What is combinatorics, and why do I love it?

When I tell people that my research area is combinatorics, I am frequently met with a blank look or a friendly shrug. I’ve found it very difficult to develop something appropriate to say in such situations. Often, I resort to rambling a bit about graph theory, a sister field in which I haven’t done any research. This piece is my attempt to make up for all those bad explanations, and also to give some insight into what I love about combinatorics.

Mathematicians are in the position of being able to define the objects of our study. The different areas of mathematics are, roughly, separated by different choices about which of these objects to investigate and what sorts of questions to ask about them. Thus, to describe a field of mathematics it is helpful to discuss the sorts of objects mathematicians in that field study.

In combinatorics, our objects of study are distinguished by usually being discrete and/or finite. For example, we often study the permutations of a set, that is, the ways that the elements of that set can be arranged in a line: the set {1, 2, 3} has six permutations: 123, 132, 213, 231, 312 and 321. The graph (that is, a set of points together with arcs connecting some pairs of them) is another frequent object of study. Often, we will be interested in permutations or graphs (or whatever) that also satisfy some other conditions (e.g., graphs with no triangle).

In enumerative combinatorics (the sub-field to which most of my work belongs), we are interested in problems of enumeration. In other words, when enumerative combinatorialists study a collection of objects, the question we seek to answer is, “How many of them are there?” Note that because we study discrete, finite objects, in a particular instance (permutations of a set of size 4; graphs with 7 vertices having no triangle) the answer is some positive integer. In principle, we could answer any such instance by brute-force search (perhaps on a computer), but what we want is not the answer to just one particular instance but rather a method (such as a simple formula, or less ideally a computer program that is guaranteed to work) that will let us know the answer for any instance (in terms of n, how many permutations are there of length n? How many graphs with n vertices that have no triangle?).

One variation on the idea of enumeration that comes up sufficiently frequently to merit special mention is that of bijective proof. Bijective proofs arise when one has two different kinds of objects (say, some special kind of graph and some special kind of permutation) that one wishes to prove are equinumerous. Rather than counting both kinds of objects (which may be difficult, e.g., if there is no nice formula for the number of them) one can try to construct a one-to-one correspondence (also known as a bijection) between the objects of one kind and the objects of the other. For reasons that I don’t think I can explain, such proofs are often more aesthetically pleasing than proofs that show the two sets are equinumerous by counting both and getting the same result.

I think that, in hindsight, I’ve always enjoyed combinatorics. I’m not absolutely certain why this is, but I now attempt to explain it. I’ve always enjoyed mathematics as an exercise in problem-solving. I began doing math team-type activities when I was 9 years old, and I continue to enjoy problems with nice solutions that one can find in a few minutes or an hour. Among areas of modern mathematics, I think combinatorics has an unusually large number of questions that appeal to that sense: I find that I can easily jump in and begin playing around with a wide variety of combinatorial questions without having to do a lot of heavy background reading. One reason for this is that combinatorics has a low “abstraction barrier.” Much of modern pure mathematics is sufficiently abstract and theoretical that only a relative handful of specialists can follow developments closely. Combinatorics, on the other hand, tends to be broadly accessible; I find that I can describe my research not only to other mathematicians, but even to bright high school students. I really enjoy being able to share my excitement with a large group of people.

Surely the previous paragraph is not enough to explain why I’ve ended up in this particular field — there are far too many other random contributors (from the quality of teaching in Budapest to the particular post-docs who were around when I returned) for that. But I do think that it captures something about the appeal of combinatorics to me.

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