numerativeombinatorics

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A Enumerative Combinatorics and Applicationsecajournal.haifa.ac.il

ECA 1:3 (2021) Interview #S3I11

Interview with Igor PakToufik Mansour

Igor Pak did his undergraduate studies at Moscow State University(1989–1993). He obtained a Ph.D. from Harvard University in 1997,under the direction of Persi Diaconis. He became an assistant profes-sor of applied mathematics at Massachusetts Institute of Technology(MIT) in 2000. From 2005 up to 2007, he continued at MIT as an as-sociate professor of applied mathematics. From 2007 to 2009 he wasan associate professor of mathematics at University of Minnesota.Since 2009 he is a full professor of mathematics at UCLA. ProfessorPak has given talks at many conferences, including an invited talkat the International Congress of Mathematicians in Rio de Janeiro,Brazil in 2018. Professor Pak has served as a member of the editorialboard in numerous journals, including Discrete Mathematics, Asso-ciate Editor (2009–2017), Transactions of American Mathematical

Society and Memoirs of American Mathematical Society, Associate Editor (2014–2018), PacificJournal of Mathematics, Editor (2016–2018), and Mathematical Intelligencer (2021-).

Mansour: Professor Pak, first of all, we wouldlike to thank you for accepting this interview.Would you tell us broadly what combinatoricsis?

Pak: Thank you, Toufik! It is an honor to beasked. I have thought a lot about this questionover the years1 and concluded that there is nogood answer, or at least there is no one goodanswer. There are three types of answers Itypically give: serious, contemplative and con-frontational, so let me give you a brief versionof each.

Serious answer: Combinatorics is a largearea of discrete mathematics comprised someinterrelated smaller subareas (enumerativecombinatorics, graph theory, probabilisticcombinatorics, algebraic combinatorics, etc.)Each of these subareas has its own rich culture,goals, and traditions, making a broad general-ization neither possible nor desirable.

Contemplative answer: I subscribe to

Rota’s approach to the question2 where hecompares areas of mathematics to warringcountries. The borders are rarely straight lines,as they tend to be produced by centuries ofbattles, making them not easily describable.Some countries have a nontrivial topology (forexample, non-simply connected or even discon-nected), some borders are disputed, and eventhe existence of several countries is subject todebate. On top of that, some countries havevarious types of federal systems, with only afew laws governing different regions which canhave different languages, uniquely distinct his-tory, specialized cuisine, etc.

Now, how is one supposed to define large di-verse countries like the USA, Russia or India?The only way is really to give a vague gen-eral description, before moving to discussionsof history how the country came to be, withits complexity of individual states and regions.Combinatorics as a field has all the complexity

The authors: Released under the CC BY-ND license (International 4.0), Published: March 5, 2021Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is tmansour@univ.haifa.ac.il

1For example, see my old blog post https://wp.me/p211iQ-bQ2G.-C. Rota, Discrete thoughts, Chapter 6.

Interview with Igor Pak

of these large diverse countries. Thus to definecombinatorics is to discuss its history of how itstarted centuries ago with elementary enumer-ative questions, and in the past decades hasrapidly expanded to acquire a vast scientificterritory3. Then you proceed to define/discussall subareas4.

Confrontational answer: Combinatorics isa large advanced area of mathematics, on parwith algebra and analysis. It is just as diverseas these areas and thus cannot be easily char-acterized. It is just as important, interesting,competitive, and highly developed as these ar-eas, and must be treated equally when it comesto hiring, peer review, or research awards. Anyattempts to suggest that it is somehow ele-mentary, trivial or recreational are highly dis-respectful and come from ignorance5. How-ever, any claim that combinatorics is now asaccepted and as prominent as other areas, that“we are all combinatorialists now”, also comesfrom willful blindness. The area has been dis-missed and mistreated for decades. Right nowit is popular and many excellent combinatori-alists were hired by the top universities in thelast few years. Unfortunately, it is rather pre-mature to declare victory. People’s attitudesdo not disappear or change overnight — theyjust stop being openly expressed for the sakeof comity and politeness6. It will probablytake some time until combinatorics is treatedas truly equal.

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

Pak: I think it is pretty good actually, betterthan ever. Despite my previous answer, thenew generation of mathematicians tends to bevery respectful of the area. These days manystudents in all fields learned to appreciate anduse combinatorial results and ideas in their re-search. This has been a dramatic transforma-tion over my lifetime, and I expect things toonly improve in the future.

Mansour: What have been some of the maingoals of your research?

Pak: I worked in many areas of combinatorics,mostly doing problem-solving rather than the-ory building7, so I am not sure there is a simpleor concise answer to this question. In general,I like to frame questions computationally, froma broad theoretical point of view. For exam-ple, when I see a formula for the number ofcertain combinatorial objects or a closed-formgenerating function, I am interested if there isa way to sample these objects uniformly at ran-dom. And if not, why not? When I see a nicecombinatorial identity, I ask if there is a bi-jective proof of this identity? And if not, whynot and what does that even mean? Similarly,I ask what does it mean that some numbershave no close formulas, or a combinatorial in-terpretation, etc.

My questions tend to be vague and often re-quire some effort to state them in the languageof computational complexity, discrete proba-bility or group theory. As a result, sometimesI get to publish in these adjacent fields. How-ever, often enough these questions lead to newunexplored combinatorial questions, which getresolved in rather conventional combinatoricspapers.

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?

Pak: I grew up in a working neighborhoodin the south of Moscow, without any inter-est in mathematics or anything else. My par-ents worked a lot and were happy when I camehome with mostly 4s (in the Russian system,the grades are 1–5), and I was happy theynever pushed me to do anything after school.I think I played a lot of ice hockey outdoorsand never did any homework, that is all I canremember.

Things changed suddenly in the fifth grade3Of the many excellent books on the history of combinatorics, I especially like R. Wilson and J. J. Watkins (editors), Combina-

torics: Ancient & Modern, Oxford Univ. Press, 2015.4This is the approach I chose when I completely rewrote the Wikipedia Combinatorics article https://en.wikipedia.org/wiki/

Combinatorics (see also a quick discussion on my blog: https://wp.me/p211iQ-2d).5For example, see quotes by Henry Whitehead, George Dantzig or Jean Dieudonne on my lengthy “What is combinatorics?”

collection of quotes https://tinyurl.com/4v7rwnn2 or a story how such views by Peter May were confronted by Laszlo Lovasz, inhis interview linked here: https://wp.me/p211iQ-t2.

6Occasionally, their views find a way to reemerge in small ways. For example, see my collections of “Just combinatorics” quotes:https://tinyurl.com/3c6v55ma.

7W. T. Gowers, The two cultures of mathematics, in Mathematics: frontiers and perspectives, AMS, Providence, RI, 2000,65–78.

ECA 1:3 (2021) Interview #S3I11 2

Interview with Igor Pak

when I discovered a “new rule” for comput-ing percentages. To compute 60% of 20, Iused (20 × 60)/100, which seemed easier than(20/100) × 60, as there was no decimal pointissue. I remember getting a 1 on some percent-ages quiz, which did not bother me. But thenthe teacher berated me in from of the wholeclass claiming that I cheated all answers sincemy “new rule” was not in the teacher’s man-ual and thus could not be true. I complainedto my father. I knew I did not cheat, so theteacher must be wrong. My father knew aboutcommutativity of multiplication and sent meto a better middle school further away frommy apartment building.

In my new school, the math teacher wasvery enthusiastic about the subject and sentus all to a local (Chertanovo) math competi-tion. This was one of my unhappiest memory.While I solved only two problems out of five,everyone else claimed to have solved four orfive, making me feel inadequate. When the re-sults were announced a few weeks later, turnedout that I got the first prize because I did solvetwo problems and everyone else was just clue-less about the meaning of the word “solve”.

My father recognized that maybe I shouldstudy math more seriously. He asked aroundand suggested I attend the free “math circle”,which happened once a week at a magnet HighSchool 57 in the center of Moscow. Eventu-ally, I became good, got accepted to that highschool, some years later to Moscow University,etc. I received a lot of help from many teach-ers, mentors, and advisors along the way, but itwas my father, who figured out what I shouldbe doing with my life.

Mansour: Were there specific problems thatmade you first interested in combinatorics?

Pak: There were no specific problems, but Idid like to read math textbooks. When I wasin college, to make ends meet I got involvedwith used book sales. This gave me access to alot of textbooks that I read obsessively, learn-ing a great deal of math along the way. OneSummer, my large extended family decided togo on vacation together, taking a boat alongthe Volga river from Moscow to Astrahan andback.

This was a very long, very cheap, and veryboring vacation. When others were entertain-

ing themselves by reading mystery novels, Iread the first volume of Stanley’s EnumerativeCombinatorics. I was completely fascinatedby the book which I read cover to cover eventhough I could not solve most exercises. I amstill working on some of them...Mansour: What was the reason you choseHarvard University for your Ph.D. and youradvisor Persi Diaconis?Pak: There were two simple reasons. First, Ienjoyed learning. I knew that in addition tocombinatorics I wanted to learn discrete prob-ability which I did not know at all at the time.I met Diaconis and he was very charming andinquisitive. It was clear to me that if I go workwith him I would have to learn a great deal.That is exactly how it worked out.

Another reason was my sorry state of fi-nances. At the time, I was a refugee immigrantto the US, who arrived in New York by him-self with about $250 and a suitcase full of mathtextbooks. I lived in rather extreme poverty onpublic assistance, studying English and prepar-ing for my entrance exams (TOEFL and GRE).

My monthly food budget was about $80,while the application fee to Harvard was ahefty $75. At the same time, many other gradschools charged only $50, so after careful con-sideration I chose to apply to only five of those.A dear friend of mine, Sasha Astashkevich,told me he believes in me. He offered to paythe $75 fee to Harvard, on condition that I ap-ply and pay him back only if I get in. I didboth.Mansour: What was the problem you workedon in your thesis?Pak: I studied mixing times of various natu-ral random walks on Sn and other related finitegroups. I used strong uniform time arguments,which are combinatorial rules to stop the ran-dom walks so that the resulting distributionis uniform. Few years prior, Aldous and Di-aconis8 introduced this approach to study themixing times, and I found it to be a nice blendof both combinatorial and probabilistic analy-sis.Mansour: What would guide you in your re-search? A general theoretical question or aspecific problem?Pak: I always start with a specific prob-lem. Even if the problem is ill-defined or in-

8D. Aldous and P. Diaconis, Strong uniform times and finite random walks, Adv. Appl. Math. 8 (1987), 69–97.

ECA 1:3 (2021) Interview #S3I11 3

Interview with Igor Pak

approachable, without the problem, I do notfeel I know what I am talking about. Whenyou have a good problem, you can try to un-derstand it, dig around, read the literature onthe subject, do some explicit calculations, lookfor bridges to other areas, etc.

Only very occasionally I completely resolvethe problem. More frequently, I either resolvesome special cases, or reformulate the problem,or discover a related but more accessible prob-lem, etc. Occasionally, more often than thatI care to admit, there no progress whatsoever.I do not consider this a failure, as I enjoy thelearning process. Sometimes, this problem cancrawl back a few years later from a differentangle, and I start anew.Mansour: When you are working on a prob-lem, do you feel that something is true evenbefore you have the proof?Pak: Yes. Always. This is partly because Iwant things to be true, or, more frequently,false9. This is often because I have tools foronly one direction, and make a convenient bet.I get things wrong sometimes and waste time.Sometimes, you just can not tell which waythings will turn out.Mansour: What three results do you considerthe most influential in combinatorics duringthe last thirty years?Pak: I do not know. I think people like myproof of the hook-length formula10 and my pa-per11 on graph liftings. Based on citations, Ithink people also enjoy my partition bijectionssurvey12.Mansour: What are the top three open ques-tions in your list?Pak: Rather than go over a safe list of every-one’s favorite million-dollar problems, let memention a few lesser-known conjectures. First,recall that Hilbert’s Third Problem remainsopen for the sphere S3 (the hyperbolic spaceH3 is just as difficult)13. Formally, the conjec-

ture claims that two spherical polyhedra arescissors congruent if and only if they have thesame volume and Dehn’s invariant. In particu-lar, is it true that all spherical tetrahedra withrational dihedral angles and the same volumeare scissors congruent? We are very far fromresolving even this special case.

Second, let me mention a curious Generat-ing Primes Problem which asks to give a deter-ministic polynomial-time algorithm for findinga prime on the interval [n, 2n], thus derandom-izing Bertrand’s postulate which has a classicalprobabilistic algorithm14. It is a good bet thatone can do this by testing primality of n + x,overall 0 ≤ x ≤ C(log n)2, but this is nowhereclose to being proved15.

Third, there is a fascinating Skolem’s prob-lem which asks if it is decidable whether a se-quence {an} defined by a linear recurrence withconstant integer coefficients and initial valueshas at least one zero:

an+1 = c0an + . . . + ckan−k, for all n ≥ k,

where a0, . . . , ak, c0, . . . , ck ∈ Z. This problemis so basic and fundamental, it is rather em-barrassing that we do not even have a goodintuition which way it will go16.

Finally, let me mention one problem I def-initely do not want to see resolved: the P vs.NP problem. I can sort of imagine how any so-lution could destroy a delicate balance in com-putational complexity, with its powerful theo-rems, beautiful reductions, and the exhaustingmultitude of complexity classes17. Fortunately,the problem is so difficult it is unlikely to beresolved in my lifetime.Mansour: What kind of mathematics wouldyou like to see in the next ten-to-twenty yearsas the continuation of your work?Pak: It would be exciting to see more connec-tions and applications of computational com-plexity to enumerative and algebraic combina-

9See my blog post why I truly enjoy disproving conjectures: https://wp.me/p211iQ-uT.10J.-C. Novelli, I. Pak and A. V. Stoyanovsky, A direct bijective proof of the hook-length formula, Discrete Math. & Theor. Comp.

Sci. 1 (1997), 53–67.11F. Chen, L. Lovasz and I. Pak, Lifting Markov Chains to Speed up Mixing, in Proc. 31-st STOC (1999), ACM, New York,

275–281.12I. Pak, Partition bijections, a survey, Ramanujan J. 12 (2006), 5–75.13For example, see Chapter 1 in J. L. Dupont, Scissors congruences, group homology and characteristic classes, World Sci., River

Edge, NJ, 2001.14For example, see Section 7.1 in A. Wigderson, Mathematics and computation, Princeton Univ. Press, Princeton, NJ, 2019.15For example, see W. Banks, K. Ford and T. Tao, Large prime gaps and probabilistic models, arXiv:1908.08613.16For example, see J. Ouaknine and J. Worrell, Decision problems for linear recurrence sequences, in Reachability problems,

Springer, Heidelberg, 2012, 21–28.17For example, see S. Aaronson, P

?= NP, in Open problems in mathematics, Springer, Cham, 2016, 1–122.

18See the expanded version in I. Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636.

ECA 1:3 (2021) Interview #S3I11 4

Interview with Igor Pak

torics. Let me refer to my ICM survey18 forthe entry point to the subject.

Mansour: Do you think that there are core ormainstream areas in mathematics? Are sometopics more important than others?

Pak: Objectively, the answer is yes. Some ar-eas are more popular and have more influencethan others, it’s a fact of life. Should this bethe case? Thats complicated. Everything in-side of me suggests the areas should be treatedequally, without a trace of discrimination. Yet,it is hard to argue with reality.

To resolve this conundrum, think of areas aslive organisms. When they are born and takingfirst steps they are largely weak and unimpres-sive. When the area matures, acquires refinedtools and applications to other fields, it reachesthe height of its influence. Sometimes the areagets so popular, parts of it specialize, developtheir own identity, tools, and problems, andeventually separate.

As new fields emerge with their own pow-erful tools and the old applications get ex-hausted, the old tools can no longer compete.Soon enough, the area goes into decline. Occa-sionally, new tools or bridges to other fields arediscovered and the area reemerges in promi-nence. That is the circle of life. But as long asyou are respectful to other areas at all stagesof their development and do not judge themout of ignorance, it is ok to acknowledge thedifferences.

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?

Pak: I think the distinction is real even if itis a relatively recent phenomenon. Clearly, itis rather important whether you are trying tounderstand the world as it is, or as you imag-ine it to be. This does not mean there is somekind of antagonism. On the contrary, pure andapplied mathematics have a sort of symbioticrelationship with obvious mutual benefits.

Note that the difference can be delicate.For example, I am thinking of computer ex-periments aiming towards understanding thenature of some combinatorial objects as pure

math, while computer experiments with real-life data sets as applied math, even if the codeand statistical analysis are fairly similar. It isthe intention that counts, not the type of work.

Personally, I am squarely in the pure camp,in a sense that in my work I do not aim forpractical applications. In the rare instanceswhen my work does have such potential, I amhappy to leave such development to others.

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

Pak: First, make friends. Develop an exten-sive network of working relationships both inyour area and across mathematical sciences.As you mature and gain expertise, so will yourfriends. You want to be the one they are go-ing to call when they have questions in yourarea. When they do, go out of your way tohelp them. On the other hand, when your re-search leads you to problems in another field,do not be shy to ask for help. Mathematicsis a highly specialized field, so collaboration isbasically the only way to overcome that.

Second, learn to write well and give goodtalks. This will help you stand out and com-municate your results both to people in thearea and to a wider audience. I am not sure Imastered either skill, but I have thought quitea bit about writing, and encourage you to readboth papers19,20 I wrote on the subject.

Third, be entrepreneurial. Organize studygroups, seminars, workshops. Offer to givetalks, do not wait to get invited. Attend talksand conferences in other areas. Ask questionsand give answers on MathOverflow. If you seean interesting problem in the paper you can re-solve in some special case, reach out to the au-thors. Ask your colleagues about their favoriteproblems. Because – hey, you never know...

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

Pak: No, I have nothing to tell. This is notbecause I want to appear so mysterious, but Ido like to keep my private life private.

Mansour: One of your great works is, Prod-uct replacement algorithm and Kazhdan’s prop-erty (T ), co-authored by Alexander Lubotzky,published in Journal of the AMS. What isthe product replacement algorithm and why

19I. Pak, How to write a clear math paper, Jour. Human. Math. 8 (2018), 301–328.20I. Pak, How to tell a good mathematical story, Notices of the AMS (2021), to appear, https://tinyurl.com/2j3cf8re.

ECA 1:3 (2021) Interview #S3I11 5

Interview with Igor Pak

is it important? Could you tell us about theKazhdan property and how these two conceptsare related? Have there been any importantfollow-up results related to that paper?

Pak: The product replacement algorithm(PRA) is a powerful practical method to gen-eral random group elements. Roughly, it isa random walk on the graph of generating k-tuples of a finite group G, for a fixed k (it iscalled the product replacement graph). Soonafter this algorithm was introduced in 199521,a number of mathematicians across differentfields attempted to prove the remarkable prac-tical performance of PRA, all with limited suc-cess.

I was fascinated by the problem, and spendseveral years as a postdoc studying it. Even-tually, I proved that PRA works in polyno-mial time using a technical Markov chain argu-ment22. This completely resolved the problemfrom the theoretical computer science pointof view, but the mystery remained, as myO(log9 |G|) upper bound was nowhere closeto the (nearly) linear time experimentally ob-served performance of the PRA.

With Lubotzky, we realized that the prod-uct replacement graphs are the Schreier graphsof the action of the group Aut(Fk). Thisimmediately implied that these graphs areexpanders if Aut(Fk) has Kazhdan’s prop-erty (T ). The latter was a well-known openproblem with a negative answer known fork = 2 and k = 3, so the implication we dis-covered was a castle built on sand.

This was back in 2001. Just a few yearsago, in a remarkable computer-assisted break-through, the property (T ) was established forall Aut(Fk), k ≥ 523,24. Combined with ourwork, this proves that the product replacementgraphs are expanders indeed, thus giving themost satisfactory ending to the PRA story.

Mansour: Kronecker coefficients, as ‘tools’from Representation theory, are used to de-scribe the decomposition of the tensor productof two irreducible representations of a symmet-ric group into irreducible representations. Howdo they come into play in combinatorics in gen-

eral and in your research in particular?Pak: Multiplication of characters gives a ringstructure to the space of characters of any finiteor compact group, but the reason the prod-uct is a nonnegative sum of characters (as op-posed to a virtual character), is fundamentallya consequence of representation theory. Thisexplains why the structure constants of char-acter multiplication play a crucial role in ourunderstanding of the nature of irreducible rep-resentations.

For GLn(C), these structure constantsare called Littlewood–Richardson (LR-) coef-ficients. They have a classical combinatorialinterpretation as either the numbers of certainYoung tableaux or as the numbers of integerpoints in polytopes (as Gelfand–Tsetlin pat-terns or Berenstein–Zelevinsky triangles). Thereal reason for their compact combinatorial de-scription is a subtle consequence of the highestweight theory, even if this is not how they areusually presented (or how they were invented).

For Sn, these structure constants are calledthe Kronecker coefficients :

χλ · χµ =∑

ν`ng(λ, µ, ν)χν .

It was shown by Murnaghan back in 1938,that in the stable case they generalize the LR–coefficients, which naturally raised the ques-tion if they have a combinatorial interpreta-tion. This continues to be one of the most cel-ebrated problems in the whole algebraic com-binatorics, which is also very close to my work.

In part due to the lack of a combinatorialinterpretations, Kronecker coefficients are in-credibly difficult to compute or even to esti-mate25. Just to give you an idea how little weknow about them, for the three staircase dia-grams δk = (k−1, . . . , 2, 1) ` n, where n =

(k2

),

the best known bounds are:

1 ≤ g(δk, δk, δk

)≤ f δk =

√n! e−O(n).

The upper bound is probably closer to thetruth, so improving the lower bound is a majorchallenge.Mansour: In one of your papers Asymptoticsof principal evaluations of Schubert polynomi-als for layered permutations, co-authored by

21F. Celler, C. R. Leedham-Green, S. H. Murray, A. C. Niemeyer and E. A. O’Brien, Generating random elements of a finitegroup, Comm. Algebra 23 (1995), 4931–4948.

22I. Pak, The product replacement algorithm is polynomial, in Proc. 41-st FOCS (2000), 476–485.23M. Kaluba, P. W. Nowak and N. Ozawa, Aut(F5) has property (T ), Math. Ann. 375 (2019), 1169–1191.24M. Kaluba, D. Kielak and P. W. Nowak, On property (T ) for Aut(Fn) and SLn(Z), Annals of Math. 193:2 (2021), to appear.25I. Pak and G. Panova, Bounds on Kronecker coefficients via contingency tables, Linear Algebra App. 602 (2020), 157–178.

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Alejandro H. Morales and Greta Panova, bystudying the asymptotic behavior of the prin-cipal evaluation of Schubert polynomials, youpartially resolved an open problem by R. Stan-ley. What were the main ingredients of whatyou called ‘surprisingly precise results’? Whatabout the general case of conjecture? Are thereother combinatorial objects, besides permuta-tions, related to that conjecture?Pak: Schubert polynomials were originally de-fined by Lascoux and Schutzenberger in 1982motivated by the geometry of flag varieties.They are now some of the central and mostheavily studied objects in algebraic combina-torics26. While their definition is technical,Stanley’s conjecture can be formulated in el-ementary terms.

Denote by a(n) the number of tilings of thetriangular (staircase) region (n, n − 1, . . . , 1)with two type of tiles as in the figure. Hereonly parallel translations of tiles are allowed,the diagonal boundary must have no crossings,and no two blue curves are allowed to intersectmore than once. The first few numbers in thesequence27 are: 1, 2, 7, 41, 393, etc.

1

2

3

4

21 3 4

Stanley28 conjectured that there is a limit α :=limn→∞

1n2 log2 a(n). This Schubert entropy is

surprisingly difficult to establish due to the in-herently nonlocal “no double crossings” condi-tion. Assuming the conjecture, Stanley showedthat 1

4≤ α ≤ 1

2, but the conjecture remains

open.Note that the curves in the tilings define

a permutation σ ∈ Sn, for example we haveσ = (1, 4, 3, 2) in the figure. Thus we can write

a(n) =∑

σ∈Sn

a(σ).

Based on experimental evidence, Merzon andSmirnov29 made a stronger conjecture, that thelargest terms a(σ) in the summation appear atlayered permutations. In our paper with Ale-jandro and Greta, we showed that this stronger

conjecture implies that α ≈ 0.2932362 is givenby a solution of a certain differential equation.

This is “surprisingly precise” in a sense thatnormally one would not expect this level of pre-cision from a qualitative assumption, perhapssuggesting that the Merzon–Smirnov conjec-ture is false for large n. Others might disagreewith this interpretation. It would be interest-ing to figure out what is going on either way.

Mansour: One recent theme in your researchactivities is “contingency tables”. Could youelaborate a little bit more on it?

Pak: Contingency tables (CT) are rectan-gular matrices of non-negative integers withrows and column sums. They play a majorrole in statistics and appear all over the place– from combinatorial optimization to socialchoice. In combinatorics, they generalize so-called “magic squares” and can be recognizedas bipartite multigraphs with given degrees, oras the RHS of the RSK correspondence.

The amount of previous work on contin-gency tables is so enormous, it cannot be eas-ily summarized. Unfortunately, several majorproblem on CTs remains out of reach, such asestimating the number of contingency tablesin full generality. The hardness of countingthe #CTs is another major open problem. Insome recent papers with my collaborators, weimprove known bounds and establish some sur-prising new phenomena in some natural spe-cial cases. This is an ongoing research projectwhich is very far from completion.

Mansour: You have a nice blog. Some ofyour articles have turned out to be very in-teresting and attracted the attention of manyreaders. How do you select topics to writeabout? In one of your articles, a few years agoThe power of negative thinking, part I. Patternavoidance you explained why people should tryto disprove conjectures more often. Is this yourusual approach towards conjectures?

Pak: Thank you for the kind words, Toufik.The blog is largely dormant as I post only whenI have something interesting to say. I stayaway from politics and other day-to-day busi-ness, and write only about math and academic

26For example, see L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS, Providence, RI,2001.

27See https://oeis.org/A33192028R. P. Stanley, Some Schubert shenanigans, arXiv:1704.00851.29G. Merzon and E. Smirnov, Determinantal identities for flagged Schur and Schubert polynomials, Europ. J. Math. 2 (2016),

227–245.

ECA 1:3 (2021) Interview #S3I11 7

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matters. My posts tend to be on a longer side,especially when I tackle a controversial topicand want to be thorough. The reason I choosesuch topics is usually not because I want toconvince anyone of anything, but to representa minority opinion, to give a piece of mind topeople with similar points of view, that theyare not alone.

As for conjectures – yes, I often like disprov-ing more than proving. In fact, I expoundedon my reasoning in What if they are all wrong?recent blog post30. Trying to disprove a con-jecture is often unglamorous and thus largelyneglected. This represents an opportunity forsomeone not swayed by the crowds, who haswhat I call “negative thinking”.

Mansour: There are many interesting and im-portant formulas in combinatorics. Which ofthem are the top three for you?

Pak: Herb Wilf31 argued that good formu-las are very efficient algorithms. From thispoint of view, the matrix-tree theorem due toKirchhoff, the Kasteleyn–Temperley–Fischerformula for the number of perfect match-ings in planar graphs, and the Lindstrom–Gessel–Viennot lemma for the number of non-intersecting arrangements of lattice paths, arethe most general and the most powerful formu-las in the area.

Mansour: You have some interesting and niceresults related to permutation patterns. Couldyou list a couple of open problems from permu-tation patterns that look interesting to you andhope to see a solution?

Pak: Again, thank you. Indeed, the permuta-tion patterns is a fascinating and very activearea with a number of interesting open prob-lems. Personally, I am mostly interested in theasymptotic and computational questions.

Denote by Avn(π) and Avn(Π) the numberof permutations in Sn avoiding permutationπ ∈ Sk and subset of permutations Π ⊂ Sk,respectively. First, is it decidable whetherAvn(Π) ≥ Avn(Π′) for all n ≥ 1, for fixedΠ,Π′? If I had to guess, the answer is NObased on our paper with Scott Garrabrant32.

Second, can one compute Avn(π) in polyno-mial time for every fixed π ∈ Sk? The answerwould be especially interesting for the notori-ous π = (1, 3, 2, 4) pattern. But again, bet onthe negative answer for general π33.

Finally, define the growth constant c(Π) :=limn→∞

1n

log Avn(Π). Can one find the set ofpermutations Π ⊂ Sk such that c(Π) exists andis not algebraic?

Mansour: “The hook-length formula” is anice combinatorial formula and there has beena search for a “natural” and “satisfactory” bi-jection which explains the formula until youand your coauthors provided one in the paper“A direct bijective proof of the hook-lengthformula” published in 1997. Would you tellus about the novelty of this work? Did any-one give a “better” bijection in the last twentyyears?

Pak: I did this jointly with Sasha Stoyanon-vsky in 1992, when we both were undergradu-ates at Moscow University. Later, an expandedversion also included J.-C. Novelli who founda cleaner proof. The original idea was to usea “two-dimensional bubble sorting” to gener-ate standard Young tableaux of a given shapeuniformly at random. The second half of thebijection, i.e. the rule for changing hook num-bers, was necessary to justify that.

This was before the internet era when wedid not know about jeu-de-taquin, or anyYoung tableaux algorithm other than RSK.Since the original note was published in Rus-sian in a non-combinatorial journal, when Icame to America I emailed an English trans-lation to a few experts who were not aware ofthe paper. They recognized some familiar ele-ments of the proof and popularized it further.

I do not know about a “better bijection”,but there are quite a few new interesting proofsof the hook-length formula by many authors.In some way, just about all of them are “betterproofs” as our bijective proof when done care-fully is actually not all that simple. In fact,I also published a couple of such proofs withother applications34,35 in mind.

30https://wp.me/p211iQ-uT.31H. S. Wilf, What is an Answer?, Amer. Math. Monthly 89 (1982), 289–292.32We showed that Avn(Π) = Avn(Π′) mod 2 for all n ≥ 1 is undecidable, see S. Garrabrant and I. Pak, Permutation patterns

are hard to count, in Proc. 27th SODA, ACM, New York, 2016, 923–936.33In the same paper with Garrabrant we showed that computing Avn(Π) mod 2 is ⊕P-hard.34I. Pak, Hook length formula and geometric combinatorics, Sem. Lothar. Combin. 46 (2001), Article B46f35I. Ciocan-Fontanine, M. Konvalinka and I. Pak, The weighted hook length formula, J. Combin. Theory, Ser. A 118 (2011),

1703–1717.

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Mansour: Your book Lectures on Discreteand Polyhedral Geometry has gone throughseveral editions and it is almost completed,at least from the eyes of us as readers. Doyou have any plans to write a book that ex-plores various aspects of Probability and Com-binatorics on Groups?Pak: No, not in the next few years. I do have alot of material on the subject based on severalcourses I taught over the years36.Mansour: In your works, you have exten-sively used combinatorial reasoning to addressimportant problems. How do enumerativetechniques engage in your research?Pak: In some sense, all combinatorics can bedivided into three overlapping parts depend-ing on the type of answer they are seeking:equalities, inequalities, or structures. I thinkof “enumerative techniques” as proving equal-ities, such as an explicit product formula, agenerating function, a bijection, etc. These areoften judged based on their beauty rather thanapplications, and the best examples are closerto art than to science.

Personally, I am very proud of some of mybijective “artworks”, for example the abovementioned hook-length formula bijection, theso-called Pak–Stanley37 labeling, the bijectiveproof of the MacMahon’s Master Theorem38,or the combinatorial proof of the Rogers–Ramanujan identities39. While there is still anoccasional value in such bijections, in the 21stcentury their time is probably over. This ispartially due to many successes of the othertwo camps, whose applications to adjacentfields are making a great impact.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”?Pak: There is nothing especially mysterioushere. For example, here is the story of theMacMahon Master Theorem (MMT) paper Imentioned earlier. Once, I invited Doron Zeil-berger to give a talk at the MIT Combina-torics Seminar which I organized at the time.He spoke about the quantum MMT which heproved recently40. I was quite excited but alsosomewhat unsettled.

The MMT is an amazing result, one of theearly combinatorial gems, with many classicalconsequences such as the celebrated Dixon’sidentity. By every indication, the new gen-eralization also looked amazing. Although itwas motivated by knot theory applications, itwas clearly of algebraic nature, creating morequestions than answers. Furthermore, as it of-ten happens to the pioneer papers, the origi-nal proof was rather technical, employing someheavy q-calculus and leaving some room forsimplification.

I was immediately convinced that once thetrue nature of the quantum MMT is under-stood, both a “proof from the book” and somefurther generalizations will come along. I had ahunch that quasideterminants is the right lan-guage for the problem. I learned them fromIsrael Gelfand and Vladimir Retakh41 some 15years earlier, back when I was an undergrad-uate. With Matjaz Konvalinka, my graduatestudent at the time, we attacked the problemand found both the generalization and the kindof proof we wanted.

As it happened, we learned we have a strongcompetition. Dominique Foata and Guo-NiuHan42 were working on their own combinato-rial proofs independently from us. They endedup writing several beautiful papers on the sub-ject, extending the classical Cartier–Foata43

approach to MMT to the quantum setting. In36For example, see my latest lecture notes here: https://tinyurl.com/mpbmpjb5.37See Section 5 in R. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Nat. Acad. Sci. USA 93 (1996),

2620–2625.38See Section 2 in M. Konvalinka and I. Pak, Non-commutative extensions of the MacMahon Master Theorem, Adv. Math. 216

(2007), 29–61.39C. Boulet and I. Pak, A combinatorial proof of the Rogers–Ramanujan and Schur identities, J. Combin. Theory, Ser. A 113

(2006), 1019–1030.40S. Garoufalidis, T. Q. Le and D. Zeilberger, The quantum MacMahon master theorem, Proc. Natl. Acad. Sci. USA 103 (2006),

13928–13931.41I. M. Gelfand and V. S. Retakh, A theory of non-commutative determinants and characteristic functions of graphs, Funct.

Anal. Appl. 26:4 (1992), 1–20.42D. Foata and G. N. Han, A new proof of the Garoufalidis–Le–Zeilberger quantum MacMahon master theorem, J. Algebra 307

(2007), 424–431, and two followup papers.43P. Cartier and D. Foata, Problemes combinatoires de commutation et rearrangements (in French), Lecture Notes in Math.

No. 85, Springer, Berlin, 1969.44P. H. Hai and M. Lorenz, Koszul algebras and the quantum MacMahon master theorem, Bull. LMS 39 (2007), 667–676.

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a different direction, Phung Ho Hai and Mar-tin Lorenz44 were using the Koszul duality togive a purely algebraic proof of the quantumMMT.

We needed another idea to stand out, andagain I knew something different from my stud-ies of representation theory with my under-graduate advisor Alexandre Kirillov. Thatwas a multiparameter (qij) generalization byYuri Manin which he defined in the contextof quantum groups and solutions the Yang–Baxter equation45. Miraculously, our approachwith Matjaz extended to this setting. Spookedby the competition, we wrapped up within sev-eral months, I think.

Mansour: Recently, we have seen some ‘un-usual and interesting events’ occurring in themath community such as resigning from theeditorial boards of some well-established jour-nals and founding similar journals. Two exam-ples: a group of former editors of the Journalof Algebraic Combinatorics founded AlgebraicCombinatorics, and recently a group of edi-tors of the Journal of Combinatorial Theory,Series A resigned and founded CombinatorialTheory46. What is your opinion about theserecent developments? Do you think this trendshould continue with other journals as well?

Pak: Personally, I strongly support thesechanges. I think it is very important to moveaway from for-profit publishing. I am veryhappy that mathematicians in my field areleading the movement. I am also at least alittle bit sad. JCTA was by far my favoritecombinatorics journal, so I take no joy in itsdemise. This is just something that had tohappen. It is a bittersweet victory.

More broadly, I hope to wake up one dayin a world where all mathematics books andjournals are electronic and free to download byanyone in the world. I learned of this dreamfrom my former Bell Labs mentor AndrewOdlyzko back in 1995 and became a firm be-liever. It is been over 25 years since Odlyzko47

published his predictions, but we are still here,celebrating the move by just two journals. Per-haps, the fact that the expanded version ofthat paper was published by an Elsevier jour-

nal, which continues to do good printing busi-ness, was both ominous and at least a littleironic.

Mansour: Suppose the academic world de-cides to eliminate all journals so that therewould be no paper submissions, referee reports,and all other-sometimes unfair and lengthy-publishing processes. Researchers just posttheir papers to arXiv, and other professionalmathematicians read and write comments withtheir names about the papers. A very trans-parent evaluation procedure. Do you thinkthat it would have been a better academicworld?

Pak: That would be terrible, in my opinion.The peer review has obvious flaws, but it is avery good system that we have learned to relyupon. It is largely a victim of its own suc-cess. The universities and government agen-cies learned to use publication records for hir-ing, promotion, and research awards. As theacademia rapidly expanded, many new jour-nals emerged with uncertain standards in theeye of these institutions, so the competition forpublishing in top journals became fierce. Thisled to various biases, hurt egos, and some re-luctance to participate in the process. But itis not a good reason to destroy the system.Rather, this suggests the need to decouple itfrom institutional use.

Unfortunately, like many other people inthe area, I have my own share of unpublishedpapers, each with its own story. For one reasonor another life intervened and they remainedin that state for years. Since most of them aredownloadable from my website, at least peoplecan see what was done there, but I regret notpublishing every one of them.

Mansour: Is there a specific problem youhave been working on for many years? Whatprogress have you made?

Pak: Yes, sure. I will mention only one, thecombinatorial interpretation of the Kroneckercoefficients problem that I discussed earlier.I have been thinking about this problem48

for about ten years, ever since Greta Panovacame to UCLA as a postdoc to work with me.Formally, the problem asks whether comput-

45Yu. I. Manin, Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier 37 (1987), 191–205.46 https://osc.universityofcalifornia.edu/2020/12/combinatorial-theory-launches/.47A. M. Odlyzko, Tragic loss or good riddance? The impending demise of traditional scholarly journals, Notices of the AMS 42

(1995), 49–53; see also an expanded version in Int. J. of Human–Computer Studies 42 (1995), 71–122.48I. Pak and G. Panova, On the complexity of computing Kronecker coefficients, Computational Complexity 26 (2017), 1–36.

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ing g(λ, µ, ν) is in #P. To no one’s surprise,I firmly believe that the answer is negative.Most people in the area strongly disagree, andsome are actively working in the positive di-rection. Greta and I have been making someunsteady progress with no end in sight, but I

am confident the problem will be resolved inmy lifetime.Mansour: Professor Igor Pak, I would like tothank you for this very interesting interviewon behalf of the journal Enumerative Combi-natorics and Applications.

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