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I wonder about the difference between the two areas combinatorics and discrete mathematics. On closer inspection, I was unable to pinpoint any concrete difference so far.

Here are a few thoughts:

  • It might be about the "advancedness". My impression is that dicrete math is mostly used in connection with freshman courses. (Discrete mathematics for computer scientists, etc.), with the notable exception of the journal Discrete Mathematics. While I know about quite a few combinatorialists, I do not remember any mathematician describing himself as a discrete mathematician. So maybe the situation is a bit like calculus vs analysis: Discrete mathematics is basic, combinatorics is advanced.

  • It might be connected to the historical development. My impression is that combinatorics was considered for quite some time as being restricted to basic counting operations based on permutations, combinations, binomial coefficients etc., resulting in combinatorics being a subfield of discrete mathematics. But if you look at modern definitions of combinatorics, they all agree that it is hard to give a precise definition, but also they agree that besides enumerative combinatorics (which is all the counting techniques lik generating functions, Pólya theory, Möbius inversion etc.), combinatorics includes the study of various combinatorial structures, like posets, graphs, matroids, finite geometries, combinatorial designs, error correcting codes etc.

So if "combinatorics is a subset of discrete mathematics" should indeed be true: I would like to see a concrete example of a subject being discrete math, but not combinatorics.

I was a bit surprised to find that the (oldschool?) viewpoint "combinatorics = counting" is also suggested by our MSE tag descriptions. From the tag description "discrete mathematics:

For questions about the study of finite or countable discrete structures, especially how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

From the tag description discrete mathematics

Consider using a more specific tag instead, such as: (combinatorics), [...]

Also note that on MSE, synonyms of combinatorics are enumerative-combinatorics (I don't agree, the existence of this refined notion clearly indicates that it is a subfield) and counting (I don't agree for the same reason, plus having the "elementary" feeling attached).

Addition: I've looked at two reputable books on combinatorics to see what they include. Here is a selection from their table of contents:

  • J. H. van Lint, R.M. Wilson. "A Course in Combinatorics": Graphs, Colorings of graphs and Ramsey's theorem, Flows in networks, Latin squares, Hadamard matrices, Codes and designs, Strongly regular graphs, Projective and combinatorial geometries, Association schemes.
  • Peter J. Cameron. "Combinatorics": Latin squares, Finite geometry, Ramsey's Theorem, Graphs, Designs, Error-correcting codes, Graph colorings.
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    $\begingroup$ I’m not sure, but I think “discrete math” often contains a bit of number theory. $\endgroup$ Commented Nov 13, 2023 at 8:03
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    $\begingroup$ I'd say Graph Theory is Discrete Math but not Combinatorics. $\endgroup$ Commented Nov 13, 2023 at 8:06
  • $\begingroup$ When you're examining finite structures you can always find things to count but if they have results unrelated to counting then they're discrete but not combinatoric. Graph theory and coding theory both have discrete structures without being inherently combinatoric. $\endgroup$ Commented Nov 13, 2023 at 8:07
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    $\begingroup$ I do find it somewhat surprising that some of the other commenters here don't view graph theory as a combinatorial topic. While it certainly isn't enumerative or algebraic combinatorics, I was under the impression that the theory of graphs and related structures was regarded as inherently combinatorial. Indeed as someone currently looking through lists of academic faculty to find potential advisors for a masters program, it seems most self-described "combinatorists" work more with graphs than with anything else (which is somewhat frustrating, as someone on the enumeration/algebra side). $\endgroup$ Commented Nov 13, 2023 at 8:42
  • $\begingroup$ @GerryMyerson For eaxmple, the popular book "A course in combinatorics" by van Lint and Wilson starts with chapter 1 "graphs". $\endgroup$ Commented Nov 13, 2023 at 8:56

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You say:

I would like to see a concrete example of a subject being discrete math, but not combinatorics.

Keeping in mind that every subject is going to touch every other subject in one way or another, I think some fairly uncontroversial examples would include:

  • Many parts of finite group theory
  • Computability theory
  • Cryptography
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  • $\begingroup$ I think I don't agree on group theory. There are quite a few combinatorial aspects. Permutations and group actions are at the core of combinatorics, the latter one for being the abstract framework for counting under symmetry. Which group related things do you consider discrete math but not combinatorics? Computability theory might be better classified as theoretical computer science, but that direction might work. Your your example of cryptography feels like the most convincing one. Is it discrete math: Probably yes. Is it combinatorics: Probably no. $\endgroup$ Commented Nov 13, 2023 at 9:31
  • $\begingroup$ @azimut you're willing to expand the definition of combinatorics to include graph theory but unwilling to expand the definition of mathematics to make computability theory part of it? When you expect specificity in some cases and vagueness in others it's cherry-picking. $\endgroup$ Commented Nov 13, 2023 at 12:01
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    $\begingroup$ @CyclotomicField No worries, my sole motivation is to understand the range and limitations of the common use of the notions. I'm not in the position to unilaterally "expand the definition .. to include graph theory", it's simply what I see important people are doing in their combinatorics books. I certainly don't think that theoretical CS and mathematics are disjoint. I'm coming more and more to the conclusion that my understanding of "discrete math" was too narrow on that side. $\endgroup$ Commented Nov 13, 2023 at 13:41
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    $\begingroup$ @azimut: "I think I don't agree on group theory. There are quite a few combinatorial aspects." There are also combinatorial aspects to differential equations, algebraic topology, etc, and conversely there are important applications of these areas in combinatorics. You'll never find a field of math that doesn't touch another. $\endgroup$ Commented Nov 13, 2023 at 15:21
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I believe that this is an important and poorly understood question with implications for education.

In my opinion, combinatorics and discrete mathematics are two different types of things, and thus defy comparison. But neither should be thought of as collections of topics, so it's not very meaningful to ask what's in their symmetric difference.

  1. Combinatorics is a field (or discipline). By which I mean it is a community of researchers who identify with each other, work on a common body of questions and techniques (asked and answered in a natural dialogue), and train new members in this disciplinary thinking.

  2. Some journals notwithstanding, discrete mathematics should be primarily understood as a course. Not only that, it is a course which is a prerequisite for other courses. Its content is determined by what people who teach those other courses want students to know before coming to their course. Discrete mathematics is not a discipline in the above sense; your observation that no working professional considers themself a discrete mathematician is on point.

Apologies in advance if there is an entire community of self-identified discrete mathematicians whom I do not know about!

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