. It seems that some tools are or will website, a set of videos (more than 100 hours), be created. For example, YouTube is creating and a set of slides. It is divided into 4 volumes chapters in videos where clicking somewhere (equivalent to four books), each volume (Part) enables you to go to another chapter. with some chapters and subchapters, each cor- One possible solution for the videos related responding to a video (about 1h30). In the to some conferences of courses is given on the same way as a book, there are (or will be) pref- website “The Art of Bijective Combinatorics” aces, introductions, references, complementary (ABjC8). Videos fit very well to explain bijec- conferences related to each Part. An advantage tions. The beauty of bijective combinatorics compared to a written book is that one can 81P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009. 82See https://cuvids.io/.

ECA 2:1 (2022) Interview #S3I4 12 Interview with Xavier Viennot add some complements, links to related talks organized by SMF (the French Mathematical or papers. But the basic material (slides and Society) and BNF (Biblioth`equeNationale de videos) should be kept unchanged with a fixed France) in Paris. Each year there are 4 confer- web address. ences (in French) for a wide audience given in This video book ABjC is based on a series the big amphit´eˆatreof the BNF in Paris, start- of courses given at IMSc during 4 years at two ing in 2005 with Don Zagier about the series of different levels at the same time (for students letters between Hardy and Ramanujan. These and for researchers). IMSc has a special video conferences are recorded by professional tech- room for recording conferences or classes. One nicians with several cameras and mixed in real- camera is for the speaker, another for the pub- time. The title of the series is called “Un texte, 84 lic or the students. Slides appear with two un math´ematicien” . The speaker starts from small rectangles in some corners of the screen. a historical paper, book or letter, gave the con- The only problem is that sometimes there are text, and shows how it is related to modern no micro for the audience and in the video, science. In particular, the reader can listen a 85 you do not hear the question, except if the video of a conference given by Pierre Cartier , speaker repeats the question. This can be done 17 January 2007, about the “return of figures” for a conference but is not natural inside a in mathematics. There have been some confer- course. It happens that for some not recorded ences related to combinatorics and I encourage conferences, I repeat the same conference in the reader to go there, in particular to a con- 86 the video room at IMSc. See for example the ference given in 2007 (300th anniversary of (high speed) talk I gave for the 60th birthday birthday) about Euler and combinatorics, con- of Christian Krattenthaler in Strobl (in 50’) ference given with two violinists and a story- and the same (with exactly the same slides) teller. (Epilogue of ABjC83) but in almost 100’. The Mansour: It seems you have started plant- next step should be to write some papers giv- ing your own vegetable garden. What was the ing an overview of each of the 24 chapters, and reason you decided to do this? I have been maybe one day to finish with written books. experiencing a complex problem at home re- This would be exactly the reverse process than cently. One of my daughters, Atil, has decided the one of Flajolet and Sedgewick. But, I know to follow a vegetarian diet based on some eth- (and also my co-authors) that I am bad at writ- ical issues. Whenever she wants to discuss it, ing papers or books. I try to escape by saying that “Ohh, this is Most combinatorics videos that are not on a challenging real-life problem; I am a math- a specific site of a University institution are on ematician, let me finish this paper, later we YouTube. A problem with videos, or a web- can talk about it!” But I have no satisfactory site for a video book, is that the URL address answer! What do you think about eating meat and the link may change with time. Another not as a nutritional issue but as a philosophical problem is for our Chinese colleagues. They issue? have no access to YouTube. A solution is to Viennot: Planting trees, making and eating put your video on bilibili, a popular chain for your own vegetables (of course organic), ob- young people who accept videos given by scien- serving Nature is really a rewarding experi- tists. It is better to go there with your harddisk ence. I hope that every researcher in science and give it to the colleague who invited you. can do the same even a tiny garden on a bal- Many thanks to Bill Chen for his kind invita- cony. For example, the below picture (selfie) tion to Tianjin University, September 2019! with two walnut trees is instructive. Another possibility is with Vimeo which The two trees are born naturally 20 years seems to be less commercial than YouTube. ago from two nuts coming from a big walnut Thus I will finish the answer to your question on our piece of land. The nuts were very close by giving some publicity for the series of videos and gave two different stems, but which fi- 83See http://www.viennot.org/abjc-epilogue.html. 84See https://smf.emath.fr/la-smf/cycle-un-texte-un-mathematicien. 85See the conference of Pierre Cartier at BNF, 17 January, 2007. The link to the video is available at https://smf.emath.fr/ smf-dossiers-et-ressources/cartier-pierre-le-symbolisme-mathematique-des-figures-aux-nombres-et. 86See http://www.viennot.org/abjc-euler.html.

ECA 2:1 (2022) Interview #S3I4 13 Interview with Xavier Viennot nally joined together at the basis of forming sues are mixed. When one eats meat, it de- a unique trunk. The two trees have exactly pends on where the meat comes from, which the same height, avoid each other, all their food was given to the animal, in which envi- branches form half circles on the left or the ronment the animal was growing, and finally right side of the picture, and between them, in which conditions was the animal killed. If not a single branch is going in direction of the the animal received many antibiotics, grow in a other. I even observed a branch on the top closed big industrial factory, with some GMO starting by going straight towards the other soy coming from huge plantations where the trunk and turning immediately before hitting forest has been destroyed for such plantation, the other trunk. How he “knows” he was going the 3 points of view are mixed. First, it will to annoy his brother? There are some scientific not be good for your health and from an eth- researches in this sense about communications ical point of view the person will participate between trees. in climate change, the change of organic agri- culture to industrial agriculture, the destruc- tion of nature, etc. Young people are very con- cerned with climate change. Humanity should immediately change radically many things and not wait. Maybe it is already too late. Eating meat instead of vegetables, even if all condi- tions for the animal are good, is sending much more carbon into the atmosphere compared to eating vegetables. The last two days, all Bordeaux vineyards were affected by the freeze. This is a classical phenomenon. But what is new, was the hot and beautiful weather the last 3 weeks, cold nights with a lot of winds. Most the vegetable gardens, fruit trees, ..., were frozen, the wind increasing the bad effect of the freeze. My neighbor (84 years old) who teach me many things about vineyards and vegetable gardens, The question related to your daughter Atil told me that it was the first time in his life is delicate, maybe going outside such an inter- he has seen such a disaster. During the night view. I do not want to give pieces of advice many winegrowers light some fires inside the or tell you what you should do in your rela- vineyard to slow down the freezing. I was con- tionship with your daughter Atil. I can only cerned with my vineyard and told you that my share an experience with my daughter. When report will be a little late. Living constantly in she was almost 18, after finishing lyc´eeshe was nature, you can really see and feel the changes. going to study abroad. Before the departure, But there are other concerns than climate we went for a trekking in the Pyr´en´eesand for change. Intensive industrial farms for grow- each stop near a lake, she told me the com- ing animals make the propagation of possible plaints she had to tell me, in particular, she viruses much more efficient. This is why bats was still remembering that when she was a kid, create more often viruses because some fifty she wanted to play with me on my keens. My thousand can live in the same grotto. Indus- answer was “I have no time, I have to finish trial farms for animals is a good vector for the some mathematics”. Every researcher work at new virus. For example, with the Covid-19 home, even if he or she works at university. We virus, in Denmark, there are intensive mink have two offices, one at university and one at breedings, some get a form of Covid. The gov- home. It is difficult to organize time at home, ernment was afraid that this virus makes muta- one for maths, one for children. tions inside these breedings, and go back to hu- About eating meat, in fact, I think that nu- mans in a variant form where the actual Covid tritional issues, philosophical, and ethical is- vaccines will not be efficient. They ask to kill

ECA 2:1 (2022) Interview #S3I4 14 Interview with Xavier Viennot the millions of minks! Of course, we do not everything in your domain. eat minks, but it may be the same in indus- Do not choose a field because it is fashion- trial farms for meat. Thus not eating meat, or able, or there are more chances to get a posi- at a least few, and checking how it has bred tion. can give more chances to avoid another virus Choose what you like the best, even if your after Covid. career will be slower at the beginning. Now another point of view is the respect If you work on a hard problem, work very of nature and of the “well-being” of the ani- hard constantly on it several days (and nights) mal. You must know how the animal travel and then stop and go back to one of your other to the slaughterhouse and how it is killed, in favorite problems. which conditions. This is for the respect of the Do not have only one problem in your mind. animal-life if it is going to give you its meat. Work on several problems, but one by one. Even if you do not care about the animal and Have conversations with friends doing re- just want to have good meat, this is impor- searches, compare your researches, maybe a tant. Here we have lampreys, a kind of ugly combination of both will leads to a solution. prehistoric fish which look like meat (we cook See my rencontre with L.-F. Pr´eville-Ratelle it in red wine, usually white wine is associ- about the discovery of ν-Tamari f . ated with eating fish). It is very expensive to Sometimes working hard on a problem you have. The first time I taste this special plate, will get the idea to solve another of you current it was in a restaurant and I did not really find problems. it exceptional, even not so good. In my vil- Maybe you will get prices, medals, and suc- lage there is a fisherman who goes fishing in cess, but it may be dangerous for your ego. Try spring for lampreys, there are no more in the to keep your usual modesty. Garonne river, but there are some in the Dor- Listen to the advice of Joseph-Louis La- dogne river. His wife told us that the animal is grange encouraging colleagues and students to very afraid. Once they are fished, they put the study Leonhard Euler: “Lisez Euler, dans ses animals in calm water for half a day, so that ´ecritstout est clair, bien calcul´e,ils regorgent the animal became calm. The preparation is de beaux exemples et parce que l’on doit tou- a whole week. The taste was completely dif- jours ´etudierles sources”. ferent. A scientist will say that when animals My friend Alain Lascoux was always going are afraid, they will secrete some chemical and back to the sources. We spent one month to- that you will eat this chemical making the taste gether at Mittag-Leffler Institute. He was sad different. Philosophers will say that you “eat to see young students spending most of their the fear” of the animal when it is going to be time on the screen of their computer without killed, which I suppose is not good for your exploring the fantastic historical library of old mental health. books. He told me to look at the book of I can continue on the spiritual side, but I Arbogast87 (1800). I was very surprised to think it is already too long. I hope this will discover Catalan numbers, the so-called bal- contribute to the discussion with your daugh- lot numbers, and the formula for them, which ter. everybody believe it was given by Witworth88 Mansour: What advice would you give to (1879), Bertrand89 (1887), Andr´e90 (1887). young people thinking about pursuing a re- But few people know that it was also given ear- search career in mathematics? lier by Delannoy91 (1886) with what he called Viennot: Try to construct your glasses, to “m´ethode de l’´echiquier”, i.e. movement of a look at the problem with your personal feel- tower on a chessboard. ing, your taste. Going back in history is really pleasant, es- Try to reprove known facts in your own way, pecially for combinatorics. At Mittag-Leffler do not wait to begin researches once you know I was also surprised to open a book of 87L. F. A. Arbogast, Du calcul des derivations (in French), Strasbourg: Levrault, 1800. 88W. A. Whitworth, Choice and Chance, 4th ed., Deighton Bell, Cambridge, 1879. 89J. Bertrand, Solution d’un probl`eme, C. R. Acad. Sci. (1887), 369. 90D. Andr´e, Solution directe du probl`emer´esolupar M. Bertrand, C. R. Acad. Sci. (1887), 436. 91H. Delannoy, Emploi de l’´echiquierpour la r´esolutionde probl`emesarithm´etiques, Assoc. Fran¸c.Nancy XV, 1886.

ECA 2:1 (2022) Interview #S3I4 15 Interview with Xavier Viennot

Eug`eneCatalan, with a handwritten dedica- mont, etc. Moreover, my favorite mountains, tion to Mittag-Leffler, and where I found a the Pyr´en´eeswere not too far. I settle in Isle- page where Catalan introduced some numbers Saint-Georges, a village 20 km south of Bor- named ”Catalan numbers” (!) (by Amiral De deaux. Near the village, I found a piece of Jonqui`ere). land to make a vegetable garden, but it was a I will finish with two more advices. vineyard and I did not want to take off every- Keep contact with Nature, view the beauty thing for the garden. This is the reason why of Maths and Nature as one, get your inspira- I start to learn to make wine, take care of the tion by walking in forests, mountains, in front vineyard. of a lake, the ocean, or under the stars by a I also practice some yoga and I am inter- beautiful night. ested in oriental spirituality. When I do in- And finally, do mathematics with your tensive research, I superpose in my mind the heart. combinatorial objects and bijections involved, Mansour: Would you tell us about your in- and concentrate on them, trying not to think, terests besides mathematics? keeping the mind in stillness. Now I under- Viennot: I like to be in contact with Nature, stand that this is a kind of meditation. to be embedded in wild and remote places. I Mansour: You also dedicate your time for the used to do climbing, mountaineering, trekking, popularization of science in general and math- and sometimes some expeditions in various ematics in particular. What do you aim for in mountains of the globe. With Pierre Leroux, these activities? almost every year we were going in winter for Viennot: I like to share the beauty of science one-week trekking with skis in some mountains and mathematics with students and the gen- of Qu´ebec, Rocky mountains and Pyr´en´ees. eral public. With bijective combinatorics you With Pierre we have done an unforgettable ex- can really touch a wide audience from elemen- pedition in canoe: 3 weeks in Nunavik ( ), tary schools to people who hate mathematics (the Inuit country far North of Qu´ebec). Now, after some trauma during their scholar educa- no more expeditions. I enjoy going with my tion. qu´ebecquois canoe on the Garonne river very For example I have explained Young close to my village. The rapids are replaced tableaux, Sch¨utzenberger “jeu de taquin” by this amazing wave called “mascaret” when and the geometric construction of Robinson- the river reverses the way it’s going. Also, I Schenstedj at elementary schools and in some like sailing. For my postdoc at UCSD in San colleges. It will take more than one session, Diego, I went by crossing the Atlantic Ocean but starting from a permutation, kids can play with a sailing boat. Again an unforgettable with these very concrete constructions on a experience (and also for the administration at grid and feel the magic coincidence of the two CNRS when they had to fill up the administra- constructions. Trees and branching structures tive papers). I love going on a bicycle, hearing in Nature, in relation with the Strahler analysis classical music, spending hours under the sky of trees mentioned in g is a perfect theme for a by night, and doing photography, especially in wide audience in order to show the mathemat- Nature. ics interacting in different domains of science. After my postdoc, when I came back from Fran¸cois Hollande’s socialist government California, I was dreaming to live in a village, a few years ago, introduced the so-called taking care of fruit trees, growing my own veg- “TAPS” (in French: Temps d’Activit´eP´eri- etables, near a big city having a university with Scolaire) at primary schools. With my wife a combinatorial group. Bordeaux was the per- Marcia Pig Lagos, storyteller, we went every fect place, with a small combinatorial group week to the primary school of our village Isle- around Robert Cori, with young people looking Saint-Georges and try to introduce children to for a “Th`esed’Etat” like Dominique Gouyou- science with astronomy, with some evenings Beauchamps, Bernard Vauquelin, Maylis De- outside under the stars (with the parents). lest, Jean-Guy Penaud, or younger for a “th`ese Children loved that. 3`emecycle” Serge Dulucq, Myriam DeSainte- In these activities, I aim to attract students Catherine, Mireille Vauchaussade de Chau- to mathematics with the beauty of bijective

ECA 2:1 (2022) Interview #S3I4 16 Interview with Xavier Viennot combinatorics. With Marcia Pig Lagos and rial objects. It is the so-called symbolic calculus G´erardDuchamp (mathematician, computer dear to Philipe Flajolet81 or in another point scientist, and violinist) we have created the as- of view the species introduced by Andr´eJoyal sociation “Cont’science”92 in which its aim is and dear to the Qu´ebec school at LaCIM. You the popularization of science, combining sci- need to define operations at the level of the ence (especially combinatorics), tales, and mu- combinatorial objects, which are a mirror of sic. The motto of our association is “Science the operations at the analytic level. All this avec Cont’science pour l’´el´evationde l’ˆame”, philosophy is now very classical and is nothing which is a paraphrase of the famous Rabelais but bijective combinatorics. “Science sans conscience n’est que ruine de This attitude can be summarized by a l’ˆame”. This reminds the thesis defended beautiful sentence suggested to me by Karine by Christian Krattenthaler “Both mathemat- Chemla, CNRS historian of ancient Chinese ics AND music are food for the soul AND the mathematics: (in French) “Dessiner des cal- 93 brain”, in a talk-performance he has given culs, calculer avec des dessins”. (“Drawing cal- in 2013 in Vienna. culus, making calculus from some drawings”). Mansour: The term bijective paradigm is of- Of course, by drawings, it is suggested (visual) ten associated to your work. How would you combinatorial objects. There is a return of describe it? figures in mathematics85. Pierre Laplace was Viennot: In a first approach, this paradigm very proud of his book on mechanics. In the is an attitude for many parts of mathematics, introduction, he claimed that you will not find proving a fact, an identity, or a theorem with a single figure. The bijective paradigm is also the construction of bijections. the return of figures, replacing calculus, but The most simple case is starting from an taking “figures” at another level, not the same identity like A = B where both sides are some as the figures illustrating for example euclidian formal power series in one variable. A bi- geometry. Finally, going back to a, the bijec- jective proof of the identity is in two steps. tive paradigm is at the basis of combinatorics First, give a combinatorial interpretation of viewed as an attitude transverse to mathemat- both sides, that is invent some combinatorial ics, theoretical physics, and other sciences. objects whose generating functions are A and Mansour:i One of the interdisciplinary re- B. This can be very difficult and strong in- search projects you led was project MARS tuition can be needed. Then the second step (mathematics, computer science, physics) will be to find a bijection (one-to-one corre- “Combinatorial Physics: Around alternating spondence) between the two classes of objects, sign matrices and the Razumov-Stroganov con- which will prove the identity. jecture on spins chains model”. Would you tell At another level, A and B can be formal us the project? What were the main outcomes power series with many variables, functions of it? coming from analysis, or algebra. The extra Viennot: I am a researcher at LaBRI, a com- variables will be considered as parameters for puter science department. When I started as the underlying combinatorial objects interpret- a CNRS researcher, combinatorics was consid- ing A and B and the identity will be a conse- ered badly, especially in France with the in- quence of a weighted preserving bijection be- fluence of Bourbaki. Thus, it was better to go tween both sides. under the umbrella of theoretical computer sci- More generally, the analytic expressions for ence. In the old-time, we did not need to sub- A and B can be decomposed into some smaller mit individual research projects, as in North pieces, each piece having its own combinato- America. Then it was time to group differ- rial interpretation. The combination of these ent teams under a common research theme, in pieces is made with some analytic operation particular, international and European (many (sum, product, integration, derivation, etc) thanks to Christian Krattenthaler) projects which will be interpreted by some combinato- which means a huge amount of energy to sub- rial constructions at the level of the combinato- mit and handle such projects. I am not good 92See http://www.contscience.org/. 93See http://www.mat.univie.ac.at/~kratt/artikel/musimate.pdf.

ECA 2:1 (2022) Interview #S3I4 17 Interview with Xavier Viennot at doing such works. But a director of ANR the end of the MARS project by physicists (analog of NSF in France) came in LaBRI and Sportiello and Cantini25 in 2010 with bijec- said that projects grouping very few people to tive techniques. We can proudly say that our solve a main conjecture would be encouraged. team gave a contribution to this success: be- Thus I decided to submit something around fore 2010, Sprotiello came to LaBRI and gave the Razumov-Stroganov (RS) conjecture com- a talk in the warm combinatorial ambiance of ing from Physics with spins chains model. RS our weekly “GT” (seminar). To my knowledge conjecture was discovered in 2001 with exper- this is the first time that a problem coming imental combinatorics and use of OEIS94 (the from physics is solved with bijective combi- On-line Encylopaedia of Integers Sequences). natorics and that no proof has been founded It describes a kind of Catalan refinement of the using the big machinery of theoretical physics enumeration of ASM with chords diagrams. I such as Yang-Baxter equation, transfer matri- called this project MARS. ces, etc. Why MARS ? First because “RS” for Razu- Mansour: One of your awards was for your mov and Stroganov, and “MA” because the work in combinatorics and control theory. How conjecture is related to ASM (alternating sign is combinatorics related to control theory? matrices), see d. Second, we submit the project What are the most important results in this in March (Mars in French) (just half an hour direction? How did combinatorics help solve before the electronic deadline!). And third, problems in control theory? there is the expression in French “promettre Viennot: The relation between combinatorics la Lune” (= promise the moon), promising and control theory is based on two methodolo- something which seems impossible to do or gies: the combinatorial theory of differential reach. What about “promettre Mars”! It equations developed with my beloved friend fits much better in the context of ASM, TSS- Pierre Leroux, and the approach of control the- CPP, DPP, and other fascinating combinato- ory with non-commutative formal power series rial objects. Finally the project95 was accepted developed by Michel Fliess and his school. (2007-2010). We were four researchers from This second approach fits very well in the the combinatorial group in LaBRI working on spirit of Sch¨utzenberger’s school in theoretical ASM and related topics (Aval, Duchon, Le computer science, replacing formal languages Borgne, Viennot, together with Lascoux from by non-commutative power series, automaton the algebraic combinatorial team in University by the representation of free monoids by ma- Paris-Est-Marne-La-Vall´ee(now called Univer- trices, rational languages (languages accepted sit´eGustave Eiffel). We did not land on Mars by finite automaton) by the so-called ratio- and did not prove the RS conjecture. In fact, nal power series in non-commutative variables. each member of our team did not work hard In control theory you have to solve differen- on this conjecture. Since the discovery of RS, tial equations with some “entries” which are physicists like Di Francesco, Zinn-Justin or Zu- some functions. For example the movements ber, in their attempt to prove the conjecture of a boat in the ocean are some functions given were turning around, producing many deep pa- by a system of differential equations, the “en- pers, with each time many new and exciting tries” are some functions that are a modeliza- conjectures. In our project, each of us did in tion of the waves and you want to know how the same way, continuing the exploration of the boat reacts. You have to solve this sys- the amazing world around RS and ASM. A tem as some functions of the entries that are list of papers and conferences, including sub- themselves functions. jects such as Six vertex model, Schubert, Jack, Forget the boat and let’s take the follow- Macdonald polynomials, symmetries in ASM, ing simple basic example with the non-linear alternative approach to ASM, and alternative differential equation y0 = y2 + u(t). This ex- tableaux for the PASEP, is on the website 95. ample is given as a toy example in a paper of The RS conjecture was proved just after Fliess, Lamnabhi, and Lamnabhi-Lagarrigue96 94See https://oeis.org/. 95See http://mars.xavierviennot.org/. 96M. Fliess, M. Lamnabhi, and F. Lamnabhi-Lagarrigue, An algebraic approach to nonlinear functional expansions, IEEE Trans. Circuits Systems, 30, (1983), 554-570.

ECA 2:1 (2022) Interview #S3I4 18 Interview with Xavier Viennot and express the behavior of a certain non- is much more efficient. In the case of the equa- linear electrical circuit. We want to express tion y0 = y2 + u(t) you do not need to use y(t) as a function of u(t). The classical analy- the algorithm. One get an explicit formula sis gives the solution as an infinite sum where for the coefficients. The iterated integrals with each term is a multiple integral (the Voltera non-zero coefficients are in bijection with Dyck kernels). This series can be rewritten as an words (!), and the corresponding coefficient is infinite sum in terms of the so-called (Chern) the product of the height of each vertex of the iterated integrals. Solving the equation means Dyck word (!). I remember the reaction of to find the coefficient in front of each iterated Michel Fliess when I showed him that. He said integral. Now come Fliess methodology. Each “I am speechless”. iterated integral is in bijection with words in But there is more than your simple question two letters. In the case of linear equations, of how combinatorics help to solve problems in you are in the world of Sch¨utzenberger ratio- control theory. It can be used to invent some nal non-commutative power series. With this new notions in control theory. I cannot resist methodology, the differential system of equa- the pleasure of telling you a funny story. tions becomes a system of non-commutative We have seen in a particular case the con- power series, where the product is replaced by nection between increasing binary trees and the shuffle product. This system can be solved weighted Dyck paths97,98. This can be done on a computer by iteration. in general. The resolution of equations in con- If you replace the entry u(t) by 1, you trol theory can be express with weighted paths. get the differential equation for the function By taking some bounds in the family of paths, tan t. The expansion into Chern iterated inte- you can define an approximant of the solution grals become a formal power series, which since of the equations in control theory, analog to the D´esir´eAndr´e,we know is the generating func- convergent of the Jacobi continued fraction as- tion for alternating permutations on an odd sociated to weighted Motzkin paths, extended number of elements. to a theory of Pad´eapproximants in Emmanuel Now comes the combinatorial theory devel- Roblet99 thesis. In a join work with Pierre Ler- oped with Pierre Leroux for (system of) dif- oux and Fran¸coiseLamanabhi-Lagarrigue, we ferential equations, written in the language of have developed this idea and define a kind of species, ”enriched” increasing trees, “´eclosions approximants in control theory, analog to clas- combinatoires”, etc, dear to Gilbert Labelle. sical Pad´eapproximants in analysis. We were In the case of the simple differential equation supposed to present these new approximants y0 = y2, we get the well-known increasing bi- in a meeting in control theory and Lamanabhi- nary trees, enumerated by n!. The crucial ar- Lagarrigue was supposed to give the talk. Just gument is that this interpretation contains the before the meeting, she discovered that these interpretation for the equation y0 = y2 + u(t). approximants were already known and asked The entry u(t) is attached to each external ver- what we should do. No problems we told her tex of the binary tree. If u(t) = 1 you get with Pierre. Replace the title “A new approx- the “complete” increasing binary trees (in bi- imants for ...” by “A combinatorial interpre- jection with alternating permutations). This tation of the approximants defined by”, and crucial argument can be developed for general change a few words inside the paper for the systems of differential equations with some en- Proceedings. tries. Sorry to make this a little long, here is Your third subquestion in your question is the relation between combinatorics and control “What are the most important results in this theory. direction?”. During my (permanent) combi- Now you ask how combinatorics help solve natorial life I just did an incursion in control problems in control theory? With the combi- theory for a few years 30 years ago. I am not at natorial approach the resolution by computer all qualified to give you an answer, even if the 97X. Viennot, Une th´eoriecombinatoire des polynˆomes orthogonaux g´en´eraux, Lecture Notes, UQAM (1983), 219 (pages II- 14,15). 98See http://www.viennot.org/abjc4-ch2.html. 99E. Roblet, Une interpr´etationcombinatoire des approximants de Pad´e, Publications du LACIM 17, UQAM, Montr´eal,(1994), 213.

ECA 2:1 (2022) Interview #S3I4 19 Interview with Xavier Viennot award in your question was the “CNRS silver zig-zag lines of different colors appear to me. medal” in 1994. In 1976 I presented this geometric version Mansour:j In combinatorics, the Viennot’s of RS at a meeting in Strasbourg organized geometric construction is named after you. by Dominique Foata. Gilbert de Beauregard What is it about? Would you mention some Robinson himself was there! Curtis Green pre- of its applications? sented his famous interpretation of the shape Viennot: The Robinson-Schensted correspon- of P and Q. They were surprised. My Master dence is a bijection between permutations and Sch¨utzenberger presented for the first time the pairs (P,Q) of Young tableaux of the same “jeu de taquin” version of RS. For me, I was shape. I presented a geometric version of the going from the world of combinatorial algebra correspondence100 at a meeting at Strasbourg to the world of algebraic combinatorics. organized by Dominique Foata in 1976. This Many years after, I learned from some com- was my first paper in combinatorics. The idea ments in mathoverflow, that this construction is to represent the permutation as a set of was equivalent, written in another language, points on a square lattice with one (and only to a construction described in 1970 Knuth’s one) point in each row and column and to put paper extending RS to matrices, (now called some “light” from one of the corners of the RSK correspondence). Morality: it happens figure. This creates successive “shadows” of many times in mathematics that the name of the set of points, a set of non-intersecting “zig- a new notion is not given to the original inven- zag lines” appear with a new set of points (the tor but to somebody who will present it in a red points) having the same initial property (at different equivalent form, maybe more beauti- most one point in each row and column) and ful, and will popularize that notion. Another giving the first row of P and Q. By repeating example is the so-called LGV Lemma101,102, re- the ”light and shadow” construction you get a lating determinants and configurations of non- set of blue points and the second row of P and intersecting paths. Linstr¨omdiscovered it and Q, etc .. until getting an empty set of colored write it in terms of matroids. Gessel and my- points and the pair (P,Q). self independently rediscovered it and joined This construction “explains” the nice sym- our efforts by introducing bijections between metry property: exchanging P and Q corre- non-crossing paths and Young tableaux, inter- spond to take the inverse of the permutation. pretation of Hankel determinants combining Maybe it is of interest for the reader to learn LGV with the interpretation of moment of or- about the genesis of this geometric construc- thogonal polynomials, and many other deter- tion. In 1974 I finished my “Th`esed’Etat” minants, which made this Lemma well known about factorizations of free monoids in relation and very popular. with free Lie algebra. I learn with great plea- The second part of your question is about sure the RS correspondence and try to look at applications. In a strict sense, I do not really it through my combinatorial algebra glasses, see other “applications” that the one explain- i.e. free monoids and Lie algebra. Young ing the symmetry (P,Q) and (Q, P ). But in tableaux are encoded by Yamanouchi words a broad sense, that is putting some “light” in and these words are one of the basic family (i.e. some planar combinatorial object, analogous code in the theory of variable length code) of constructions can be done and lead to new re- a factorization of free monoids with only two sults. factors X∗ = A∗B∗ (called bisection). Let’s mention the nice construction of Ste- I played with the insertion process of RS fan Felsner given at FPSAC’00. Edelman taking Yamanouchi words instead of Young and Green103 gave a bijective proof of Stan- tableaux. Displaying these words on the grid, ley’s conjecture about the number of maximal playing with examples, one day, suddenly the chains in the weak Bruhat order of Sn. Fel- 100 X. Viennot, Une forme g´eom´etriquede la correspondance de Robinson–Schensted, in: D. Foata (Ed.), Combinatoire et Repr´esenatationdu Groupe Sym´etrique,in: Lecture Notes in Math., vol. 579, Springer, (1977), 29–58. 101I. Gessel and X. Viennot, Binomial determinants, paths, and hook length formulae, Adv. Math. 58 (1985), 300–321. 102B. Lindstr¨om, On the vector representations of induced matroids, Bull. London Math. Soc. 5 (1973), 85–90. 103P. Edelman and C. Green, Balanced tableaux, Adv. Math. 63 (1987), 42–99. 104S. Felsner, The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene, Electron. J. Combin. 8:1 (2001), Article #R10.

ECA 2:1 (2022) Interview #S3I4 20 Interview with Xavier Viennot sner104 gave an extension and said “As it is jective and fruitful pollination. How to choose the case with the classical correspondence the which flower is more important than another? planarized proofs have their own beauty and I am rather interested in the beauty. simplicity”. In this spirit, I would list the hard square Another example is given by Burstein and gas model. Is there an explicit expression for Lankham105 with the Patience Sorting al- the density of the model, in the same way as gorithm, having some similarities with the the hard hexagons model? Is it algebraic? A Robinson-Schensted correspondence. The au- solution may come from a bijective explanation thors use this similarity and gave a nice cor- of the algebricity of the hard hexagons model, respondence between the piles formed under see e. Amazingly, the model on the hexagonal Patience Sorting and the shadow lines of the lattice can be solved106, but is completely open geometric construction of RS. This was pre- on the square lattice. sented at FPSAC’05. Another example is the Ising model. This is The construction with light and zig-zag far from being open and hundred of solutions lines also appears in some works related to sta- giving the Onsager partition function (1944) tistical mechanics. has been given. Well-known combinatorial so- In fact, here I should talk about the no- lutions have been given by Kastelyn107 and tion of “essence” of bijections, that is some- Fisher-Temperley108 in the ’60s, giving bijec- thing, an idea, common to different bijections, tion between the configurations of loops in a even if there are operating on different combi- rectangle of the square lattice (interpreting the natorial objects, enumerated by different num- partition function of the model) and perfect bers. An idea of this concept is given in the matchings on a planar graph whose solutions talk I gave for the 60th birthday of Christian is given by a Pfaffian and thus a determinant. Krattenthaler, SLC81 (S´eminaireLotharigien There is a connection between the Ising de Combinatoire) at Strobl. Video, slides, and model on a planar graph and non-backtracking some extraits of Christian concert are available walks introduced by Kac and Ward109 in 1952 at 83. Thus it is the “essence” of the geomet- and make rigorous by Sherman110 in 1960. ric RS construction which gives nice “applica- Tyler Helmuth111 presented an illuminating al- tions”. ternating proof using the theory of heaps of Mansour: One of your areas of interest is sta- pieces. The paper gives a new self-contained tistical physics. We have seen great progress and simple proof that the partition function of in many statistical mechanical models in re- the Ising model on a finite graph can be ex- cent years. Would you list a couple of open pressed in terms of weighted non-backtracking questions of great importance from statistical walks. The core of this work is the use of heaps physics which you think combinatorics would of pieces (the logarithmic lemma) in the stan- play an important role in their solutions? dard theory of the so-called Mayer expansion. Viennot: I am not an expert in statistical Such connections between heaps theory and physics and I am embarrassed to give you this Ising model, or the use of heaps as an efficient list of open questions you are asking. I feel source of formulas for Mayer coefficients should like a bee going from flowers to flowers in the be used to solve some open problems in this garden of statistical mechanics models, taking domain. some pollen here and there, and going back to Surprise, surprise, in this story Ising, Feyn- the combinatorial hive making honey. Then man and Sherman came Sch¨utzenberger, who the bee will come back in the garden for bi- made to Sherman the remark that one of his 105A. Burstein and I. Lankham, A geometric form for the extended patience sorting algorithm, Adv. in Appl. Math. 36:2 (2006), 106–117. 106R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London, 1989. 107P. W. Kastelyn, The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica 27:12 (1961), 1209–1225. 108H. N. V. Temperley and M. E. Fisher, Dimer problem in statistical mechanics-an exact result, The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics 6:68 (1961), 1061–1063. 109M. Kac and J. Ward, A combinatorial solution of the two-dimensional Ising model, Phys. Rev. 88 (1952), 1332–1337. 110S. Sherman, Combinatorial aspects of the Ising model for ferromagnetism. I. A conjecture of Feynman on paths and graphs, J. Math. Phys. 1(3), 202 (1960), Article 202. 111T. Helmuth, Ising model observables and non-backtracking walks, J. Math. Phys. 55 (2014), Article 083304.

ECA 2:1 (2022) Interview #S3I4 21 Interview with Xavier Viennot identity in his paper is the same as an iden- so-called Cartier-Foata115 monoids introduced tity of Witt related to the dimension of some in 1969 as an “algebrization” of the work of D. elements in free Lie algebra. They published a Foata going back to his thesis in 1965 about paper together112. We are back to the begin- combinatorial problems of rearrangements and ning of this interview with free groups and free permutations with repetitions, with applica- Lie algebra and so. tions to probability theory. Mansour: The title of one of your 2015 talks Cartier-Foata monoids (or free partially was The birth of new domain: Combinatorial commutative monoids) have been used in com- Physics. Would you tell us about this new puter science as a theoretical model for paral- field? lelism and concurrency. They have been intro- Viennot: In fact the relation between combi- duced by Mazurkiewicz116 in 1977 under the natorics and physics is not new. I just related name trace monoids. A trace is just an equiv- in the previous question to the work of Kaste- alence of words, i.e. words up to commutation lyn and Fisher-Temperley in the ’60s giving a of some letters. The commutations are defined combinatorial resolution of the Ising model in by a symmetric relation on the letters of the two-dimension with no external field. alphabet. The purpose of the talk113 was to present This equivalence class is interpreted by a with some examples this new field putting in heap of pieces. Thus heaps of pieces are equiv- strong relation combinatorics and physics (sta- alent to commutation monoids. The (big) ad- tistical, quantum mechanics, quantum grav- vantage is that you get a powerful spatial rep- ity, etc). What is new is the first journal resentation which is missing in the linear vi- devoted to this field, created by its founding sion of words up to the commutation of letters. editors G´erardDuchamp, Vincent Rivasseau, Imagine some solid dimers falling on a chess- Alan Sokal and Adrian Tanas˘a,in the presti- board, one by one, such that the projection of gious series of the Annals of the Poincar´eIn- the dimers on the chessboard is two adjacent stitute. cells. When a dimer falls down it may fall on Physicists like Philippe Di Francesco calls the floor (the chessboard) or on another dimer the domain “combinatoire int´egrable” “= solv- and will glue to that dimer. The heap is what able combinatorics” about systems in physics you see at the end. where you can find an explicit expression for The theory is based on 3 basic lemmas: the the partition function (or other similar func- inversion lemma N/D, the logarithmic lemma, tion “solving” the model). and the lemma “paths = heaps”. With these 3 In this review, there are some examples lemmas a huge variety of classical and new re- of this relation between combinatorics and sults can be proved bijectively. Some classical physics, PASEPd, Razumov-Stroganov conjec- theories get a new life as most of linear algebra turei, heaps of piecesk, and directed animals. (See Chapter 2b and 2c of ABjC, Part II). Mansour:k Theory of heaps of pieces is a I do not see connections with Young theme that appears in many of your papers. tableaux, except a connection about the What is this theory about? What are con- Temperley-Lieb algebra. You give me the nections with Young tableaux, Cartier-Foata opportunity to say a few words about this commutation monoids, statistical physics, 2D beautiful topic. The dimension is the Cata- Lorentzian quantum gravity, and chromatic lan number. There are many Catalan bijec- polynomials of graphs? tions and relations between the following con- Viennot: Heaps of pieces114 were introduced cepts: fully commutative elements in Coxeter in 1985 as a “geometric” interpretation of the groups, (321)-avoiding permutations, paral- 112M. P. Sch¨utzenberger and S. Sherman, On a formal product over the conjugate classes in a free group, J. Math. Anal. Appl. 7:3 (1963), 482–488. 113See https://www.youtube.com/watch?v=tcGNLbRR4KU. 114X. Viennot, Heaps of pieces. I. Basic definitions and combinatorial lemmas, In Combinatoire ´enum´erative (Montreal, Que., 1985/Quebec, Que., 1985), Lecture Notes in Math. 1234, 321–350. Springer, Berlin, 1986. 115P. Cartier and D. Foata, Probl`emes combinatoires de commutation et r´earrangements, Lecture Notes in Mathematics 85, Springer-Verlag, Berlin, 1969. 116A. Mazurkiewicz, Introduction to Trace Theory, 3–41, in The Book of Traces, V. Diekert, G. Rozenberg, eds., 1995, World Scientific, Singapore.

ECA 2:1 (2022) Interview #S3I4 22 Interview with Xavier Viennot lelogram polyominoes (= staircase polygons), Connes’s approach with non-commutative ge- heaps of dimers. In ABjC, A (new) bijec- ometry. Other approaches, such as loop quan- tion between parallelogram polyominoes and tum gravity, are based on a discrete vision, Kauffman generators of the Temperley-Lieb which I like much more, see a. Another ap- algebra appears naturally in the context of proach is with a discrete triangulation of space- heaps of dimers. An exercise given to stu- time. “Lorentzian” means you make a distinc- dents is to prove that this bijection is nothing tion between time and space, in opposite to but the Robinson-Schensted correspondence in “Euclidian”. the case of (321)-avoiding permutations. See One day my friend Deepak Dhar told me: Chapter 6b, slide 58 of ABjC, Part II, http: Xavier you should look at a paper of Am- //www.viennot.org/abjc2-ch6.html where bjorn and Loll121 about “non-perturbative you will also find the link to the video. Lorentzian quantum gravity” published in Temperley-Lieb algebra117 are related to the 1998 in Nuclear Physics B in a journal I would so-called totally commutative elements in a never look at. In that paper, Catalan num- Coxeter group. A fundamental characteriza- bers appear (!) related to the enumeration tion is due to John Stembridge118. We take of the so-called Lorentzian triangulations. In heaps114 on a Dynkin diagram. Here it is bet- the paper of Ambjorn and Loll, the model can ter to take an equivalent definition of heaps. be “solved”, but only in 2 dimensions (one The binary relation defining the commutations for the time and one for the space). Com- is defined by a graph. A heap is a certain binatorial proofs were given by the physicists poset ”above” the graph satisfying some ax- Di Francesco, Guitter, and Kristjansen122 in ioms. Many papers have been written, such as 2000 using bijection with heaps of dimers on the recent one quoted in f . see Chapter 6 of a line. Thus I became really excited about ABjC Part II. the connection between heaps and 2D quan- Chromatic polynomials are replaced by tum gravity, see Chapter 7b of ABjC http: chromatic power series and correspond to the //www.viennot.org/abjc2-ch7.html intuitive idea of painting several times the ver- Mansour:l Enumeration of polyominoes is an tices of a graph with successive layers (in each interesting topic in enumerative combinatorics. layer some vertices are colored with the usual But we still do not know how to enumerate condition that no two adjacent vertices are in them for the general case? Why is this prob- the same layer). Thus a vertex can receive lem very difficult? Which results do you con- different color during the process. In partic- sider the best so far? Are you still interested ular one can deduce the very classical theo- in polyomino enumeration? rem of Stanley119 relating acyclic orientations Viennot: Underlying your question about the and chromatic polynomials at value −1. This difficulty of the problem of enumerating poly- proof with heaps has been done by Gessel120 ominoes is the question: what means to solve in terms of commutation monoids. see Chap- the enumeration problem? In general, we think ter 5a of ABjC Part II http://www.viennot. about finding a formula for the number of poly- org/abjc2-ch5.html ominoes according to some parameters (num- About Lorentzian quantum gravity: Quan- ber of squares, perimeter, length, height, ...). tum gravity is a very active field in theoret- But what means an “enumerative formula?” I ical physics. It is a tentative to unify two will not discuss this interesting debate but I incompatible theories: general relativity and will go back to the most central problem: enu- quantum mechanics. Various candidates are merate polyominoes on a square lattice accord- well known such as string theory or Alain ing to the area (number of squares). 117H. N. V. Temperley and E. H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A, 322 (1549), 251–280, 1971. 118J. R. Stembridge, The enumeration of fully commutative elements of Coxeter groups, J. Algebraic Combin. 7:3 (1998), 291–320. 119R. P. Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171–178. 120I. M. Gessel, Acyclic orientations and chromatic generating functions, Discrete Math. 232:1-3 (2001), 119–130. 121J. Ambjorn and R. Loll, non-perturbative Lorentzian quantum gravity, Nuclear Physics B 536:1-2 (1998), 407–434. 122P. Di Francesco, E. Guitter, and C. Kristjansen, Generalized Lorentzian gravity in (1+1)D and the Calogero Hamiltonian, Nucl. Phys. B608 (2001), 485–526.

ECA 2:1 (2022) Interview #S3I4 23 Interview with Xavier Viennot

The problem seems to be very difficult be- M´elouthesis, when Marco asked (he was one cause maybe a “formula” does not exist. In of the referees) Mireille to retype her whole that case the problem in your question is manuscript with the name DSV. solved. I do not mean that the difficulty to Going back to the enumerative problem for solve a combinatorial problem increases with convex polyominoes we can cut a convex poly- the complexity of the formula. See the “ex- omino in 3 parts: a parallelogram polyomi- treme” Example m. noes (equivalently called staircase polygon by As for the question, which result I consider physicists). The middle part is enumerated by the best is very subjective. For me the “best” Catalan numbers and the two other parts are means a result solving a difficult problem with enumerated by Fibonacci numbers, each part beautiful proof using new tools which can be has an algebraic or rational generating func- interesting elsewhere. Thus I have chosen, this tion. The problem will be solved if you invent is a purely personal choice which I hope is ob- an “algebraic glue” to encode the whole poly- jective, even if one of the two authors is one of omino. It is not so easy, and we introduce a my former Ph.D. students and the other a good new bijection between parallelogram polyomi- friend which I met several times in meetings all noes and Dyck paths123, and after long calcu- over the world and also spend some time in his lus by computer we get the surprising formula Melbourne department and apartment. for the number of such polyominoes according The story starts in 1984 with a paper by to the perimeter. This example of the reso- Delest and Viennot123. where we give a for- lution of an enumerative problem using some mula for the number of convex polyominoes bijections gives me a nice argument when col- n with given perimeter p2n+8 = (2n + 11)4 − leagues in physics ask for the interest of bijec- 2n
tive proofs. When we start with Maylis work- 4(2n+1) n . The proof is in the spirit of what Sch¨utzenberger called the DSV methodology. ing on the problem, we did not know the ex- This methodology is the following. In the istence of an exact formula. But following our case where the generating function is alge- knowledge of Catalan bijections, we construct braic, the algebricity is explained by coding some new bijections and arrive at a system of the combinatorial objects by some words of equations, via the coding with algebraic lan- (non-ambiguous) algebraic languages, a no- guages, and finally find and prove the formula. tion coming from theoretical computer sci- We did not know at the same time, Guttmann ence, language which are words accepted by a and Enting were doing some experiments, and pushdown automaton or an algebraic grammar from the first values of the sequence, applying from which you get a system of equations in techniques of approximants, guess (and conjec- non-commutative variables. In case the gram- ture) the formula124. Because of DSV method- mar is not “ambiguous”, sending everything ology, our paper was published in a journal in to commutative variables, you get the proof of theoretical computer science and could not be the algebricity (with the system of equations) seen by our colleagues in statistical mechanics. for the ordinary generating function. This This was the starting point of nice and fruitful DSV methodology has been used by Cori and cooperation between our combinatorial group Vauquelin46 for the bijective proof of the equa- in LaBRI and the department of statistical me- tions enumerating planar maps, see e. I asked chanics at Melbourne University. Sch¨utzenberger the meaning of DSV. He told It may qualify to the prize for the “best” re- me that “D” was for “Dyck” (language) the ba- sult at that time: a nice amazing formula, dif- sic example of algebraic language, “S” was for ficult to prove, and with a new bijection, from Sch¨utzenberger, and he let me guess the mean- parallelogram polyominoes to binary trees and ing of “V”. He wanted to introduce a name Dyck paths39, which has played a key role on a methodology, analog to some principles much later in defining an extension of the in physics. But the name DSV did not sur- Tamari lattice to ν-Tamari lattice (work with vive after that. Even after Mireille Bousquet- Pr´eville-Ratelle). By modesty, I cannot give 123M.-P. Delest and X. Viennot, Algebraic languages and polyominoes enumeration, Theoretical Computer Science, 34 (1984) 169-206. 124A. J. Guttmann and I. G. Enting, The number of convex polygons on the square and honeycomb lattices, J. Phys. A: Math. Gen. 21:8 (1988), L467–L474.

ECA 2:1 (2022) Interview #S3I4 24 Interview with Xavier Viennot the prize to this formula. At that time, for a me take this opportunity to mention Mireille bijective proof of the formula for the number Bousquet-M´eloufor a digression about prize, of convex polyominoes with given perimeter, I medals and the progress of combinatorics, es- gave a prize of ten bottles of wine I was mak- pecially in France. In this interview, you ing in my village Isle-Saint-Georges South of mention the CNRS silver medal for myslef in Bordeaux (“Domaine des Mattes” 1981). The “Combinatorics and Control Theory”. The name “les Mattes” (!) is a pure coincidence CNRS is divided in institutes, there is one for with my domain of research. It comes from Physics, one for Computer Science, one for the local name place where is my vineyard, Mathematics. As explained above combina- close to the Garonne river, which means it is at torists get a decent life as being under the um- the lower level in the classification of Bordeaux brella of “mathematics for computer science” wines. (= appellation “Bordeaux sup´erieure”). also called “theoretical computer science”. I Offering bottles of wine for a problem is better considered the medal I received a medal for the than offering money. With time, the wine be- recognization of combinatorics in Computer comes older and quality improves (“il se boni- Science. Mireille, CNRS research director at ifie”), but after a certain number of years, LaBRI combinatorial group, also received the it becomes too old, and the quality decrease. silver medal. My surprise was, not that she Time for solving the problem has a limit. received this prestigious medal, but the fact The first solution to my problem came and that the medal was received from the “Insti- the authors asked for their ten bottles of wine. tut National des Sciences Math´ematiqueset de It was a short paper with a series of identities leurs Interactions” of CNRS. Of course, this and calculus. Not a single bijection explained! medal is an official reconnaissance of the fact The authors claim that each of these identities that she is an international star in the differ- or line of calculus had a bijective proof. The ent facets of combinatorics (enumerative, alge- above paper gave a “bijective” proof, but is it braic, asymptotic, and physics, etc). From my really what we expect of “nice” bijective proof point of view, these two medals in the same which explains the formula? combinatorial group at LaBRI, one in com- From this experience, I learn that is dan- puter science and one in mathematics is the gerous to give some price for a bijective proof recognition of the efforts of many colleagues to of an identity. I should add “nice” bijective elevate the field of combinatorics to one of the proof and a jury would meet and decide if the major domain at the frontier in computer sci- bijection is really ”nice”. In the ambiguity of ence, maths (and physics). Soon after Mireille the question I gave only 5 bottles of wine and was admitted ro the French Academy of Sci- 5 other bottles were still remaining for a really ence (in Mathematics). Philippe Flajolet also “nice” bijective proof. get the CNRS silver medal of CNRS in 2004 Then came the paper125 of M. Bousquet- and Michel Fliess in 1991. Another recogni- M´elou and Tony Guttmann. The paper tion of Sch¨utzenberger’s school! contains many enumerative formulas for 3- This section would not be complete with- dimensional paths, and as a by-product give, out mentioning that the famous “Coll`egede maybe not the expected bijection, but an “ex- France” open a ”chaire de Combinatoire”. planation” of the rational part of the formula, Timoty Gowers, fields medalist, was invited of the algebraic part, and of the difference be- to begin this “chaire”. This is another side tween the two. Thus, after a long delibera- of combinatorics called “Hungarian combina- tion the jury composed of G´erardViennot and torics” after Paul Erd¨os,B´elaBollob´as,L´aszl´o Xavier Viennot, decided to give the last other Lov´asz,etc. 5 bottles of wine. (In fact time to solved the The last part of your question is about my problem was too long, the bottles became not present interest in polyominoes enumeration. good, and my colleagues received some good Polyominoes are equivalent to animals, squares quality official Bordeaux wine) are replaced by points. The last time I did My answer is maybe already too long but let something on that topic was to give a bijective 125M. Bousquet-M´elouand A. J. Guttmann, Enumeration of three-dimensional convex polygons, Ann. Combin. 1 (1997), 27–53. 126X. Viennot, Multi-directed animals, connected heaps of dimers and Lorentzian triangulations, J. Phys. Conf. Ser. 42 (2006), 268–280.

ECA 2:1 (2022) Interview #S3I4 25 Interview with Xavier Viennot proof126 of an amazing formula (the generat- imals (animals with several source points, but ing function is not D-finite) of Rechnitzer and consecutive on a line perpendicular to the di- Bousquet-M´elouabout the so-called multidi- agonal). I do not remember how he got this rected animals, which are equivalent to enu- awesome idea. Enumerating by experiment, we merate connected heaps of dimers on a line, get another amazing formula. No need to look equivalent to enumerate (general) Lorentzian at the Sloane book encyclopedia of sequences, triangulations. This was presented at the the number of such animals is 3n (!!). We work conference “Counting complexity” in honor of hard to find a bijection between this class of Tony Guttmann for his 60th birthday, Dunk animals and words in 3 letters A, B, C. Equiv- Island, Australia. alently defining 3 operators acting on animals, Mansour:m Would you tell us about your adding a new point in the animal, which you thought process for the proof of one of your can reverse the construction. favorite results? How did you become inter- We start by defining some natural parame- ested in that problem? How long did it take ters on both objects: compact source directed you to figure out a proof? Did you have an animal and words in 3 letters viewed as a path “eureka moment”? with North-East, East, and South-East steps. Viennot: I will choose the enumeration of For two parameters on both classes of objects, compact source directed animals on a square we found by a computer that the distribution lattice. At that time the problem was open was the same for the first value of n. For and not easy at all. It comes from statisti- directed animals the first parameter was, of cal physics in relation to percolation model, course, the number of source points and for branch polymer, etc. A directed animal is a the second parameter, we invented a new pa- set of points on the square lattice such that rameter we called it “lower width”. each point can be reached from the origin (the Experimental combinatorics lead to a source point) by a path included in the animal search for 3 operators A, B, C with the follow- with an elementary step North or East. ing property. An operator A will add a source The starting point are the papers of Dhar, point, an operator C will reduce by one the Phani, and Barma127 and Nadal, Derrida, and number of source points, except if there is only Vannimenus128 in 1982. In the first paper, a one source point, in that case, the number of conjectured enumeration formula was given, in source point remains one. For the operator the second an extension conjecture formula for B, the number of source points should be in- the number of such animal bounded on the cir- variant. By experiment, we know what should cular strip. There are also the so-called critical be the behavior of this second parameter with exponent for the average with 1/2 and a con- the operators A, B, C. Finally, we get easily a jecture for the length with 9/11. At that time possible definition for operators A and C, but I knew nothing about statistical physics. The operator B strongly resists unveilling his mys- three physicists were at the Physics laboratory tery. of ENS Ulm and asked for a solution of their I stay several days, isolated in my village, conjecture to various mathematicians of this completely concentrated on B. One day the same ENS, in particular Adrien Douady. “eureka moment” arrives and long after, the The problem arrived in Bordeaux and with writing of the paper2. The length of the proof Dominique Gouyou-Beauchamps we start im- is inversely proportional to the simplicity of the mediately to work on these combinatorial con- formula 3n. Finally, a two-dimensional prob- jectures. The formula for directed animals is lem is put in bijective correspondence with a simple and is algebraic. The problem was also one-dimensional path problem. related to the width and length of the animal. From this bijection, we get a proof that the After some manipulation and experiment, Do- critical exponent for the width is 1/2. To my minique extended the class of directed animals knowledge, the critical exponent for the length to a larger class: compact source directed an- is still open (but by experimentation seems not 127D. Dhar, M. K. Phani, and M. Barma, Enumeration of directed site animals on two-dimensional lattices, J. Phys. A 15 (1982), L279. 128J. P. Nadal, B. Derrida, and J. Vannimenus, Directed lattice animals in 2 dimensions: numerical and exact results, J. Phys. (Paris) 43 (1982), 1561.

ECA 2:1 (2022) Interview #S3I4 26 Interview with Xavier Viennot to be 9/11). At the same time, the idea of 3 when the diameter increase. Next, another heaps of pieces emerge in my mind, directed researcher was doing some experiments about animals are in bijection with some pyramids of zeros of chromatic polynomials of a sequence of heaps of dimers, which gives simpler proof and graphs, which limit was 4. (the 4 colors theo- bijection for the number of directed animals rem is not true in the “infinite limit”). For jok- on a square lattice. From the basic lemma on ing, they compare their sequence of real num- heaps of pieces, it is easy to get a proof of the bers, they were the same up to an addition by formula of Nadal, Derrida, and Vannimenus128 1! From that, the physicist guessed the other for the number of directed animals on a circu- eigenvalues and conjectured the formula which lar strip, involving the zeros of the Tchebycheff is proved with dimers. polynomials (first kind). A second anectode [about the position of Betrema and Penaud129 gave a simpler combinatorics among the French mathemati- proof of the formula 3n with heaps of dimers on cians]. To my surprise, after this resolution of a line, which is deduced from a system of equa- the directed animal problem with the combi- tions. Long after, Zeiberger130 transformed natorial tool, I was invited to give a talk in this system into a system involving equations 1984 on the subject at the famous “S´eminaire with only positive coefficients, from which a Bourbaki” at the Institut Poincarr´ein the cen- “bijectivisation” is possible, leaving the task ter of Paris. After the talk, I heard a famous to the reader. I guess that this “bijectivisa- mathematician asking Adrian Douady: Is the tion” would give the same bijection as the one speaker a mathematician or a physicist? An- we discovered with Gouyou-Beauchamps. swer: neither, he is a computer scientist! This More anectodes: [about experimental re- talk131 has been published in the proceeding in searches 40 years ago] Jean Vannimenus told the category “statistical physics”. me that they made a computer experiment to compute the biggest eigenvalue of the tran- Mansour: Professor Xavier Viennot, I would sition matrix related to directed animals on like to thank you for this very interesting in- a bounded circular strip of given diameter. terview on behalf of the journal Enumerative Some real numbers appear and were close to Combinatorics and Applications.

129J. Betrema and J.-G. Penaud, Mod`eles avec particules dures, animaux dirig´eset s´eriesen variables partiellement commuta- tives, arXiv:math/0106210 [math CO], 2001 (LaBRI Report, May 1993). 130See https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/bordelaise.html. 131G. Viennot, Probl`emes combinatoires pos´espar la physique statistique, Ast´erisque, SMF, tome 121–122 (1985), S´eminaire Bourbaki, expos´e626, 225–246. Available at http://www.numdam.org/article/SB_1983-1984__26__225_0.pdf.

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