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How I chose Enumerative Combinatorics

Apologies for not writing anything for awhile. After Feb 24, the math part of the “life and math” slogan lost a bit of relevance, while the actual events were stupefying to the point when I had nothing to say about the life part. Now that the shock subsided, let me break the silence by telling an old personal story which is neither relevant to anything happening right now nor a lesson to anyone. Sometimes a story is just a story…

My field

As the readers of this blog know, I am a Combinatorialist. Not a “proud one”. Just “a combinatorialist”. To paraphrase a military slogan “there are many fields like this one, but this one is mine”. While I’ve been defending my field for years, writing about its struggles, and often defining it, it’s not because this field is more important than others. Rather, because it’s so frequently misunderstood.

In fact, I have worked in other (mostly adjacent) fields, but that was usually because I was curious. Curious what’s going on in other areas, curious if they had tools to help me with my problems. Curious if they had problems that could use my tools. I would go to seminars in other fields, read papers, travel to conferences, make friends. Occasionally this strategy paid off and I would publish something in another field. Much more often nothing ever came out of that. It was fun regardless.

Anyway, I wanted to work in combinatorics for as long as I can remember, since I was about 15 or so. There is something inherently discrete about the way I see the world, so much that having additional structure is just obstructing the view. Here is how Gian-Carlo Rota famously put it:

Combinatorics is an honest subject. […] You either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. [Los Alamos Science, 1985]

I agree. Also, I really like to count. When prompted, I always say “I work in Combinatorics” even if sometimes I really don’t. But in truth, the field is much too large and not unified, so when asked to be more specific (this rarely happens) I say “Enumerative Combinatorics“. What follows is a short story of how I made the choice.

Family vacation

When I was about 18, Andrey Zelevinsky (ז״ל) introduced me and Alex Postnikov to Israel Gelfand and asked what should we be reading if we want to do combinatorics. Unlike most leading mathematicians in Russia, Gelfand had a surprisingly positive view on the subject (see e.g. his quotes here). He suggested we both read Macdonald’s book, which was translated into Russian by Zelevinsky himself just a few years earlier. The book was extremely informative but dry as a fig and left little room for creativity. I read a large chunk of it and concluded that if this is what modern combinatorics looks like, I want to have nothing to do with it. Alex had a very different impression, I think.

Next year, my extended family decided to have a vacation on a Russian “river cruise”. I remember a small passenger boat which left Moscow river terminal, navigated a succession of small rivers until it reached Volga. From there, the boat had a smooth gliding all the way to the Caspian Sea. The vacation was about three weeks of a hot Summer torture with the only relief served by mouth-watering fresh watermelons.

What made it worse, there was absolutely nothing to see. Much of the way Volga is enormously wide, sometimes as wide as the English channel. Most of the time you couldn’t even see the river banks. The cities distinguished themselves only by an assortment of Lenin statues, but were unremarkable otherwise. Volgograd was an exception with its very impressive tallest statue in Europe, roughly as tall as the Statue of Liberty when combined with its pedestal. Impressive for sure, but not worth the trip. Long story short, the whole cruise vacation was dreadfully boring.

One good book can make a difference

While most of my relatives occupied themselves by reading crime novels or playing cards, I was reading a math book, the only book I brought with me. This was Stanley’s Enumerative Combinatorics (vol. 1) whose Russian translation came out just a few months earlier. So I read it cover-to-cover, but doing only the easiest exercises just to make sure I understand what’s going on. That book changed everything…

Midway through, when I was reading about linear extensions of posets in Ch. 3 with their obvious connections to standard Young tableaux and the hook-length formula (which I already knew by then), I had an idea. From Macdonald’s book, I remembered the q-analogue of #SYT via the “charge“, a statistics introduced by Lascoux and Schützenberger to give a combinatorial interpretation of Kostka polynomials, and which works even for skew Young diagram shapes. I figured that skew shapes are generic enough, and there should be a generalization of the charge to all posets. After several long days filled with some tedious calculations by hand, I came up with both the statement and the proof of the q-analogue of the number of linear extensions.

I wrote the proof neatly in my notebook with a clear intent to publish my “remarkable discovery”, and continued reading. In Ch. 4, all of a sudden, I read the “P-partition theory” that I just invented by myself. It came with various applications and connections to other problems, and was presented so well, much nicer than I would have!

After some extreme disappointment, I learned from the notes that the P-partition theory was a large portion of Stanley’s own Ph.D. thesis, which he wrote before I was born. For a few hours, I did nothing but meditate, staring at the vast water surrounding me and ignoring my relatives who couldn’t care less what I was doing anyway. I was trying to think if there is a lesson in this fiasco.

It pays to be positive and self-assure, I suppose, in a way that only a teenager can be. That day I concluded that I am clearly doing something right, definitely smarter than everyone else even if born a little too late. More importantly, I figured that Enumerative Combinatorics done “Stanley-style” is really the right area for me…

Epilogue

I stopped thinking that I am smarter than everyone else within weeks, as soon as I learned more math. I no longer believe I was born too late. I did end up doing a lot of Enumerative Combinatorics. Much later I became Richard Stanley’s postdoc for a short time and a colleague at MIT for a long time. Even now, I continue writing papers on the numbers of linear extensions and standard Young tableaux. Occasionally, I also discuss their q-analogues (like in my most recent paper). I still care and it’s still the right area for me…

Some years later I realized that historically, the “charge” and Stanley’s q-statistics were not independent in a sense that both are generalizations of the major index by Percy MacMahon. In his revision of vol. 1, Stanley moved the P-partition theory up to Ch. 3, where it belongs IMO. In 2001, he received the Steele’s Prize for Mathematical Exposition for the book that changed everything…

How to fight the university bureaucracy and survive

The enormity of the university administration can instill fear. How can you possibly fight such a machine? Even if an injustice happened to you, you are just one person with no power, right? Well, I think you can. Whether you succeed in your fight is another matter. But at least you can try… In this post I will try to give you some advise on how to do this.

Note: Initially I wanted to make this blog post light and fun, but I couldn’t think of a single joke. Somehow, the subject doesn’t inspire… So read this only if it’s relevant to you. Wait for future blog posts otherwise…

Warning: Much of what I say is relevant to big state universities in the US. Some of what I say may also be relevant to other countries and university systems, I wouldn’t know.

Basics

Who am I to write about this? It is reasonable to ask if any of this is based on my personal experience of fighting university bureaucracies. The answer is yes, but I am not willing to make any public disclosures to protect privacy of all parties involved. Let me just say that over the past 20 years I had several relatively quiet and fairly minor fights with university bureaucracies some of which I won rather quickly by being right. Once, I bullied my way into victory despite being in the wrong (as I later learned), and once I won over a difficult (non-personal) political issue by being cunning and playing a really long game that took almost 3 years. I didn’t lose any, but I did refrain from fighting several times. By contrast, when I tried to fight the federal government a couple of times (on academic matters), I lost quickly and decisively. They are just too powerful….

Should you fight? Maybe. But probably not. Say, you complained to the administration about what you perceive to be an injustice to you or to someone else. Your complaint was denied. This is when you need to decide if you want to start a fight. If you do, you will spend a lot of effort and (on average) probably lose. The administrations are powerful and know what they are doing. You probably don’t, otherwise you won’t be reading this. This blog post might help you occasionally, but wouldn’t change the big picture.

Can you fight? Yes, you can. You can win by being right and convince bureaucrats to see it this way. You can win by being persistent when others give up. You can also win by being smart. Big systems have weaknesses you can exploit, see below. Use them.

Is there a downside to winning a fight? Absolutely. In the process you might lose some friends, raise some suspicions from colleagues, and invite retribution. On a positive side, big systems have very little institutional memory — your win and the resulting embarrassment to administration will be forgotten soon enough.

Is there an upside to losing a fight? Actually, yes. You might gain resect of some colleagues as someone willing to fight. In fact, people tend to want being friends/friendly with such people out of self-preservation. And if your cause is righteous, this might help your reputation in and beyond the department.

Why did I fight? Because I just couldn’t go on without a fight. The injustice, as I perceived it, was eating me alive and I had a hunch there is a nonzero chance I would win. There were some cases when I figured the chances are zero, and I don’t need the grief. There were cases when the issue was much too minor to waste my energy. I don’t regret those decision, but having grown up in this unsavory part of Moscow, I was conditioned to stand up for myself.

Is there a cost of not fighting? Yes, and it goes beyond the obvious. First, fighting bureaucracy is a skill, and every skill takes practice. I remember when tried to rent an apartment in Cambridge, MA — some real estate agents would immediately ask if I go to Harvard Law School. Apparently it’s a common practice for law students to sue their landlords, an “extra credit” homework exercise. Most of these lawsuits would quickly fail, but the legal proceeding were costly to the owners.

Second, there is a society cost. If you feel confident that your case is strong, you winning might set a precedent which could benefit many others. I wrote on this blog once how I dropped (or never really started) a fight against the NSF, even though they clearly denied me the NSF Graduate Fellowship in a discriminatory manner, or at least that’s what I continue to believe. Not fighting was the right thing to do for me personally (I would have lost, 100%), but my case was strong and the fight itself might have raised some awareness to the issue. It took the NSF almost 25 years to figure out that it’s time to drop the GREs discriminatory requirement.

Axioms

1. If it’s not in writing it never happened.
2. Everyone has a boss.
3. Bureaucrats care about themselves first and foremost. Then about people in their research area, department and university, in that order. Then undergraduates. Then graduate students. You are the last person they care about.

How to proceed

Know your adversary. Remember — you are not fighting a mafia, a corrupt regime or the whole society. Don’t get angry, fearful or paranoid. Your adversary is a group of good people who are doing their jobs as well as they can. They are not infallible, but probably pretty smart and very capable when it comes to bureaucracy, so from game theory point of view you may as well assume they are perfect. When they are not, you will notice that — that’s the weakness you can exploit, but don’t expect that to happen.

Know your rights. This might seem obvious, but you would be surprised to know how many academics are not aware they have rights in a university system. In fact, it’s a feature of every large bureaucracy — it produces a lot of well meaning rules. For example, Wikipedia is a large project which survived for 20 years, so unsurprisingly it has a large set of policies enforced by an army of admins. The same is probably true about your university and your department. Search on the web for the faculty handbook, university and department bylaws, etc. If you can’t find the anywhere, email the assistant to the Department Chair and ask for one.

Go through the motions. Say, you think you were slighted. For example, your salary was not increased (enough), you didn’t get a promotion, you got too many committee duties assigned, your sabbatical was not approved, etc. Whatever it is, you are upset, I get it. Your first step is not to complain but go through the motions, and email inquiries. Email the head of the department, chair of the executive committee, your faculty dean, etc., whoever is the decision maker. Calmly ask to explain this decision. Sometimes, this was an oversight and it’s corrected with a quick apology and “thanks for bringing this up”. You win, case closed. Also, sometimes you either get a convincing explanation with which you might agree — say, the university is on salary freeze so nobody got a salary increase, see some link. Again, case closed.

But in other cases you either receive an argument with which you disagree (say, “the decision was made based on your performance in the previous year”), a non-answer (say, “I am on sabbatical” or “I will not be discussing personal matters by email”), or no answer at all. These are the cases that you need to know how to handle and all such cases are a little different. I will try to cover as much territory as possible, but surely will miss some cases.

Ask for advice. This is especially important if you are a junior mathematician and feel a little overwhelmed. Find a former department chair, perhaps professor emeritus, and have an quiet chat. Old-timers know the history of the department, who are the university administrators, what are the rules, what happened to previous complaints, what would fly and what wouldn’t, etc. They might also suggest who else you should talk to that would be knowledgeable and help dealing with an issue. With friends like these, you are in a good shape.

Scenarios

Come by for a chat. This is a standard move by a capable bureaucrat. They invite you for a quick discussion, maybe sincerely apologize for “what happened” or “if you are upset” and promise something which they may or may not intend to keep. You are supposed to leave grateful that “you are heard” and nothing is really lost from admin’s point of view. You lost.

There is only one way to counter this move. Agree to a meeting — play nice and you might learn something. Don’t record in secret — it’s against the law in most states. Don’t ask if you can record the conversation — even if the bureaucrat agrees you will hear nothing but platitudes then (like “we in our university strive to make sure everyone is happy and successful, and it is my personal goal to ensure everyone is treated fairly and with respect”). This defeats the purpose of the meeting moving you back to square one.

At the meeting do not agree with anything, never say yes or no to anything. Not even to the routine “No hard feelings?” Just nod, take careful notes, say “thank you so much for taking time to have this meeting” and “This information is very useful, I will need to think it over”. Do not sign anything. If offered a document to sign, take it with you. If implicitly threatened, as in “Right now I can offer you this for you, but once you leave this office I can’t promise… ” (this is rare but does happen occasionally), ignore the threat. Just keep repeating “Thank you so much for informing me of my options, I will need to think it over.” Go home, think it over and talk to somebody.

Get it all in writing. Within a few hours after the meeting, email to the bureaucrat an email with your notes. Start this way: “Dear X, this is to follow up on the meeting we had on [date] regarding the [issue]. I am writing this to ensure there is no misunderstanding on my part. At the meeting you [offered/suggested/claimed/threatened] …. Please let me know if this is correct and what are the details of …”

A capable bureaucrat will recognize the move and will never go on record with anything unbecoming. They will accept the out you offered and claim that you indeed misunderstood. Don’t argue with that — you have them where you want it. In lieu of the misunderstanding they will need to give a real answer to your grievance (otherwise what was the point of the meeting?) Sometimes a bureaucrat will still resort to platitudes, but now that they are in writing, that trick is harder to pull off, and it leads us to a completely different scenario (see below).

Accept the win. You might receive something like this: “We sincerely apologize for [mistake]. While nothing can be done about [past decision], we intend to [compensate or rectify] by doing…” If this is a clear unambiguous promise in writing, you might want to accept it. If not, follow up about details. Do not pursue this any further and don’t make it public. You got what you wanted, it’s over.

Accept the defeat. You might learn that administration acted by the book, exactly the way the rules/bylaws prescribe, and you were not intentionally discriminated in any way. Remain calm. Thank the bureaucrat for the “clarification”. It’s over.

Don’t accept a draw. Sometimes a capable bureaucrat might reply to the whole list on CC and write “We are very sorry [your grievance] happened. This is extremely atypical and related to [your unusual circumstances]. While this is normally not appropriate, we are happy to make an exception in your case and [compensate you].” Translation: “it’s your own fault, you brought it on yourself, we admit no wrongdoing, but we are being very nice and will make you happy even though we really don’t have to do anything, not at all.” While other bureaucrats will recognize the move and that there is an implicit admission of fault, they will stay quiet — it’s not their fight.

Now, there is only one way to counter this, as far as I know. If you don’t follow up it’s an implicit admission of “own fault” which you don’t want as the same issue might arise again in the future. If you start explaining that it’s really bureaucrat’s fault you seem vindictive (as in “you already got what you wanted, why do you keep pushing this?”), and other bureaucrats will close ranks leaving you worse off. The only way out is to pretend to be just as illogical as the bureaucrat pretends to be. Reply to the whole CC list something like “Thank you so much for your apology and understanding of my [issue]. I am very grateful this is resolved to everyone’s satisfaction. I gratefully accept your sincere apology and your assurances this will not happen again to me nor anyone else at the department.”

A capable bureaucrat will recognize they are fighting fire with fire. In your email you sound naïve and sincere — how do you fight that? What are they going to do — reply “actually, I didn’t issue any apology as this was not my fault”? Now that seem overly defensive. And they would have to reply to the whole CC list again, which is not what they want. They are aware that everyone else knows they screwed up, so reminding everyone with a new email is not in their interest. And there is a decent chance you might reply to the whole CC list again with all that sugarcoated unpleasantness. Most likely, you won’t hear from them again, or just a personal (non-CC’d) email which you can ignore regardless of the content.

Shifting blame or responsibility. That’s another trick bureaucrats employ very successfully. You might get a reply from a bureaucrat X to the effect saying “don’t ask me, these are rules made by [people upstairs]” or “As far as I know, person Y is responsible for this all”. This is great news for you — a tacit validation of your cause and an example of a bureaucrat putting their own well-being ahead of the institution. Remember, your fight is not with X, but with the administration. Immediately forward both your grievance and the reply to Y, or to X’s boss if no names were offered, and definitely CC X “to keep your in the loop of further developments on this issue”. That immediately pushes bureaucracy into overdrive as it starts playing musical chairs in the game “whose fault is that and what can be done”.

Like with musical chairs, you might have to repeat the procedure a few times, but chances are someone will eventually accept responsibility just to stop this embarrassment from going circles. By then, there will be so many people on the CC chain, your issue will be addressed appropriately.

Help them help you. Sometimes a complaint puts the bureaucrat into a stalemate. They want to admit that injustice happened to you, but numerous university rules forbid them from acting to redress the situation. In order to violate these rules, they would have to take the case upstairs, which brings its own complications to everyone involved. Essentially you need to throw them a lifeline by suggesting some creative solution to the problem.

Say, you can write “while I realize the deadline for approval of my half-year sabbatical has passed, perhaps the department can buyout one course from my Fall schedule and postpone teaching the other until Spring.” This moves the discussion from the “apology” subject to “what can be done”, a much easier bureaucratic terrain. While the bureaucrat may not agree with your proposed solution, your willingness to deal without an apology will earn you some points and perhaps lead to a resolution favorable to all parties.

Now, don’t be constrained in creativity of when thinking up such a face saving resolution. It is a common misconception that university administrations are very slow and rigid. This is always correct “on average”, and holds for all large administrative systems where responsibility is distributed across many departments and individuals. In reality, when they want to, such large systems can turn on a dime by quickly utilizing its numerous resources (human, financial, legal, etc.) I’ve seen it in action, it’s jaw-dropping, and it takes just one high ranking person to take up the issue and make it a cause.

Making it public. You shouldn’t do that unless you already lost but keep holding a grudge (and have tenure to protect you). Even then, you probably shouldn’t do it unless you are really good at PR. Just about every time you make grievances public you lose some social points with people who will hold it against you, claim you brought it on yourself, etc. In the world of social media your voice will be drowned and your case will be either ignored or take life of its own, with facts distorted to fit a particular narrative. The administration will close ranks and refuse to comment. You might be worse off than when you started.

The only example I can give is my own combative blog post which remains by far my most widely read post. Everyone just loves watching a train wreck… Many people asked why I wrote it, since it made me a persona non grata in the whole area of mathematics. I don’t have a good answer. In fact, that area may have lost some social capital as a result of my blog post, but haven’t changed at all. Some people apologized, that’s all. There is really nothing I can do and they know it. The truth is — my upbringing was acting up again, and I just couldn’t let it go without saying “Don’t F*** with Igor Pak”.

But you can very indirectly threaten to make it public. Don’t do it unless you are at an endgame dealing with a high ranking administrator and things are not looking good for you. Low level university bureaucrats are not really afraid for their jobs. For example, head of the department might not even want to occupy the position, and is fully protected by tenure anyway. But deans, provosts, etc. are often fully vested into their positions which come with substantial salary hike. If you have a sympathetic case, they wouldn’t want to be featured in a college newspaper as denying you some benefits, regardless of the merit. They wouldn’t be bullied into submission either, so some finesse is needed.

In this case I recommend you find an email of some student editor of a local university newspaper. In your reply to the high ranking administrator write something like “Yes, I understand the university position in regard to this issue. However, perhaps [creative solution]”. Then quietly insert the editor’s email into CC. In the reply, the administrator will delete the email from CC “for privacy reasons”, but will google to find out who is being CC’ed. Unable to gauge the extend of newspaper’s interest in the story, the administrator might chose to hedge and help you by throwing money at you or mollifying you in some creative way you proposed. Win–win.

Final word

I am confident there will be people on all sides who disagree collectively with just about every sentence I wrote. Remember — this blog post is a not a recommendation to do anything. It’s just my personal point of view on these delicate matters which tend to go undiscussed, leaving many postdocs and junior faculty facing alone their grievances. If you know a good guide on how to deal with these issues (beyond Rota’s advice), please post a link in the comments. Good luck everyone! Hope you will never have to deal with any of that!

In this post I will try to evaluate Gian-Carlo Rota‘s predictions on the future of Combinatorics that he made in this 1969 article.  He did surprisingly well, but I am a tough grader and possibly biased about some of the predictions.  Judge for yourself…

It’s tough to make predictions, especially about the future

It is a truth universally acknowledged that humans are very interested in predicting the future. They do this incessantly, compiling the lists of the best and the worst, and in general can’t get enough of them. People tend to forget wrong predictions (unless they are outrageously wrong).  This allows a person to make the same improbable predictions over and over and over and over again, making news every time.  There are even professional prognosticators who make a living writing about the future of life and technology.  Sometimes these predictions are rather interesting (see here and there), but even the best ones are more often wrong than right…

Although rarely done, analyzing past predictions is a useful exercise, for example as a clue to the truthiness of the modern day oracles.  Of course, one can argue that many of the political or technology predictions linked above are either random or self-serving, and thus not worth careful investigation. On the other hand, as we will see below, Rota’s predictions are remarkably earnest and sometimes even brave.  And the fact that they were made so long ago makes them uniquely attractive, practically begging to be studied.

Now, it has been 45 years since Rota’s article, basically an eternity in the life span of Combinatorics. There, Rota describes Combinatorics as “the least developed branches of mathematics“. A quick review of the last few quotes on this list I assembled shows how much things have changed. Basically, the area moved from an ad hoc collection of problems to a 360-degree panorama of rapidly growing subareas, each of which with its own deep theoretical results, classical benchmarks, advanced tools and exciting open problems. This makes grading rather difficult, as it suggests that even random old predictions are likely to be true – just about anything people worked on back in the 1960 has been advanced by now. Thus, before turning to Rota, let’s agree on the grading scale.

To give you the feel for my curve, I will use the celebrated example of the 1899-1901 postcards in the En L’An 2000 French series, which range from insightful to utter nonsense (click on the titles to view the postcards, all available from Wikimedia).

Electric train.  Absolutely.  These were introduced only in 1940s and have been further developed in France among other countries.  Note the aerodynamic shape of the engine.  Grade:  A.

Correspondance cinema.  Both the (silent) cinema and phonograph were invented by 1900; the sound came to movie theaters only in 1927.  So the invention here is of a home theater for movies with sound.  Great prediction although not overly ambitious. Grade:  A-.

Military cyclists.  While bicycle infantry was already introduced in France by 1900, military use of motorcycles came much later.  The idea is natural, but some designs of bikes from WW2 are remarkably similar.  Some points are lost due to the lack of widespread popularity in 2000.  Grade: B+.

Electric scrubbing.  This is an electric appliance for floor cleaning.  Well, they do exist, sort of, obviously based on different principles.  In part due to the modern day popularity, this is solid prediction anyway.  Grade:  B.

Auto-rollers.  Roller skates have been invented in 18th century and by 1900 became popular.  So no credit for the design, but extra credit for believing in the future of the mean of transportation now dominated by rollerblades. Thus the author’s invention is in the category of “motorized personal footwear”. In that case the corresponding modern invention is of the electric skateboard which has only recently become available, post-2000 and yet to become very popular. Grade: B-.

Barber.  The author imagines a barber operating machinery which shaves and cuts customer’s hair.   The design is so ridiculous (and awfully dangerous), it’s a good thing this never came about.  There are however electric shavers and hair cutters which are designed very differently.  Grade:  C.

•  Air cup.  The Wright brothers’ planes had similar designs, so no credit again.  The author assumes that personal air travel will become commonplace, and at low speeds and heights.  This is almost completely false.  However, unfortunately, and hopefully only very occasionally, some pilots do enjoy one for the road.  Grade:  D.

Race in Pacific.  The author imagines that the public spectacle of horse racing will move underwater and become some kind of fish racing.  Ridiculous.  Also a complete failure to envision modern popularity of auto racing which already began in Paris in 1887.  Grade:  F.

Rota’s predictions on combinatorial problems:

In his paper, Rota writes:

Fortunately, most combinatorial problems can be stated in everyday language. To give an idea of the present state of the field, we have selected a few of the many problems that are now being actively worked upon.

We take each of these “problems” as a kind of predictions of where the field is going.  Here are my (biased and possibly uninformed) grades for each problem he mentions.

1)  The Ising Problem.  I think it is fair to say that since 1969 combinatorics made no contribution in this direction.  While physicists and probabilists continue studying this problem, there is no exact solution in dimension 3 and higher.  Grade: F.

2)  Percolation Theory.  The study of percolation completely exploded since 1969 and is now a subject of numerous articles in both probability, statistical physics and combinatorics, as well as several research monographs.  One connection is given by an observation that p-percolation on a complete graph Kn gives the Erdős–Rényi random graph model. Even I accidentally wrote a few papers on the subject some years ago (see one, two and three).  Grade: A.

3)  The Number of Necklaces, and Polya’s Problem.  Taken literally, the necklaces do come up in combinatorics of words and free Lie algebra, but this context was mentioned by Rota already. As far as I can tell, there are various natural (and interesting) generalizations of necklaces, but none surprising.  Of course, if the cyclic/dihedral group action here is replaced by other actions, e.g. the symmetric group, then modern developments are abundant.  But I think it’s a reach too far, since Rota knew the works of Young, MacMahon, Schur and others but does not mention any of it.  Similarly, Polya’s theorem used to be included in all major combinatorics textbooks (and is included now, occasionally), but is rarely taught these days.  Simply put, the g.f. implications haven’t proved useful.  Grade: D.

4)  Self-avoiding Walk. Despite strong interest, until recently there were very few results in the two-dimensional case (some remarkable results were obtained in higher dimensions). While the recent breakthrough results (see here and there) do use some interesting combinatorics, the authors’ motivation comes from probability. Combinatorialists did of course contribute to the study of somewhat related questions on enumeration of various classes of polyomino (which can be viewed as self-avoiding cycles in the grid, see e.g. here).  Grade: C.

5)  The Traveling Salesman Problem. This is a fundamental problem in optimization theory, connected to the study of Hamiltonian cycles in Graph Theory and numerous other areas. It is also one of the earliest NP-hard problems still playing a benchmark role in Theoretical Computer Science. No quick of summary of the progress in the past 45 years would give it justice. Note that Rota’s paper was written before the notions of NP-completeness. In this light, his emphasis on algorithmic complexity and allusions to Computability Theory (e.g. unsolvable problems) are most prescient.  So are his briefly mentioned connections to topology which are currently a popular topic.  Well done!  Grade: A+.

6)  The Coloring Problem.  This was a popular topic way before Rota’s article (inspired by the Four Color Theorem, the chromatic polynomial, etc.), and continues to be even more so, with truly remarkable advances in multiple directions.  Note Rota’s mentioning of matroids which may seem extraneous here at first, but in fact absolutely relevant indeed (in part due to Rota’s then-ongoing effort).  Very good but unsurprising prediction.  Grade: A-.

7)  The Pigeonhole Principle and Ramsey’s Theorem. The Extremal Graph Theory was about to explode in many directions, with the the Erdős-Stone-Simonovits theorem proved just a few years earlier and the Szemerédi regularity lemma a few years later.  Still, Rota never mentions Paul Erdős and his collaborators, nor any of these results, nor potential directions.  What a missed opportunity!  Grade: B+.

Rota’s predictions on combinatorial areas:

In the concluding section “The Coming Explosion”, Rota sets this up as follows:

Before concluding this brief survey, we shall list the main subjects in which current work in combinatorial theory is being done.

Here is a list and more of my comments.

1)  Enumerative Analysis.  Sure.  But this was an easy prediction to make given the ongoing effort by Carlitz, Polya, Riordan, Rota himself and many other peope.  Grade: A-.

2)  Finite Geometries and Block Designs.  The subject was already popular and it did continue to develop but perhaps at a different pace and directions than Rota anticipated (Hadamard matrices, tools from Number Theory).  In fact, a lot of later work was connection with with Group Theory (including some applications of CFSG which was an ongoing project) and in Coding Theory (as Rota predicted).  Grade: B-.

3)  Applications to Logic.  Rota gives a one-sentence desctiption:

The development of decision theory has forced logicians to make wide use of combinatorial methods.

There are various important connections between Logic and Combinatorics, for example in Descriptive Set Theory (see e.g. here or more recent work by my future UCLA colleague there).  Note however, that Infinitary Combinatorics was already under development, after the Erdős-Rado theorem (1956).  Another very interesting and more recent connection is to Model Theory (see e.g. here).  But the best interpretation here I can think of here are the numerous applications to Game Theory, which already existed (Nash’s equilibrium theorem is from 1950) and was under rapid development.  Either way, Rota was too vague in this case to be given much credit.  Grade: C.

4)  Statistical Mechanics.   He mentions the Ising model again and insists on “close connections with number theory”.  One can argue this all to be too vague or misdirected, but the area does indeed explode in part in the directions of problems Rota mentions earlier. So I am inclined to give him benefit of the doubt on this one. Grade: A-.