Herbert S. Wilf (1931–2012)

Fan Chung, Curtis Greene, , Coordinating Editors

received both the Steele Prize for Seminal Contri- butions to Research (from the AMS, 1998) and the Deborah and Franklin Tepper Haimo Award for Dis- tinguished Teaching (from the MAA, 1996). During his long tenure at Penn he advised twenty-six PhD students and won additional awards, including the Christian and Mary Lindback Award for excellence in undergraduate teaching. Other professional honors and awards included a Guggenheim Fellow- ship in 1973–74 and the Euler Medal, awarded in 2002 by the Institute for and its Applications. ’s mathematical career can be divided into three main phases. First was numerical analysis, in which he did his PhD dissertation Photo courtesy of Ruth Wilf. (under at Herb Wilf, Thanksgiving, 2009. in 1958) and wrote his first papers. Next was complex analysis and the theory of inequalities, in particular, Hilbert’s inequalities restricted to n Herbert S. Wilf, Thomas A. Scott Emeritus Professor variables. He wrote a cluster of papers on this topic, of Mathematics at the University of Pennsylvania, some with de Bruijn [1] and some with Harold died on January 7, 2012, in Wynnewood, PA, of Widom [2]. Wilf’s principal research focus during amyotrophic lateral sclerosis (ALS). He taught at the latter part of his career was combinatorics. Penn for forty-six years, retiring in 2008. He was In 1965 Gian-Carlo Rota came to the University widely recognized both for innovative research of Pennsylvania to give a colloquium talk on his and exemplary teaching: in addition to receiving then-recent work on Möbius functions and their other awards, he is the only to have role in combinatorics. Wilf recalled, “That talk was so brilliant and so beautiful that it lifted me right is distinguished professor of mathematics, and out of my chair and made me a combinatorialist professor of computer science and engineering, Univer- sity of California at San Diego. Her email address is The text of this article draws from several sources, includ- [email protected]. ing a tribute written by Chung and Hutchinson for Wilf’s Curtis Greene is J. McLain King Professor of Mathematics, 65th birthday celebration in 1996 (see [9]) and also essen- Haverford College. tial input from Ruth Wilf, who also took most of this article’s His email address is [email protected]. photos. Some of the material in quotes is drawn from Wilf’s Joan Hutchinson is professor of mathematics and computer personal notes. science, Emerita, Macalester College. Her email address is All references to the website of [21] refer to a version of this [email protected]. article with a full list of references, which are not included DOI: http://dx.doi.org/10.1090/noti1247 here due to space limitations.

346 Notices of the AMS Volume 62, Number 4 on the spot.” The next year Wilf took a sabbatical a distinguished academic leave in London, where he spent his time in the career at Howard and library at Imperial College exploring combinatorics Clark Atlanta Universi- by reading “every issue of Math Reviews starting ties. After completing from the creation of the world.” By the end of the his PhD coursework at year he had proved a result [4] which he described Columbia, Wilf worked as his “debut”: The chromatic number of a graph full-time for NDA and G is at most 1 plus the largest eigenvalue of its later for Fairchild En- adjacency matrix, with equality if and only if G is gine Division on Long a complete graph or an odd circuit. Island, where he headed In fact, an earlier paper [3], written with N. J. the computing section Fine, shows signs of Wilf’s developing interest in with a small staff of one combinatorics. It contains the following result: If (himself). Wilf’s work at Fairchild included calcu- two infinite periodic sequences {an} and {bn} have Photo courtesy of Ruth Wilf. lating eigenvalues of a periods p and q respectively, and an = bn for p+q− Herbert Wilf, high school proposed jet engine de- gcd(p, q) consecutive integers n, then an = bn for graduation photo, age 16. all n. Furthermore, p +q −gcd(p, q) is the smallest sign on the IBM Card possible value making the statement true. This Programmed Calculator (CPC). result and its continuous generalizations became Long before most researchers and educators, collectively known as the “Fine-Wilf Theorem,” and Wilf recognized the important connections be- [3] has been widely cited in both the math and tween computers and mathematics. In the summer computer science literature. of 1954 he worked for IBM in Manhattan, where From then on, Wilf had a profound influence in the first mainframe IBM 701 had recently been unveiled. He and his MIT roommate Tony Ralston numerous areas of combinatorics, including graph worked in the display window of IBM’s head- theory, discrete algorithms, enumeration, asymp- quarters on Madison Avenue, an experience he totics, and generating functions. Wilf’s pioneering described as “heady stuff.” They spent the first work [5] with on computerized month learning how to program, since neither discovery and proof of hypergeometric identities had ever seen a computer before. Eventually, they resulted in the book A = B (with Marko Petkovšek), wrote a program implementing a multistage linear as well as the AMS Steele Prize citation in 1998. regression analysis, following a method developed In addition, Wilf made notable contributions to by Mike Efroymson, then of Esso (now Exxon) the study of permutation patterns, for example, Corp. The program was used by Esso to control the Stanley-Wilf conjecture (see [6]), which stim- the quality of refined gasoline. Together, Wilf and ulated much interest until it was settled in 2004 Ralston edited the first volume of Mathematical by Marcus and Tardos [21]. Much of his later Methods for Digital Computers, followed by two work was on generating functions and enumera- more volumes a few years later. tive problems, and he was especially known for After receiving his PhD from Columbia in 1958, his work on integer partitions. For example, his Wilf taught for three years at the University typically succinct and gemlike paper with Neil of Illinois. In 1962, after visiting Penn during Calkin [7] contains this result: The positive rational a year-long celebration of Hans Rademacher’s numbers are uniquely enumerated by the sequence seventieth birthday, he received an offer to stay on {b(n)/b(n−1)}, where b(n) is equal to the number permanently, which he accepted. of ways of writing n as a sum of powers of 2, each Wilf was a superb teacher. His lectures were full repeated at most twice. To prove this theorem, of delight, always finding the easy and graceful the authors invent a structure which subsequently way to reveal the inner workings of mathematics. became famous in its own right as the “Calkin-Wilf Wilf’s philosophy in teaching was reflected in his tree.” article “On buckets and fires” [8], named after While still an undergraduate at MIT, Wilf began the quote “A mind is a fire to be kindled, not taking summer jobs at Nuclear Development a bucket to be filled.” Throughout his years at Associates, which did consulting work for utility Penn, Wilf lit that fire in numerous undergraduates, companies trying to build electric-power-nuclear graduate students, colleagues, friends, and other reactors. He described the young staff as “incredibly researchers. In particular, his PhD students have brilliant and idealistic as they faced the new directly benefitted not only by his teaching but also frontier.” Wilf recalled being greatly influenced from his example of how to be a mathematician. by Gerald Goertzel, a physicist then at NYU, and The computer continued to play a key role in J. Ernest Wilkins, a pioneering African American both Wilf’s research and teaching. He and Albert mathematician and physicist who later enjoyed Nijenhuis masterfully integrated FORTRAN with

April 2015 Notices of the AMS 347 348

Photo courtesy of Ruth Wilf. eoe esa al 1990s. early Cessna, beloved ebadRt ifwt his with Wilf Ruth and Herb a ihyrgre etrradtaee ex- traveled and lecturer regarded highly a was an was He . sixty than more with nerircneec a enhl n19 at 1996 in held been had conference earlier An 18–1,h otiue eiso luiaigex- illuminating of series a contributed he (1987–91), xeddfrbyn ahmtc.Frabout For mathematics. beyond far extended his celebrate [9]). to (see birthday Pennsylvania sixty-fifth of University the Sciences of Academy National the of Proceedings [ biology mathematical problems in statistical and combinatorial Warren on colleague two Ewens biology on Penn talk with a papers gave recent he Waterloo, conference in that University At Ontario. Laurier Wilfred his at in conference honor he birthday 2011 eightieth May an In attended world. and the conferences throughout at colloquia talks invited he giving tensively career his Throughout illness mobility. when even his email limited by with intensively colleagues five collaborating last research and life, course the his in a of papers years teaching ten Penn, publishing from regularly, retiring after ics Algo- and generatingfunctionology there, Algorithms, Complexity, and Combinatorial accessible rithms Sciences, remain ical books his including of Many website most from downloading Penn make free his for to available publishers books his his of with worked journals and and books to access open of advocate early process the in collaborating articles, hundred research one sixty than more Wilf and career, books research seven prolific published and long his In editorial duties. his fulfilling to addition in articles pository the for as editor tenure his the During editor-in-chief. an as served he Combinatorics h nest n iest fWl’ interests Wilf’s of diversity and intensity The mathemat- with engaged deeply remained Wilf A = B . ahmtclMtosfrtePhys- the for Methods Mathematical n19 ihNi akn o which for Calkin, Neil with 1994 in mrcnMteaia Monthly Mathematical American www.math.upenn.edu/~wilf/. 10 ihDnl nt and Knuth Donald with and Petkovšek with work the Algorithms of journals: major proof. mathematical as ognized rec- been previously had what of probed boundaries the and way a novel in em- computers ployed which Zeilberger, celebrated his in and flourish grow to computers continued in interest Algorithms torial book their in seen as combinatorics, ,[ ], ifcfuddtwo cofounded Wilf lcrncJunlof Journal Electronic 11 ,pbihdi the in published ], oie fteASVolume AMS the of Notices Combina- n1980 in Journal Wilf’s . , . if fe eihe ntkn usso boating on guests taking in delighted often Wilfs hmh are n ot fe receiving after month one married he whom with times various at himself devoted he which hc copn hsarticle. this accompany which o nyprilyrtrdatrejyn ogand long a enjoying after is retired partially and only in nurse-midwife now certified shifted PhD a become subsequently her to She gears completed Columbia. she at other as and zoology Ruth moral to provided also support he PhD Columbia, own at his finishing juggling David, four and While and responsibilities later. Within family years Susan seven 1952. children, Peter in by two followed MIT had they from years degree BS his boating. and bird-watching, bread making, telescopes, and astronomy developing, and photography to large-format included passion Hobbies similar places. other among Bahamas, Alaska, and to Newfoundland, the wife Scotia, his Nova with California, trips Baja and US coast-to-coast the many in included flights pilot and a as conferences travels Cessna. His single-engine to his in himself events mathematical flying amateur avid often an was pilot, he 2002, until years, twenty rgamn,Safr University. Computer Stanford of Art Programming, The of emeritus professor from is Knuth learn E. Donald I’ll things to of now number forced the I’m that Alas, to realize jokes. me favorite introduced my he of and some fullest, the about to me life taught living he leadership, about me pedagogy, me about taught taught he me He taught him. he from reminded mathematics, learned am about still I I that day things each of times astonished many I’m how death, at three untimely reminiscences, his these after write months I As in life. figures” also significant own “most he my the met, of one ever was as out I only stood Not mathematician ways. tallest many the in he giant a was Wilf Herb Knuth E. Donald collaborators and friends mathematical his of remembrances the throughout These resonate kinds. qualities all of interactions personal mathematical his and of ingredients essential were erosity watch and and stargaze showers. birds to meteor night spot at and to wildlife Bay other Barnegat on expeditions their the occasions, at these On or Jersey. New Pennsylvania, in house Narberth, shore in were home either and hospitality, at gracious friends their Wilfs’ of beneficiaries the became inevitably and children his grandchildren. in six interest great his took to and devoted family was Wilf career. professional active ifi uvvdb i ie uhTmnWilf, Tumen Ruth wife, his by survived is Wilf ebWl’ lyu uo,wrt,adgen- and warmth, humor, playful Wilf’s Herb students and colleagues, friends, Mathematical 62 Number , 4 him is bounded above. But my good fortune has been that the lower bound is already quite high. I actually met Herb rather late in life, although of course I’d heard his name rather often before we ever connected. In the spring of 1974 he published an abstract announcing a new way to generate uniformly random partitions of a given integer, and I wrote to ask for a preprint. By the fall of that year he had sent me a complete draft manuscript of his book with , Photo by Jill Knuth. Combinatorial Algorithms, and our decades of Don Knuth and Herb Wilf at Knuth’s 64th correspondence began in earnest. By the 1990s we birthday conference, 2002. were writing letters back and forth about once per month, mostly trading conjectures or solutions to problems of common interest. Journal of Combinatorics, in 1994. Although way At the beginning of 1979 he presented me a ahead of its time, it was another magnificently sweet offer: He foresaw the need for a new journal, successful venture. to be entitled Journal of Algorithms (a perfect title!), Let me close with one other indelible memory. and he asked me to serve as founding coeditor, On the occasion of my sixty-fourth birthday— although he promised to do 99 percent of the work. that’s THE BIG ONE for a computer scientist!—Herb Naturally I agreed. Typical of my own involvement was the keynote speaker, and he even presented was the following quote from a letter that I sent me with an original theorem (see www.math. him in April: “In any case I wish to leave the final upenn.edu/~wilf/website/tributes.html). decisions about these matters to you, and I will Thus it’s no surprise that when I lectured to sign what you tell me to. (If the typography of the Stanford’s theory students on January 19, 2012, journal turns out to be unacceptable, I’ll scream I dedicated my talk to Herb’s memory and wore though.)” His vision was indeed prophetic, and JoA a yellow T-shirt that exhibits the Calkin-Wilf immediately began to thrive. sequence [7]. One of my fond memories from the ’80s is the time that he sent me a bouquet of roses. I had Aviezri S. Fraenkel submitted a paper under a pseudonym—a female Herbert Saul Wilf was a great alchemist. Anything name, in fact—in hopes of getting referee reports he touched turned into pure gold, be it spreading that were based entirely on the content of my the word, applications of math, or pure math. Here work rather than on my inflated reputation. Herb are a few examples out of many. agreed to this experiment (and later did a similar Spreading the Word. Herb recognized early the thing with a paper of his own that he submitted importance of the budding algorithmics era and to the American Mathematical Monthly). When he swiftly founded the J. of Algorithms, together with wrote to my alter ego, acknowledging receipt of Don Knuth; he noticed the emerging Internet and the manuscript, he added, “By the way, if you’re promptly pioneered, jointly with Neil Calkin, the free next Saturday night, I have a couple of tickets Electronic Journal of Combinatorics, which became to ….” And the roses arrived after the paper was highly influential and boasts nowadays over a accepted. thousand submissions annually; he was the editor- He also contributed a priceless guest lecture in-chief of the American Mathematical Monthly, to a course called Mathematical Writing that I and was on the advisory board of Integers and taught at Stanford in 1987. Among other tidbits, the Electronic Journal of Combinatorial Number he demonstrated how to write up a typical theorem Theory. in two dramatically different styles: one, for Acta Applications of Math. We note Herb’s two recent Hypermathica, was called “Enumeration of orbits ventures into the life sciences, both of which lean of mappings under action of C , the cyclic group;” n heavily on math. For both he teamed up with the other, for MathWorld, was called “Counting biologist Warren Ewens. First there is the contribu- necklaces” (see [21]). tion to epidemiology [10], which suggests that a We enjoyed submitting a paper together [12] to certain outbreak of childhood leukemia in a small Crelle’s venerable Journal für die reine und ange- community is not uncommon, with high probabil- wandte Mathematik, extending a result that E. E. ity, whereas the outbreak of acute lymphocytic Kummer had contributed to the same publication some 137 years earlier. Aviezri S. Fraenkel is professor of computer science and Herb’s continuing foresight led him to launch applied mathematics, Weizmann Institute of Science. His one of the very first eJournals, the Electronic email address is [email protected].

April 2015 Notices of the AMS 349 350

omn fmt,wr h okfudtosthat foundations rock the were global math, his of on command drawing integrity, and efficiency, strength precision, inner elegance, talent, of His latter. eccentricity the the much without constantly, Erd˝os but math Paul did like doing math, for love and mankind. and and kind friends, a family, sharing his with while soul gates feeling those skeleton both. all the for of created key he command locksmith, complete master a a integrated Being an with had Herb soul soul. global discrete a or continu- soul a ous either have mathematicians Most gates. a sciences. ingredient It’s number disciplines. dominating increasing an of the of research as successful for more and more [ sciences” the natural in mathematics of effectiveness “The all, wrote unreasonable Wigner After Eugene physicist surprising. laureate that Nobel all aren’t applications impact? lasting yet immediate having gold, pure in application [21]. unexpected theory This game an way. combinatorial finds new beautiful even a result in [ elegant, rationals Calkin the short, Neil count with the joint created paper, influential and and rationals look for a the Prize had at he Steele Research; P. to Leroy Contributions AMS Seminal which an WZ, authors ground-breaking the the got into it made Math. and Pure Z to Applications and Math Pure time. much too theories take evolutionary thus that processes, obstacle parallel the in removing by series in of replacing theory of the into [ evolution insight genetic profound the is Second, probability. there high with be chance, cannot by explained community small another in leukemia Photo courtesy of Ruth Wilf. hsi o h niesoy ebhdapassion a had Herb story. entire the not is This these to keys the has everybody not However, math to contributions Herb’s it, of face the On became touched Herb everything come How ver rekladHr if a 2010. May Wilf, Herb and Fraenkel Aviezri 11 21 ,avnigbl e ideas new bold advancing ], .Mt sbigrecognized being is Math ]. gate-opener oie fteASVolume AMS the of Notices otenatural the to ebsaw Herb 7 ,to ], Frhmrsac n ecigwr intertwined.” were teaching and research him “For .H a h etrhl f“Zpi”ad“WZ and pair” “WZ of half better the was He W. od hs ulte nbe i osrk gold. strike to him enabled qualities These word. ihAbr iehi h il fcombinatorial of bible the Nijenhuis Albert with ah h atrmc h aewya Bertrand as way did: same [21] the Russell his much and latter family the his math, loving usual usual, while as me,” his life kill in continuing will me illness “This told manner: under He friendly work constraints. to illness him as enable the condition and his possible ease as to much ward hospice-like a home a into their made turning had lovingly transformation, Ruth, major wife, His departure. untimely the spreading in as well promising as truly research the in directions into unfailingly him guided ahmtc,RtesUiest.Hsealadesis of address Professor email Governors His of [email protected]. University. Board Rutgers is Mathematics, Zeilberger Doron pioneered he started, Internet the imple- When and mentation. theory intertwined which algorithms, cowrote he later, (be- years pseudocode Fifteen FORTRAN!). and fore flowcharts with computers, complete with problems mathematical how he solve people Anthony to telling used, with “cookbook” seminal together get a fifties, Ralston, to late the starting in edited, just were computers forth. so and on so personal, and a the and he professional was the the boy, expositor!), great (and, and exposition picture and research or big code, FORTRAN the of details applied, nitty-gritty and pure conceptual, the the and and computational teaching The only research. not but agree, wholeheartedly eulogy, I her in said W., Tumen wife Ruth beloved describe years, His sixty to “intertwined.” of word be one would choose it to him, had I If theory.” about solicited something write to me for hard is It Zeilberger Doron come. to generations for live, shine to light. and continue Herb spread, will meteor special and and contributions gentle, His warm, a radiated iie i bu w otsbfr his before months two about him visited I ewsas a ha fhstm!When time! his of ahead way also was He but austere nor cold neither was himself Herb ihu h ogostapnso painting of trappings gorgeous the without xlain h es fbigmr than more being of the delight, sense of the spirit exaltation, true The greatest show. the can only art as such perfection of stern capable a and pure, sublimely yet nature, music, or weaker our without of part sculpture, any of to appeal that and cold like beauty austere, supreme but truth, only not possesses viewed, rightly Mathematics, ssrl si poetry. in mathematics as in surely found as be highest to the is of excellence, touchstone the is which Man, 62 Mathematica Number , 4 the Electronic Journal of Combinatorics, a free(!) my message that I had an algorithm for finding online journal. He convinced the publishers of his hypergeometric solutions. Immediately I got a numerous books to post his books freely on the most enthusiastic reply, which changed my life: Internet, thereby being a pioneer of open access Herb explained to me that this was just the missing and open source code. When Tim Gowers asked piece of the puzzle that he and Doron Zeilberger him to write the entry for the Princeton Compan- needed to solve the old open problem of whether ion on Mathematics “Enumerative and Algebraic a definite hypergeometric sum could be expressed Combinatorics,” he politely refused but suggested in closed form. In short, without knowing it, I that he write instead an entry “Mathematics: An already had a very good result, and I was saved! Experimental Science,” thereby introducing this But Herb did not satisfy himself with that: he avant-garde subject into the canon of mathematics. wrote a circular letter to George Andrews, Richard And his mathematics was divine! See also Askey, , and Dana Scott, telling them www.math.rutgers.edu/∼zeilberg/ for lec- what a great thing I had done. My advisor was, of tures and talks, especially the lecture “Herb Wilf, course, happy about this turn of events too. At in memoriam,” February 2, 2012. my defense in September, before any questions were posed, he stood up and read Herb’s letter Marko Petkovšek out loud. After that, the question period was a breeze for me. … Herb also invited me to visit the I first got in touch with Herb while working on University of Pennsylvania and give a talk there my thesis in computer science at Carnegie Mellon after my defense in Pittsburgh. He said that he University under the supervision of Dana Scott in would invite Doron as well, adding that he was the first half of 1990. By then I had implemented a not sure if Doron would come. But Doron came, package for solving recurrences by the method of and after my talk he posed two questions. I don’t generating functions in Mathematica, which at the remember what they were, but I do remember that time was just two years old. My plan was to design they were right to the point and that I was amazed an algorithm for finding Liouvillian solutions at how insightful Doron was. This was the first of linear difference equations with polynomial time that I met Herb and Doron in person. coefficients, in analogy to the well-known Kovacic- The second time that Herb affected my career Singer algorithm in the differential case. I realized was in September 1993 at the Symbolic Compu- that one should work with difference rings rather tation in Combinatorics 1 meeting at Cornell than fields but was unable to develop the requisite ∆ University. One day Galois theory (Singer and van der Put did that in during dinner at the 1997). So I tried the “small steps” approach. First famous Moosewood I looked for polynomial solutions, which was not Restaurant in Ithaca, too difficult. Then I tried to find rational solutions, NY, Herb took me and but could not get a suitable denominator. I was Doron aside and pro- not aware that Sergei Abramov had solved this posed that we write problem in 1989. But I had just learned about a book about au- Gosper’s algorithm and hypergeometric terms tomated proofs of from the newly published Concrete Mathematics identities involving hy- by Graham, Knuth, and Patashnik, pointed out to pergeometric sums. me by my fellow graduate student Todd Wilson. We agreed, and Herb I managed to come up with an algorithm for drafted the table of Photo courtesy of Ruth Wilf. finding hypergeometric solutions but was still in contents for nine chap- Doron Zeilberger, Herb Wilf, the depths of despair: my original goal seemed ters, which we divided Gainesville, 2005. just as remote as when I started, I felt that I had among us. In two years, Doron and I wrote two no good results to put in my thesis, and my time chapters each, and Herb wrote the remaining five was running out, since I had to return home by chapters. I would have kept working on the draft October that year. forever, correcting errors and polishing the details, Luckily for me, Herb’s generatingfunctionology but Herb urged us to hurry because things were “in appeared in 1990 as well, and my advisor presented the air” and there was the possibility that others me with a copy fresh from the press. At the end might overtake us. He selected the publisher—Alice of the preface Herb invites readers to send him and Klaus Peters, his good friends. Herb prepared corrections. After reading the whole book avidly, a list of fifty or so possible titles for the book, one I did so. As an afterthought, I mentioned in wittier than the next. In the end, we settled on Marko Petkovšek is professor of discrete and computational the eye-catching A = B. The book was published mathematics, University of Ljubljana. His email address is in December 1995. Although I am listed as the [email protected]. first author because Herb and Doron insisted on

April 2015 Notices of the AMS 351 alphabetical order, Herb was without a doubt the learning how to set up a web server, write HTML principal author. and CGI scripts, and put together a mock-up of The third time that Herb made a difference in the journal, including fake papers such as “On the my life was in 2006 when I was in the middle of structure of the empty set,” by Neil J. Calkin and my two-year stint as department head in Ljubljana. Herbert S. Wilf, and “On the number of subsets I was desperate about how I was going to survive of the empty set,” by Herbert S. Wilf and Neil J. it, and he offered me the chance to “recuperate” Calkin. on a sabbatical afterwards. So I spent the spring On seeing the mock-up, Herb immediately term of 2008 as a visiting scholar at the University recognized that the technology was mature enough of Pennsylvania, which helped me get my research to realize our vision, and so he set about the much back on track. But most of all, Herb inspired me harder task of assembling a stellar editorial board with his unfailingly positive attitude, his wit and and twisting arms to get some excellent papers good humor, his warm personality, his kindness for the first volume of the Electronic Journal of and care for his students and coworkers. When I Combinatorics (E-JC). We announced the journal was doubting my mathematical abilities, he gave to the world a mere three months later, in April me his paper “Self-esteem in mathematicians” [20], 1994, and it has gone from strength to strength intended for young mathematicians with just this since then. kind of insecurity, and it helped me a great deal. For a while we were seen as leaders in the new To put it simply, he was the rare kind of person field of electronic publishing in mathematics and who makes the people around him happy. He was a were invited to various gatherings to discuss our great mathematician and could have behaved like a endeavors. At one meeting, at MSRI, I referred to prima donna, but he never did. Just the opposite— the E-JC as being like “my child,” and I remember he was a wonderful human being, uncomplaining how upset he was at the time: ever after, I referred and unselfish, sensitive to the needs and feelings of to it as “our child”! others, and that is how he lives on in my memory. There was a great deal of skepticism among Neil Calkin some in the publishing community that E-JC could survive: People told us we didn’t have a sustainable Some memories of my friend and mentor, Herbert S. model, that it was bound to fail for lack of interest, Wilf: I first met Herb at a conference on the West that people wouldn’t take it seriously, and most Coast, I believe in 1985 when I was still a beginning worryingly, that once Herb and I were not involved, graduate student. He was incredibly generous with it would fall apart. At least partly with this in mind, his time and attention, and especially with his we both made efforts to bring other people into ideas. the fold and to step back when the time was right The next time I met him, a couple of years so that others could demonstrate that the E-JC had later, he remembered me and what sorts of things an existence beyond either one of us. And I think I was interested in and took care to pose some that, looking at how strong the journal is now, we questions he thought might interest me. Over the have proved that point! next few years we met up at conferences often and Herb and I wrote three papers together. It could would talk and developed ( I feel, at least) a close have been many more had we been in closer friendship. physical proximity—for all my love of electronic In 1993, at Herb’s former postdoc Jamie Rad- communication, I still work better in the same cliffe’s wedding, I learned of the World Wide room with my collaborators. That said, working Web and of the beta release of NCSA’s Mosaic with Herb via email was a delight: bouncing ideas web browser. Consequently, the following January, back and forth until we’d obtained a beautiful when Herb suggested that we needed a mailing result or discovering that we’d just reinvented list in combinatorics, I was ready to suggest that something in Chapter 4 of Comtet’s Combinatorial the time was right to start an online journal in Analysis. combinatorics, a much more ambitious project. Our first paper, “On counting independent sets The very same day (I don’t know whether it was in grid graphs,” remains one of my most cited before or after I talked to him) Fan Chung also papers: it was the convergence of work that each of mentioned Mosaic to him, and so the project us was doing independently (in my case, building on seemed like it might be interesting. work I’d done some years earlier in my thesis; in his, Herb was travelling to a meeting in Bordeaux a building upon a paper on counting nonattacking few days later on a two-week trip and said he’d consider the idea upon his return. So I got to work configurations of kings). It’s apparently of great interest in the IEEE community, and as the results Neil Calkin is professor of mathematics, Clemson University. are not yet sharp, it’s a problem I still return to His email address is [email protected]. frequently.

352 Notices of the AMS Volume 62, Number 4 Our second paper, “Recounting the rationals,” has garnered more attention for me than anything else I’ve been involved with. I’ve had more than twice as many emails and comments about that paper than on all my other work combined. There is a Wikipedia page about the “Calkin-Wilf Tree” (which we didn’t name!) and one can even buy a T-shirt with a wonderful picture of the tree and rendition of our proof. I learned a very important lesson on that paper: not to give up! When we submitted the paper, a referee pointed to a paper in a birthday conference proceedings which proved some of the results we had proved. Naturally we withdrew the paper. Herb, however, recognized that although the countability of the rationals was incredibly well known, it was worth writing up our proof of that, and so the note in the Monthly came about. I tell all my students this story as a life lesson! Our final paper together, “On a shooting game of Lampert and Slater,” with Rod Canfield, was a delight to research and a joy to write. It showed Herb’s touch and grace: when we couldn’t analyze the original question, he suggested a simpler model which we were able to analyze exactly. This led us first to being able to understand what the behavior of the shooting game would be and then to developing a technique to prove it—incidentally, requiring fairly heavy computation along the way, making it very much of the flavor of experimental Photo courtesy of Ruth Wilf. mathematics. Herb Wilf, Boca Raton, 2002. Herb and I would frequently send emails back and forth about various topics in mathematics: I remember one long exchange in which I was Rodney Canfield developing all sorts of ideas, with Herb sending encouragement after encouragement after encour- I treasure time spent one-on-one with Herb, usually agement until finally he realized that I had recreated at conferences, visiting and chatting, mostly about Faà di Bruno’s not-well-enough-known formula for math, of course, but over the years I learned a little the derivative of a function of a function. He made about his interesting personal history. He told me sure that I knew not to be disheartened by this, to about living with his Aunt Martha while his parents understand that to rediscover something gorgeous were overseas for a while and about going off to is good in and of itself. This was typical of his MIT as an undergraduate. He painted a picture of generosity of spirit and nurturing manner. a rather naive young man but one who had many Herb was tremendously supportive of me in adventures. all ways, not least by writing letters of reference Even before graduate school at Columbia, Herb and recommendation for me for jobs and (I began his acquaintanceship with computers. The believe) tenure and promotion. He was a wonderful delightful story of how in 1951 he came to be mentor to me, an inspiration mathematically and employed by Nuclear Development Associates pedagogically, and a great man. I am very fortunate while searching for duplicate bridge opportunities to have called him my friend and feel incredibly is told on his website in an essay honoring his honored that he called me friend in return. He is, mentor, Gerald Goertzel. (Also among his fellow and will remain, sorely missed. programmers was Alicia Nash.) Throughout his career Herb was enthusiastic about new ways to use computers in mathematics. He was among the first I knew to use TEX or email or to embrace the

Rodney Canfield is professor of computer science, University of Georgia. His email address is [email protected].

April 2015 Notices of the AMS 353 World Wide Web. Regarding the latter, in 1994, in including works on identities, algorithms and com- collaboration with Neil Calkin, he founded an online plexity, numerical analysis, mathematical methods journal, the Electronic Journal of Combinatorics. for the physical sciences, integer partitions, and Then and now, in addition to being delivered via generating functions. I’d like to say something the WWW, the journal adheres to the dual goals about the last. of being cost free to readers and authors and of The generating function, historically as old at leaving copyrights with the authors. Presently, the least as Laplace, is one of the most powerful and future of scientific publication and the role of dependable tools in combinatorics. As a graduate commercial publishing companies therein are very student I soaked up the tricks and techniques of much in the limelight. (See the June/July 2012 issue generating functions, but later in an undergraduate of the Notices, for example.) In the preelectronic course my students were perplexed by the topic. publishing days (1979) Herb was also the founding When they asked for a good reference on the topic, editor, along with Donald Knuth, of the Journal of I had no ready response—until, that is, 1990, when Algorithms. the first edition of Herb’s generatingfunctionology My first personal contact with Herb came when appeared. A beautifully written, appropriately he was the editor of the Monthly, in the form of titled, and totally useful text, it is a perfect starting a request for a referee’s report. Although I had a place for ambitious students and well worth the lot to say about the paper, I lacked the confidence occasional visit from the expert. to say it and kept putting the report off. I got a In the last chapter of generatingfunctionology stern reminder that it was due, and so I didn’t Herb comes to analytic and asymptotic methods. know how else to proceed except to write up what In the preface he likens using generating functions I had been thinking. I got back a prompt thank-you only in the formal sense without analysis to note saying, “Well-worth waiting for,” and it was listening to a symphony in stereo but with only most gratifying. Shortly after this, thanks to Ed one channel turned on. Given his mathematical Bender’s friendship with Herb, I had the chance to roots, it is no surprise that as Herb turned his attend an informal math workshop at Herb’s beach attention to combinatorial problems, he brought house on the New Jersey shore. Besides Ed and me, along an abundance of analytic techniques. Having Bob Robinson, Bruce Richmond, and Nick Wormald myself been once congratulated on having the only were in attendance. We did math endlessly, and it contour integral at a combinatorics conference, is truly a fond memory. this love of analytic methods was a common bond For about the last twenty-five years I had fairly for us. In 1990, when generatingfunctionology regular correspondence with Herb. Any time I had appeared, it had been twenty-two years since a question or wanted an opinion, I could write and Herb published the paper “A mechanical counting receive an immediate useful reply. I tried not to method and combinatorial applications.” In that overdo my welcome and also to reciprocate by paper he combined Cauchy’s formula for coefficient being a good correspondent. It was not so hard extraction with inclusion-exclusion. In his review for me, because I did not have people all over the of the paper, David Klarner wrote, in part, “Using world writing to ask me my opinion, as I am sure these results the author (a) describes a formula for Herb did. I suspect he was just as prompt with the number of Hamilton circuits in a connected virtually all inquiries, and I am mystified as to how graph involving about n42n operations, (b) finds he did it. Part of the answer is that he had such a the number of 1-factors of a graph, (c) gives a new fine mind, and the other part is a testimony to his formula for the permanent of a square matrix and dedication. derives Ryser’s formula, (d) solves the generalized When it comes to talking about Herb’s accom- ‘problème des ménages,’ and (e) gives inequalities plishments in research, it is literally impossible to for the permanent function which are superior know where to start. Surely one of the most lasting in some cases to the old bounds. It should be and significant breakthroughs was his work with clear from this impressive list that the elegant Doron Zeilberger on the WZ pairs. For me person- generating function technique developed in this ally, certain other of Herb’s interests are especially paper is worthy of further study.” appreciated. His original book with A. Nijenhuis Our favorite topic was integer partitions. We Combinatorial Algorithms, published in 1978, is often wrote and talked about these and worked on one of my prized possessions. I have spent an a number of problems. Here is one. Notice that if incredible amount of time with that book. The you allow yourself the binary powers 1, 2, 4,... as various problems associated with efficient listing parts and only 0 and 1 as multiplicities, then there and navigating through combinatorial objects, I is exactly one partition of each positive integer think, are totally absorbing. Besides this work n. Problem. Do there exist an infinite set of parts on combinatorial algorithms, Herb has authored A = {1 ≤ a1 < a2 < ···} and an infinite set of an impressive number of other influential books, multiplicities M = {0 = µ1 < µ2 < ··· } such that

354 Notices of the AMS Volume 62, Number 4 p(n; A, M), the number of partitions of n whose parts belong to A and whose multiplicities belong to M, that satisfy: • p(n; A, M) > 0 for all n sufficiently large, • there exists constant C such that p(n; A, M) = O(nC ), n → ∞? We were unable to resolve this question, and it is listed as unsolved problem #2 on Herb’s website. Recently, Noga Alon has given a complete solution; see Integers, Volume 13, 2013. Besides his monumental body of work and his generous service to the profession, Herb had a profound effect on many younger persons who love mathematics. He posed intriguing problems

of just the right level of difficulty. He was a great Photo courtesy of Ruth Wilf. source of encouragement, and we are all left with Albert Nijenhuis, Herb Wilf, 1996. a deep sense of personal loss at his passing. Albert Nijenhuis as number-theoretical questions. Herb decided When Herb Wilf, then a junior faculty member at to become a combinatorialist and spent much of U. Illinois, came to Penn in the spring of 1962 1966–67, while on leave at the London Imperial as part of the year-long Hans Rademacher retire- College, acquainting himself with the field. ment conference, I. J. Schoenberg, the department During the year 1973–74 Herb was on leave at chairman, was so impressed by this young man Rockefeller University on a Guggenheim Fellowship. that, in consultation with the department and When he returned to Penn, he brought with him an the university dean, he offered him an associate outline for a book. It was an early version of what professorship with tenure. Herb accepted with would be our book, Combinatorial Algorithms. pleasure, looking forward to an active relationship. Each chapter would contain a mathematical dis- Things worked out differently. Within a year, cussion, followed by an algorithm and a FORTRAN Schoenberg moved elsewhere. Herb continued program. The book was therefore both a textbook working on his other numerous interests but was suitable for a combinatorics course and also a looking for another focus. reference for immediate application. In 1964 I joined the Penn mathematics de- The chapters that deal with “combinatorial partment, still actively involved with deformation families” have a novel feature. Each time the theory, but I sensed this was approaching its end. algorithm is called, it chooses another member of I was also looking for a new focus and had had the family. The sequencing rule for the choices occasional contact with Herb. depends on the nature of the family. One option is Around that time, the faculty and students at uniform randomness; another interesting one is Penn gained access to the mainframe computer the so-called “revolving door” method. on campus. The procedure was clumsy, but Herb In spite of some mixed reviews, the book was and I were both interested. Herb had previous well received. The first edition in 1975 was followed experience with computers: In his graduate student by the second in 1978. Gradually, FORTRAN went days he had participated in a project simulating out of style, and we did not envision another the operation of a nuclear reactor. At that time edition. However, when Herb made a number of machine language was used; now FORTRAN was his books freely available online, this book was available. Here we found an area of mutual interest. among them, and from then on it has enjoyed an Herb was the first to teach a section of Freshman unexpectedly large number of “hits.” Calculus that required the use of a computer; During our twenty-or-so years together at Penn, others followed. He also wrote an introductory there was steady traffic between our respective booklet on FORTRAN, less detailed but more offices, resulting in published papers and chapters accessible to beginners than standard texts. We in the book. At the same time, Herb kept up with his discovered how suitable computers are for the other interests, resulting in papers on a variety of exploration of combinatorial structures, as well subjects with a variety of coauthors. His teaching Albert Nijenhuis was affiliate professor of mathematics, at Penn earned him a distinguished campuswide . He passed away on February 13, award and his guest lecturing a national MAA 2015. award. The latter activity was aided by Herb’s

April 2015 Notices of the AMS 355 356 ie uh hs nlddflig idn,boating, birding, flying, included These Ruth. wife, al .Svg spoesro optrscience, is computer address of email Her professor University. is [email protected] State Carolina Savage North D. Carla also he But mathematics. to contributions many projects years. interesting the several through Ruskey, on Frank expert collaborator code put my Gray He you with gems. touch of as in full so, me are legendary, emails those is imagine, wit can Herb’s results. problems and about correspondence email an started years. the has through he encouraged mathematicians and of inspired dozens the by of him one about any written been have could paragraph talk. Pennsylvania a of give University to the to me invited and encouraging enthusiastically was Herb to it. about wrote him and one solved I codes. Gray combinatorial SIAM 1989 his monograph. of combi- subject the about algorithms, address natorial invited He spectacular Francisco. a Confer- San SIAM gave in the Mathematics at Discrete 1988 on in ence Wilf Herbert met first I Savage D. Carla humor. of lightning-quick sense in Herb’s present: house Always shore Jersey. their New at informally friends they their as entertained environment hospitable warm a and created Ruth and He gadgets. photography, preferred his travel. provided of mode which airplane, private tteW0brha eerto,Wtro,2011. Waterloo, celebration, birthday W80 the at cneec oraie) uhWl,SsnWilf, Susan Wilf, Ruth coorganizer), (conference ebWl a ra rbe ovrwomade who solver problem great a was Wilf Herb we and talk, a give to invitation Herb’s accepted I previous the that is Herb about special so is What about questions intriguing several posed He his with shared often hobbies, many had Herb akrw ai if ee if ueeZima Eugene Wilf, Peter Wilf, David row: Back Photo courtesy of Ruth Wilf. . oie fteASVolume AMS the of Notices ihtegop eoal xeinefru all. us for experience memorable a group, the with hymd oegetdsoeisadbcm our became and discoveries great some made They ie fiqiy eeaeafweape rmthe from partitions. examples integer few of a new realm are Here entirely inquiry. of started lines and objects mathematical meet and talk a give to State N.C. to down plane his [ coauthors Corteel). in Sylvie students was few them a of (one include discussions to our me allowed he and partitions, integer about questions some suggested berger. like titles, functionology playful with was books he of ideas And inside mathematical results. hard explaining new at about masterful hear to one first Combinatorics of Journal Monthly Mathematical enjoyed American the most with he work things his the about of one think mathe- I about matics. communicating and learning loved • • • • • ebakduuulqetosaotfamiliar about questions unusual asked Herb Herb 1996. until not was collaboration first Our 111111 where 1 integer partitions Consider of m partition random part a chosen in randomly size a that probability the is As following: the considered Herb cannot. [ Spencer and Bona, diagrams Ferrers the to the corresponding by tiled be can enough rectangle large for if, asked Herb of consisting diagram, Ferrers of partition A as 0 to goes Several [ Erd˝os graph? Richmond including and simple hand, their a tried a of that 2 sequence integer probability degree even an the of is partition random What asked: [21]. Herb yes is answer the Surprisingly, requirements: example, of the For a partitions 1.) of or list to 0 following increase to the may decrease 0 may size 1 of size part of decreased part has (A part part 1? one one by and that 1 its by in from increased only differs has list element the an on that predecessor integer way of a partitions such always the it list Is to possible Conference: 1988 Math the at Discrete talk his SIAM in question this posed Herb 21 /(m(m ttrsotta h rbblt approaches probability the that out turns It ? .Fnly itlsoe htteprobability the that showed Pittel Finally, ]. x , 1 m 16 + 21111 ≥ o o aypriin of partitions many how for , ,[ ], 1 and x )) n 2 17 · · · ≥ ∞ → as , A=B .Drn hstm,Hr flew Herb time, this During ]. n 3111 n a erpeetdb its by represented be can ∞ → [21]. ihPtoškadZeil- and Petkovšek with , x ora fAlgorithms of Journal 222 21 p(n) x k , a etn ob the be to getting was , 1 > eeal oso it show to able were ] 2211 [15]. + , 0 321 attosof partitions . x ie nonnegative Given 2 n the and , , n + · · · + , a multiplicity has n 33 62 n n ntsquares. unit an , n , Number , = generating- 42 ∞ → satisfies 6 Electronic n , x n n k × 411 Alon, . what , sthe is of p(n) the , n n . in is n 4 m = x2 + x4 + x6 + · · ·? Herb observed that for n ≥ 3m, there is a very nice answer and it is independent of n (!) We were able to explain and generalize this in [14] with Rod Canfield. One of the side benefits of collaborating with Herb was getting to know his wife, Ruth, a renowned midwife and childbirth educator, and organizer extraordinaire, who is now my own friend. I am grateful to Herb and Ruth for their hospitality during my many visits to Philadelphia. I am full of memories of joyful meals with them at conferences all over the world, talking about math,

life, and childbirth. Some standouts: Philadelphia Photo by Albert Nijenhuis. (food trucks, cheesesteak subs), the Southeastern Herb Wilf, Paul Erd˝os,1996. Conference in Boca Raton (lunch with Sister Celine), MADCAP in Baltimore (steamed crabs), the Clem- son miniconference (he flew me there in his define recursively the result S(w) of passing w plane!), Herb’s PIMS lectures on integer parti- through a stack as follows. Write w = unv, so u tions (see www.math.upenn.edu/~wilf/PIMS/ and v are permutations of complementary subsets PIMSLectures.pdf) in Victoria (Butchart Gardens, of {1, 2, . . . , n − 1}. Then S(w) = S(u)S(v)n.A more steamed crabs), FPSAC in Sicily (gelato), permutation w is stack-sortable if S(w) = 12 ··· n. CanaDAM in Montreal (bagels), and Chapel Hill We also say that w is 231-avoiding if there does (Ruth’s 80th birthday dinner). not exist a subsequence of w of length three Our last paper together was begun in March whose terms are in the same relative order as 231 2011 when Herb and Ruth were in Chapel Hill for a (smallest term last, second-smallest term first, and childbirth conference that Ruth was chairing. While largest term in the middle). Equivalently, there Ruth attended the conference, Herb and I sat at a does not exist i < j < k with ak < ai < aj . Knuth table in the lobby talking about hypergeometric [Exercise 2.2.1.5 in The Art of Computer Program- identities associated with binary words. Herb wrote ming, Vol. 1] showed that w is stack-sortable if and to George Andrews about it, and with George’s only if w is 231-avoiding. He also showed [Exercise response, our joint paper was born [13]. 2.2.1.4 in The Art of Computer Programming, Vol. With Herb, doing mathematics was not only 1] that the number of 231-avoiding permutations exciting but sometimes hilarious. I will greatly miss w ∈ S is again the Catalan number C . sharing it with him. n n With the benefit of hindsight it seems pretty Richard P. Stanley obvious to consider permutations avoiding a permutation v ∈ Sk for any k ≥ 3 or, more One of Herb Wilf’s most influential contributions generally, avoiding a set of permutations in various to mathematics is the theory of pattern avoidance. Sk’s. Thus, for instance, 3614725 does not avoid The first result in this area is due to P. A. MacMahon 2143, since the subsequence 3175 has its elements [21], who showed that the number of permutations in the same relative order as 2143. The first person to come up with this “obvious” insight was Herb w ∈ Sn (the symmetric group of all permutations of 1, 2, . . . , n) with no increasing subsequence of Wilf. Previously, pattern-avoiding permutations 1 2n arose in connection with other problems, but no length three is equal to the Catalan number . n+1 n one thought of investigating these permutations Like almost all of MacMahon’s work, this striking for their own sake. The first paper [21] on general result was forgotten or ignored until the combina- pattern avoidance per se was published in 1985 by torial renaissance of the 1960s. MacMahon’s result Herb’s PhD student Rodica Simion in collaboration was rediscovered by Hammersley [21] around 1970. with Frank Schmidt. They considered even and odd Hammersley’s result arose from a famous problem permutations avoiding sets of permutations in S . posed by Ulam [21], asking for the expected length 3 Write s (v) for the number of permutations in of the longest increasing subsequence of a random n S avoiding the permutation v. In unpublished permutation in S . Meanwhile, Donald Knuth in n n work in the 1980s, Wilf raised the question of when the 1960s considered the question of sorting a two permutations u, v ∈ S satisfy s (u) = s (v) permutation w = a a ··· a on a stack. We can k n n 1 2 n for all n ≥ 1. These permutations are now called Richard P. Stanley is professor of applied mathematics, Wilf-equivalent, and the equivalence classes are Massachusetts Institute of Technology. His email address is called Wilf classes. Probably the first paper to use [email protected]. the term “Wilf-equivalent” (in fact, in the title) is

April 2015 Notices of the AMS 357 358 a nlygvna xetoal lgn proof elegant exceptionally an given finally was whether asked I while Amc togrrsl a rvdb ee [ Regev by proved was result stronger much (A h nes cemn ucin h conjecture The function. Ackermann inverse the Friedgut and [ Alon doubt because little conjecture, a the expressed about Wilf that interesting is It as known the became questions equivalent two these of inequality existence the that the constant asked some for whether others of question and weaker seemingly Bóna Miklos Meanwhile him by [ published in first was conjecture modified number a L exists there whether asking by conjecture all for whether Regev) of 1980 large around of behavior asked the (independently) about I and Herb that a but lacking. equivalence, is Wilf understanding of definitive theory the into gone in has classes Wilf three are in [ permutations West and Babson by 21

v Photo courtesy of Ron Graham. ti ipecneuneo h RSK-algorithm the of consequence simple a is It on nuprbound upper an found ] < tne-ifconjecture Stanley-Wilf 6 elatrfis icltn h problem. the circulating first after well ] n ∞ ebfis se we nwr ftework the of unaware (when asked first Herb . v ebWl n a hn ttesixty-fifth the at Chung Fan and Wilf Herb ∈ n lim uhthat such →∞ S s n k s v c < (v) ebqikymdfidhsoriginal his modified quickly Herb . n ( 1 S , 3 2 s r ifeuvln,wiethere while Wilf-equivalent, are k) , . . . , c n n v (k < (v) lim = s naraiease to answer affirmative An . itdyCneec,1996. Conference, Birthday 21 n lim (v) c(v) L n .Tu,frisac,all instance, for Thus, ]. S v →∞ aecie yBóna. by coined name a , n 4 1 →∞ /n ra elo work of deal great A . o general for seuvln othe to equivalent is rai [ Arratia . c s nα(n) n = + s (v) n 1 (k (v) oie fteASVolume AMS the of Notices ) 1 n where , − /n 1 o all for 1 /n ) = 21 v 2 s = . n L ∈ proved ] v c < (v) (k v α(n) v S This . − k ∈ 21 and 1 S ) is ].) k n 2 , ptenaodneWl. nana conference annual An Wilf.” avoidance “pattern what anti utmr ocalnesuet with students strange— challenge he was to said That customary and this. it about surprised wasn’t sorry looked really he was him, told we lecture. next the for waiting hsseshgl nieybtrmisoe.For open. remains but unlikely highly seems This 21 ute plctoso atr avoidance pattern of applications Further 4231. eymc w-eetrcus ncombinatorics in course two-semester a much very GL variety plcto,Lkhia n ady [ Sandhya and vast Lakshmibai a sample a application, with As applications. right, and variations own of number its in subject flourishing permutations. multiset and compositions to permutations [ from avoidance Savage pattern Carla generalizing with The paper, permutations. second pattern-restricted certain subsequence [ in increasing longest Hildebrand the Deutsch of J. distribution Emeric A. with and first, The developments. future anticipated which of both avoidance, pattern permutation identity of the values nor smallest largest the neither surprisingly, Somewhat numbers the conjecture. open are Stanley-Wilf instance, many For the are to there related classes, problems Wilf [ of Tardos theory G. the and Marcus A. by ei aenki rfso fmteais nvriyof is University address email mathematics, His of Delaware. professor is Lazebnik Felix he did nor motivation, give and not views panoramic did broad Herb Furthermore, problems? hard one When no problem. when homework assigned day an a in solve remember or could I immediately lecture. either next quickly, the arose, difficulty it a resolved ever he if lectured and He smoothly, stimulating. and so calmly was that course the in became I that impatient further more The the 1983–84. progressed, in course him from took I that liked I that was advisor my be main to him The asked University 1987. I the reason until at 1984 from advisor Pennsylvania PhD of my was Wilf Herb Lazebnik Felix fecund and proliferant a forest. into developed Wilf Herb have by planted [ seedlings 2011 words avoidance in and pattern published was permutations Kitaev in Sergey patterns by on and 2003, book in a begun was patterns search permutation on the for hits [ 1,200,000 database over Tenner’s gives Bridget Google in found if be may only and if smooth is variety Schubert a that h hoyo atr viac snwa now is avoidance pattern of theory The [ to addition in papers two had Herb uhltrItidt nlz hteatyh did he exactly what analyze to tried I later Much ,v u, ∈ (n, S C k )/B ow have we do L (where v for v = X(w) v 18 [email protected] 12 B ∈ ,i eoe othe to devoted is ], w · · · L saBrlsubgroup) Borel a is S ntecmlt flag complete the in L u k 21 v vis31 and 3412 avoids L < sahee ythe by achieved is k lasalgebraic? always n20.A in As 2004. in ] . 62 v 19 or Number , 6 ,concerns ], eae to related ] 21 L 21 u show ] = .The ]. 21 L v 4 ? ]. . mention history—features that I valued in courses. So why did I like his course so much? Here is one reason I can name: it was not just combinatorics, it was mathematics. Simple analysis, operators involving partial derivatives, light number theory, or linear algebra were used without warnings. A long transformation was carried out on the board and was not left for a dull homework problem. Some needed facts (like Euler’s summation formula or the Gauss-Lucas Photo courtesy of Ruth Wilf. theorem, etc.) were not mentioned as “well known” Back row: Ellen Gethner, Carsten Thomassen, but presented and proved. And it was clear that Ralph Stanton, Joan Hutchinson, Roy Levow, Herb enjoyed talking about this mathematics to us. Ruth Wilf, Lis Thomassen. Front row: Debra It was not an onerous duty for him; it was sharing Boutin, Michael Albertson, Herb Wilf, Esther of love. Another characteristic of his lectures was Levow after the Euler Medal award ceremony, his ability to present hard proofs in such a clear Boca Raton, 2002. way that a listener was left with an impression that nothing extraordinary was happening. Until one of Herb’s problems (and also Linial’s [21]) that I tried to reconstruct these proofs. like and have worked on. It is about thirty years Later, when I took a reading course with him, old and is still a subject of active research. What is Herb asked me to read and do problems (selected the greatest number of proper vertex colorings in by him) from D. E. Knuth’s The Art of Computer λ colors that a graph with n vertices and m edges Programming, Vol. 1 and L. Lovász’s Combinatorial can have? I obtained some initial results, more Problems and Exercises, both books with helpful recently good progress has been made, yet the solutions included. He asked me whether I did problem remains unsolved (see, for example, [21]). what he assigned, whether I had questions, and Doron Zeilberger once said that if he had to then assigned new reading and new problems. characterize Herb using one word only, it would During the first year of my research, Herb wanted be elegance. I agree. (See also D. Zeilberger’s to see me regularly once a week. Often I had very contribution to realize that one word could be little to report and so felt less than enthusiastic chosen in more than one way.) Restraint and grace going to these meetings. When on those days I was of style—this was Herb. And humor. Attending leaving his office, I felt great. My spirit was up, his eightieth birthday conference, together with and I was full of energy and optimism for another others, I tried to help push his wheelchair. He said week. What he said or did for this I have no idea! with a smile, “Well, I used to push my students, All our meetings rarely lasted more than fifteen and now they are pushing me!” minutes and usually took place in late afternoon. We became closer during the years. Ruth and His parting words were typically “peace, enjoy…," Herb were always adventurous with food. The with a hand raised slightly and a smile on his face. last dinner my wife, Luba, and I had with them When Herb agreed to be my advisor, he said was at Uzbekistan, a restaurant in far northeast that his task was to suggest a good problem and Philadelphia, and this was their idea. Four months that he expected me to be independent. Later, before his death, Herb and Ruth invited us to visit his most common answer to my questions was them in their vacation home on the Jersey shore. a simple “I do not know,” and somehow it was Something made us reschedule it, but then it was comforting. I believe that he did help me to become too late. We will always regret it. professionally independent. Much later I tried to convince him to write at References length about his teaching philosophy and methods, [1] N. G. de Bruijn, H. S. Wilf, On Hilbert’s inequality in since they resonated well with so many of his n dimensions, Bull. Amer. Math. Soc. 68 (1962), 70–73. students. His answer was a short “No,” with a [2] H. Widom, H. S. Wilf, Small eigenvalues of large remark that teaching was an art and could not be Hankel matrices, Proc. Amer. Math. Soc. 17 (1966), replicated. Having taught for many years, I agree. I 338–344. also agree with his view that “ …teaching is mostly [3] N. Fine, H. S. Wilf, Uniqueness theorems for periodic a chemical activity that goes on between Human A functions, Proc. Amer. Math. Soc. 16 (1965), 109–114. and Human B.” More of Herb’s views on teaching [4] H. S. Wilf, The eigenvalues of a graph and its chromatic number, J. London Math. Soc. 42 (1967), can be found on his Penn website. 330–332. Herb was very reserved with giving unsolicited [5] H. S. Wilf, D. Zeilberger, Rational functions certify pieces of advice. I remember only this one: Never combinatorial identities, J. Amer. Math. Soc. 3 (1990), work on a problem you do not like. Here is one 147–158.

April 2015 Notices of the AMS 359 [6] H. S. Wilf, The patterns of permutations, Discrete Math. 257 (2002), 575–583. [7] N. Calkin, H. S. Wilf, Recounting the rationals, Amer. Math. Monthly 107 (4) (2000), 360–363. [8] H. S. Wilf, On buckets and fires, Notices of the American Mathematical Society 57 (6) (2010), 750–751. [9] The Wilf Festschrift Volume, F. Chung (ed.), Electron. J. Combin. 4 (2) (1997). [10] W. J. Ewens, H. S. Wilf, Computing the distribution of the maximum in balls-and-boxes problems, with ap- plication to clusters of disease cases, Proc. Nat. Acad. Sci. USA 104 (52) (2007), 11189–11191. [11] H. S. Wilf, W. J. Ewens, There’s plenty of time for evolution, Proc. Nat. Acad. Sci. USA 107 (52) (2010), 22454–22456. [12] D. E. Knuth, H. S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math. 396 (1989), 212–219. [13] G. E. Andrews, C. D. Savage, H. S. Wilf, Hypergeo- metric identities associated with statistics on binary words, Advances in Combinatorics, Waterloo Workshop in Computer Algebra 2011, W80, I. S. Kotsireas, E. V. Zima (eds.), Springer-Verlag, Berlin, 2013. [14] E. R. Canfield, C. D. Savage, H. S. Wilf, Regularly spaced subsums of integer partitions, Acta Arith. 115 (3) (2004), 205–216. [15] S. Corteel, B. Pittel, C. D. Savage, H. S. Wilf, On the multiplicity of parts in a random partition, Random Structures Algorithms 14 (2) (1999), 185–197. [16] S. Corteel, C. D. Savage, H. S. Wilf, D. Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (2) (1998), 186–192. [17] J. Nolan, C. D. Savage, H. S. Wilf, Basis partitions, Discrete Math. 179 (1–3) (1998), 277–283. [18] E. Deutsch, A. J. Hildebrand, H. S. Wilf, The distribution of longest increasing subsequences in pattern-restricted permutations, Electron. J. Combin. 9 (2003), #R12. [19] C. D. Savage, H. S. Wilf, Pattern avoidance in compo- sitions and multiset permutations, Adv. Appl. Math. 26 (2006), 194–201. [20] H. S. Wilf, Self-esteem in mathematicians, College Math. J. 21 (4) (1990), 274–277. [21] References can be found in the full version of the article at www.math.ucsd.edu/~fan/wilf/hw-ref. pdf

360 Notices of the AMS Volume 62, Number 4