numerativeombinatorics

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A Enumerative Combinatorics and Applicationsecajournal.haifa.ac.il

ECA 2:1 (2022) Interview #S3I1https://doi.org/10.54550/ECA2022V2S1I1

Interview with Edward A. BenderToufik Mansour

Photo by Arlene Bender

Edward Bender completed his undergraduate studies at theCalifornia Institute of Technology in 1963 and obtained a Ph.D.there in 1966, under the direction of Olga Taussky-Todd. Hispost-doctoral position 1966-1968 was at Harvard University.Between 1968-1974, he was full-time at the Institute for DefenseAnalyses/Communication Research Division in Princeton; nowcalled the Center for Communications Research (CCR). Since1974, he was a Professor of Mathematics in the Departmentof Mathematics at UC San Diego (UCSD) except for a yearoff to help set up a branch of CCR in La Jolla. Since 2008he is Professor Emeritus. He is interested in enumeration andasymptotics in combinatorics, graph theory, and populationbiology. In these areas, he wrote several textbooks: An Intro-duction to Mathematical Modeling; Mathematical Methods inArtificial Intelligence; Foundations of Applied Combinatorics;Mathematics for Algorithms and Systems Analysis; A Short

Course in Discrete Mathematics; Foundations of Combinatorics with Applications; Lists, Deci-sions and Graphs - With an Introduction to Probability. The combinatorial texts were writtenjointly with S. Gill Williamson.

Mansour: Professor Bender, first of all, wewould like to thank you for accepting this in-terview. Would you tell us broadly what com-binatorics is?

Bender: It is the study of leftover subjectsthat do not have too much structure and havesome aspect of finiteness. What I mean by“leftover” is that combinatorics is the newplayer on the block and some combinatorialthings had been claimed by other fields. Forexample, partitions of numbers began in num-ber theory and some aspects of algebra arebasically combinatorial. Since these leftoverswere often considered uninteresting or puzzles(like the seven bridges of Konigsberg1), it tooksome time for combinatorics to gain respectin the mathematical community. “Too muchstructure” is not something I will even try to

define. So what aspects of these things docombinatorialists study? What comes to mindare enumeration, additional structure (e.g. thesize of the largest component in a randomgraph), existence (e.g. Hamiltonian paths),and construction (e.g. block designs). Speak-ing of block designs, they are an interesting ex-ample of a leftover. Statisticians wanted themto use but not study these leftovers, so combi-natorics was able to claim them for study.

Mansour: What do you think about the de-velopment of the relations between combina-torics and the rest of mathematics?

Bender: Predictably, combinatorics has madeuse of tools in the rest of mathematics. But asa late arrival, combinatorics has had a lesser ef-fect on most other areas of mathematics. Moreinteresting to me is the rapid use combinatorics

The authors: Released under the CC BY-ND license (International 4.0), Published: March 19, 2021Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is tmansour@univ.haifa.ac.il

1L. Euler, Solutio problematis ad geometriam situs pertinentis, Comment. Acad. Sci. U. Petrop 8 (1736), 128–40.

Interview with Edward A. Bender

has found in other sciences, which is also pre-dictable given that combinatorics deals withfinite matters. Don Knuth2 is much more ablethan I to discuss its connections with computerscience, so I will ignore that. Ecology, epi-demiology, and sociology are among the areasthat make use of graph theory. Parts of chem-istry and genetics make use of combinatorics. Iam sure there are other applications I am notaware of. Since the computational difficultyof combinatorial problems often grows rapidlywith size, many of these applications have beenmade possible by the availability of faster andfaster computers.Mansour: What have been some of the maingoals of your research?Bender: I am primarily a problem solver,which may have grown out of a desire to do wellon tests. Of course, I try to see if my approachhas a broader application. In fact, sometimesI look at a known result and wonder if thereis another way to solve the problem that willlead to more. My earlier research was in vari-ous areas and then settled down to be primarilyenumeration—perhaps because the more I haddone the easier it was to do. Problem-solvinginfluenced my teaching, too. For example, Iwas bothered that the calculus text I was us-ing at Harvard did not have real applications,so I decided to add some which eventually ledto a math modeling course and text at UCSD.Mansour: We would like to ask you aboutyour formative years. What were your earlyexperiences with mathematics? Did that hap-pen under the influence of your family or someother people?Bender: My earliest mathematics memorywas in third grade where long division sur-prised me because dividing one number by an-other could go on forever after the decimalpoint. Other than that, math was easy butnot interesting, until I got to algebra in highschool. Mathematics could be fun! I finishedreading the textbook by Christmas.Mansour: Were there specific problems thatmade you first interested in combinatorics?Bender: My early work was in algebra andsome of it had a combinatorial flavor. I foundcombinatorics interesting but do not recall anyspecific problems that got me started.

Mansour: How did you get interested in theasymptotic enumeration?Bender: Jay Goldman and I were teachinga combinatorics course jointly at Harvard andthought we might write a text. I remarked thatwe should include something about asymp-totics based on what Erdos and others haddone. He said that since it was my idea, Ishould learn about it. The book never hap-pened, but my interest did. Admission to can-didacy at Caltech is an oral exam after which Iwas told that I was weak in analysis and musttake another course in it. Fortunately, combi-natorial asymptotics was in its infancy whenI started, so my weak background was not aproblem and later I had co-authors who werestronger than I.Mansour: What was the reason you chosethe California Institute of Technology for yourPh.D. and your advisor Olga Taussky-Todd?Bender: Since I was an Air Force brat, sub-consciously I may not have wanted one moremove. My conscious rational reason was thatI was interested in three areas based on under-graduate courses. I thought I might work withone of those faculty members at Caltech. Hereis what happened:• Elementary number theory with MorganWard: Unfortunately, he died of cancer thesummer I finished my undergraduate work.• Lattices with Robert Dilworth: His classdealt primarily with finite lattices and led tomy second undergraduate paper.3 Sadly forme, his interests had shifted to infinite latticesand I had little interest in them.• Enumerative combinatorics with MarshallHall, Jr.: Although his combinatorics classcovered various areas, I learned too late thathis research interest was design theory.

Fortunately, I found out in my first year asa graduate student that Olga Taussky-Todd’sinterest in finite matrices fit well with my in-terests.Mansour: What was the problem you workedon in your thesis?Bender: It was in algebra, although parts ofit had a combinatorial flavor. The title of mythesis was Symmetric Representation of an In-tegral Domain over a Subdomain4. The rep-resentation was by finite matrices. Charac-

2See http://ecajournal.haifa.ac.il/Volume2021/ECA2021_S3I9.pdf.3Numerical identities in lattices with an application to Dirichlet products. Proc. Amer. Math. Soc. 15 (1964) 8-13.4See https://thesis.library.caltech.edu/9157/.

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teristic polynomials and number theory wereinvolved. Some of the work was with 2 × 2matrices over Z.Mansour: What would guide you in your re-search? A general theoretical question or aspecific problem?Bender: As mentioned earlier, I’m mostlya problem solver, so I usually started with afairly specific problem.Mansour: When you are working on a prob-lem, do you feel that something is true evenbefore you have the proof?Bender: Yes—but I am not always right! Aneasily described early example: I was lookingat recursions for numbers a(n+1, k) in terms ofa(n, j) with nonnegative coefficients dependingon n, k, j. Based on some common combinato-rial numbers I thought that the distributionwould tend to a normal as n→∞. After get-ting nowhere I found a simple bimodal coun-terexample.Mansour: What three results do you considerthe most influential in combinatorics duringthe last thirty years?Bender: I don’t think specific results were themost influential. The development of fastercomputers that allowed new applications ofcombinatorics and consequently spurred devel-opment of the field was quite influential. Inenumerative combinatorics, I think the booksby Erdos and Spencer (Probabilistic Methods inCombinatorics5) and Flajolet and Sedgewick(Analytic Combinatorics6) have been influen-tial for people entering the field. I do not feelcompetent enough to make statements aboutgraph theory or algebraic combinatorics.Mansour: What are the top three open ques-tions in your list?Bender: I do not have a list. Of course like alot of people I would like to know the answer toP = NP and that may involve combinatorics.Mansour: Do you think that there are core ormainstream areas in mathematics? Are sometopics more important than others?Bender: Yes to both, but things change. Also,while most people might agree on the core ar-eas, they would disagree on the importance ofvarious topics. As a general rule, I would say

that the more assumptions that are imposedabout what is being studied, the less impor-tant the results.

Mansour: What do you think about the dis-tinction between pure and applied mathemat-ics that some people focus on? Is it mean-ingful at all in your case? How do you see therelationship between so-called “pure” and “ap-plied” mathematics?

Bender: I assume here you are talking aboutresearch, not just an application such as usinga statistical test to study some data. Giventhat, I think it is more of a distinction betweenpure and applied mathematicians—what is themotivation for studying something. For exam-ple, if one person studies large random graphsbecause he is interested in their implicationsfor the internet, and another studies them be-cause he is interested in what the graphs looklike, they may do the same work but one isapplied and the other is pure.

Mansour: What advice would you give toyoung people thinking about pursuing a re-search career in mathematics?

Bender: Do you love it enough? Do you havea talent for it? Job opportunities are not asplentiful as they once were, so what can youdo if you do not get a research position?

Mansour: Would you tell us about your in-terests besides mathematics?

Bender: I enjoy reading fiction and nonfic-tion. The fiction is mostly mysteries and sci-ence fiction. The nonfiction is mostly science,current affairs, and history. Before Covid, I en-joyed going to plays. I enjoy gardening, but Iam doing less due to age and some nerve dam-age in my spine.

Mansour: Pizza slicing, phi’s, and the Rie-mann hypothesis7? It is an interesting titleof one of your papers, co-authored with OrenPatashnik and Howard Rumsey Jr. How doesthe Riemann Hypothesis enter the picture?What was the motivation for this paper?

Bender: The paper is partially expository.My recollection is that it was written to enter-tain and educate students. It deals with thenumber of regions a unit square is divided intoby certain lines with integral slopes. (Yes a

5P. Erdos and J. Spencer, Probabilistic methods in combinatorics, Probability and Mathematical Statistics, Volume 17, AcademicPress [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974.

6P. Flajolet and R. Sedgewick, Analytic combinatorics, Cambridge University Press, Cambridge, 2009.7E. A. Bender, O. Patashnik, and H. Jr. Rumsey, Pizza slicing, phi’s, and the Riemann hypothesis, Amer. Math. Monthly

101:4 (1994), 307–317.

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Interview with Edward A. Bender

square pizza, no π here.) As I recall, the ti-tle and much of the writing were due to Orenwho may have come up with the question. Thecount depends on the sum of the Euler phifunction and the accuracy of asymptotics forthat depends on the Riemann hypothesis.

Mansour: There is a nice formula8 on yourhome page. Would you tell us about this for-mula and why you chose it specifically for yourweb page? Which formulas from combinatoricsare the top three for you?

Bender: I chose it because it is a simple ques-tion with a fairly simple nonintuitive formulaand the paper is expository. So I thought itcould be a way of getting someone interestedin enumerative combinatorics. I do not haveany favorite formulas I just take them as theyare. If I had to choose a formula for a T-shirt.It would not be from combinatorics. It wouldbe eiπ + 1 = 0 because of the amount of math-ematical understanding involved from simplearithmetic to a sophisticated understanding ofbasic constants and exponentiation.

Mansour: In your work, you have extensivelyused combinatorial reasoning to address im-portant problems. How do enumerative tech-niques engage in your research?

Bender: I am not sure I understand youclearly. Much of my research was in the enu-meration. (Yes, “was” not “is”. I have notbeen doing research since I do not think I amup to doing the level of research I would want.)

Mansour: You have also published some pa-pers in biology journals. How were you drivento work on such aspects of mathematical biol-ogy? Would you tell us about some questionsfrom biology which could take the attention ofcombinatorialists?

Bender: Because some ecology graduate stu-dents took my math modeling course, I met theecology professors Ted Case and Mike Gilpin.Our joint work was forty years ago and is notcombinatorial, so I can not make any recom-mendations of combinatorial problems. Any-one who is interested should talk to someone inbiology perhaps about gene sequencing, ecol-ogy, or epidemiology.

Mansour: On your home page, in a fewlines, you have explained “Why do We NeedProofs?” What do you think of computer-

assisted proofs?

Bender: Suppose a person gives a relativelyshort proof. If someone reliable looks at it andsays it is fine, I think it is reasonable to ac-cept it. But if the proof is very long or in-tricate, I would like more than one person tolook at it. The same thing applies to computerassisted proof, but now we have to considermore: the method, the program, the compiler,and maybe the computer. So, after checkingthe method, we might want someone else toprogram it in another language.

Mansour: You have some notes regardingHealth/Nutrition on your web page. What wasthe reason you have decided to do this? I havebeen experiencing a complex problem at homerecently. One of my daughters, Atil, has de-cided to follow a vegetarian diet based on someethical issues. Whenever she wants to discussit, I try to escape by saying that “Ohh, this isa challenging real-life problem; I am a math-ematician, let me finish this paper, later wecan talk about it!” But I have no satisfactoryanswer! What do you think about eating meatnot as a nutritional issue but as a philosophicalissue?

Bender: A while ago I was collecting informa-tion for use by my wife and me. Then I thoughtsomeone else might be interested in it so I putit on my web page. Of course, it is out of date,not just because of new information but alsobecause a large fraction of results change whenpeople attempt to replicate studies. Some ani-mals, including humans, eat other animals—itis the way the world is. So I have no prob-lem with the idea of eating meat. I do havea problem with how most animals are treatedto produce meat or milk or eggs as cheaply aspossible. Most of the time in the grocery storeI pay attention to that. I think a side benefitis that more humanely produced products arehealthier.

Mansour: In a very recent short arti-cle, published at Newsletter of the EuropeanMathematical Society, professor Melvyn B.Nathanson9, while elaborating on the eth-ical aspects of the question “Who Ownsthe Theorem?” concluded that “Mathemati-cal truths exist, and mathematicians only dis-cover them.” On the other side, there are opin-

8See http://www.math.ucsd.edu/~ebender/reprints/pubs.html.9See https://www.ems-ph.org/journals/newsletter/pdf/2020-12-118.pdf.

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ions around that “mathematical truths are in-vented”. As a third way, some people claimthat it is both invented and discovered? Whatdo you think about this old discussion?

Bender: I have not seen the article. The ques-tion sounds simple, but there are some twists.What are the rules of the game? Do we al-low or deny the continuum hypothesis? Arewe limited to constructible real numbers? Etcetera. So the rules of the game are invented.We might then conclude that, given the rules,the truths are discovered. But Godel’s Incom-pleteness Theorem10 muddies the waters if weconfound the ideas of truth and provabilitywithin a given set of rules. Is it okay to saysomething is true if we know it is but can notprove it with the set of rules we are using? Myresearch topics allowed me to solve these prob-lems by ignoring them.

Mansour: Would you tell us about yourthought process for the proof of one of yourfavorite results? How did you become inter-ested in that problem? How long did it takeyou to figure out a proof? Did you have a “eu-reka moment”?

Bender: I am assuming “favorite” does notmean “important”. In that case, I might pickan early result—it is like a first love. ButI do not remember the details and they arenot combinatorics. I know that I had a “eu-reka moment” when I was working on theasymptotic enumeration of rooted graphs onsurfaces—basically how much various terms inthe recursion mattered, but again I forget thedetails. So I will tell you about a simple prob-lem that I enjoyed. A composition c1, c2, · · ·of n is called Carlitz11 if ck 6= ck+1 for all k.They were studied by setting up a recursionbased on adding one more part. It took a lit-tle work to show that the circle of convergencehad a single, simple pole. All this was doneand published by someone. I wondered “Whatabout different restrictions?” The “eureka mo-ment” came fairly quickly: Instead of addingan additional part, add one to each part. Thismeant I could study things like ck − ck+1 /∈ Sfor some set S, and the nature of the singular-ity was easy to see.

Mansour: Is there a specific problem youhave been working on for many years? What

progress have you made?Bender: Here is a simple problem that Ithought about off and on for a while with noreal progress. Call the composition c1, c2, · · · ofn d-Carlitz if ck 6= ck+i for all k and 1 ≤ i ≤ d.So Carlitz compositions are 1-Carlitz. I cannot even say much about 2-Carlitz composi-tions.Mansour: Some entrepreneurs and scientistshave very ambitious plans to go to Mars. Manypeople criticize such endeavors by proposingthat the money needed for this project couldimmediately help to fight against diseases,poverty, climate change and instead supportother practical scientific projects. What isyour point of view on going to Mars?Bender: I look at it from the point of viewof the benefits of going to the moon. Here iswhat I see.• It led to the development of technology use-ful here. I think that is less likely with goingto Mars.• It got people interested in doing things inspace. That interest has waned and we needinterest if space exploration is going to befunded. I simply do not know if planning atrip to Mars will help that much.• It enhanced U.S. prestige in the cold war.Does that matter so much now?• The rock samples brought back led to knowl-edge. The same would be true of going toMars, but do we need people there? Therehas been robotic retrieval of samples from themoon. NASA’s first phase of robotic retrievalrecently landed on Mars. The argument canbe made that a scientist on Mars could makea better choice. On the other hand, roboticskeeps improving and humans on Mars havebuilt-in limitations. Besides, working on im-proving robotics will be useful in exploring be-yond Mars.

Thus, cost aside, I think sending people toMars may be the wrong approach.Mansour: My last question is philosophical:have you figured out why we are here?Bender: If “why” means the purpose, then Ithink the question makes an assumption I donot accept. If “why” is more along the lines of“what makes life satisfying”, then that is upto the individual to decide. If it is a scientific

10For example, see the entry by Panu Raatikainen in the Stanford Encyclopedia of Philosophy, 2013.11L. Carlitz, Restricted compositions, The Fibonacci Quart. 14 (1976), 254–264.

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question, evolution takes care of the biologicalpart of it, but physics has not explained whyour universe exists. There is a joke about thatwhose source I do not know:• Suppose nothing exists.• There would be no rules since rules are some-thing.• If there are no rules, then nothing is impos-sible.

• This contradicts the assumption that noth-ing exists.

• So something must exist and our universehappens to be at least part of it.

Mansour: Professor Edward A. Bender, Iwould like to thank you for this very interestinginterview on behalf of the journal EnumerativeCombinatorics and Applications.

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