The problem with combinatorics textbooks

Every now and then I think about writing a graduate textbook in Combinatorics, based on some topics courses I have taught. I scan my extensive lecture notes, think about how much time it would take, and whether there is even a demand for this kind of effort. Five minutes later I would always remember that YOLO, deeply exhale and won’t think about it for a while.

What’s wrong with Combinatorics?

To illustrate the difficulty, let me begin with two quotes which contradict each other in the most illuminating way. First, from the Foreword by Richard Stanley on (his former student) Miklós Bóna’s “A Walk Through Combinatorics” textbook:

The subject of combinatorics is so vast that the author of a textbook faces a difficult decision as to what topics to include. There is no more-or-less canonical corpus as in such other subjects as number theory and complex variable theory. [here]

Second, from the Preface by Kyle Petersen (and Stanley’s academic descendant) in his elegant “Inquiry-Based Enumerative Combinatorics” textbook:

Combinatorics is a very broad subject, so the difficulty in writing about the subject is not what to include, but rather what to exclude. Which hundred problems should we choose? [here]

Now that this is all clear, you can probably insert your own joke about importance of teaching inclusion-exclusion. But I think the issue is a bit deeper than that.

I’ve been thinking about this when updating my “What is Combinatorics” quotation page (see also my old blog post on this). You can see a complete divergence of points of view on how to answer this question. Some make the definition or description to be very broad (sometimes even ridiculously broad), some relatively narrow, some are overly positive, while others are revoltingly negative. And some basically give up and say, in effect “it is what it is”. This may seem puzzling, but if you concentrate on the narrow definitions and ignore the rest, a picture emerges.

Clearly, these people are not talking about the same area. They are talking about sub-areas of Combinatorics that they know well, that they happen to learn or work on, and that they happen to like or dislike. Somebody made a choice what part of Combinatorics to teach them. They made a choice what further parts of Combinatorics to learn. These choices are increasingly country or culture dependent, and became formative in people’s mind. And they project their views of these parts of Combinatorics on the whole field.

So my point is — there is no right answer to “What is Combinatorics?“, in a sense that all these opinions are biased to some degree by personal education and experience. Combinatorics is just too broad of a category to describe. It’s a bit like asking “what is good food?” — the answers would be either broad and bland, or interesting but very culture-specific.

Courses and textbooks

How should one resolve the issue raised above? I think the answer is simple. Stop claiming that Combinatorics, or worse, Discrete Mathematics, is one subject. That’s not true and hasn’t been true for a while. I talked about this in my “Unity of Combinatorics” book review. Combinatorics is comprised of many sub-areas, see the Wikipedia article I discussed here (long ago). Just accept it.

As a consequence, you should never teach a “Combinatorics” course. Never! Especially to graduate students, but to undergraduates as well. Teach courses in any and all of these subjects: Enumerative Combinatorics, Graph Theory, Probabilistic Combinatorics, Discrete Geometry, Algebraic Combinatorics, Arithmetic Combinatorics, etc. Whether introductory or advanced versions of these courses, there is plenty of material for each such course.

Stop using these broad “a little bit about everything” combinatorics textbooks which also tend to be bulky, expensive and shallow. It just doesn’t make sense to teach both the five color theorem and the Catalan numbers (see also here) in the same course. In fact, this is a disservice to both the students and the area. Different students want to know about different aspects of Combinatorics. Thus, if you are teaching the same numbered undergraduate course every semester you can just split it into two or three, and fix different syllabi in advance. The students will sort themselves out and chose courses they are most interested in.

My own teaching

At UCLA, with the help of the Department, we split one Combinatorics course into two titled “Graph Theory” and “Enumerative Combinatorics”. They are broader, in fact, than the titles suggest — see Math 180 and Math 184 here. The former turned out to be quite a bit more popular among many applied math and non-math majors, especially those interested in CS, engineering, data science, etc., but also from social sciences. Math majors tend to know a lot of this material and flock to the latter course. I am not saying you should do the same — this is just an example of what can be done.

I remember going through a long list of undergraduate combinatorics textbooks a few years ago, and found surprisingly little choice for the enumerative/algebraic courses. Of the ones I liked, let me single out Bóna’s “Introduction to Enumerative and Analytic Combinatorics“ and Stanley’s “Algebraic Combinatorics“. We now use both at UCLA. There are also many good Graph Theory course textbooks of all levels, of course.

Similarly, for graduate courses, make sure you make the subject relatively narrow and clearly defined. Like a topics class, except accessible to beginning graduate students. Low entry barrier is an advantage Combinatorics has over other areas, so use it. To give examples from my own teaching, see unedited notes from my graduate courses:

Combinatorics of posets (Fall 2020)

Combinatorics and Probability on groups (Spring 2020)

Algebraic Combinatorics (Winter 2019)

Discrete and Polyhedral Geometry (Fall 2018) This is based on my book. See also videos of selected topics (in Russian).

Combinatorics of Integer Sequences (Fall 2016)

Combinatorics of Words (Fall 2014)

Tilings (Winter 2013, lecture-by-lecture refs only)

In summary

In my experience, the more specific you make the combinatorics course the more interesting it is to the students. Don’t be afraid that the course would appear be too narrow or too advanced. That’s a stigma from the past. You create a good course and the students will quickly figure it out. They do have their own FB and other chat groups, and spread the news much faster than you could imagine…

Unfortunately, there is often no good textbook to cover what you want. So you might have to work a little harder harder to scout the material from papers, monographs, etc. In the internet era this is easier than ever. In fact, many extensive lecture notes are already available on the web. Eventually, all the appropriate textbooks will be written. As I mentioned before, this is one of the very few silver linings of the pandemic…

P.S. (July 8, 2021) I should have mentioned that in addition to “a little bit about everything” textbooks, there are also “a lot about everything” doorstopper size volumes. I sort of don’t think of them as textbooks at all, more like mixtures of a reference guide, encyclopedia and teacher’s manual. Since even the thought of teaching from such books overwhelms the senses, I don’t expect them to be widely adopted.

Having said that, these voluminous textbooks can be incredibly valuable to both the students and the instructor as a source of interesting supplementary material. Let me single out an excellent recent “Combinatorial Mathematics” by Doug West written in the same clear and concise style as his earlier “Introduction to Graph Theory“. Priced modestly (for 991 pages), I recommend it as “further reading” for all combinatorics courses, even though I strongly disagree with the second sentence of the Preface, per my earlier blog post.