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Maybe some could be interested in reviewing it (Toby?).
Seems that the first paragraph is mostly about “enumerative combinatorics”, so I wonder whether that more accurately reflects what you were intending. If so, maybe a change of title would be appropriate?
This is the central part easiest to define. The counting and bijections combine in building mroe complicated algorithms and object. So the idea (which is not mine) is to define combinatorics more narrowly but precisely and then to allow natural and reasonable extensions.
The definition in wikipedia is on the other hand too inclusive – I mean study of countable discrete structures. Not all countable discrete mathematics is combinatorial. For example the entirety of the theory of finite groups is NOT part of combinatorics nor is the study of finite axiomatic systems.
I agree that it would be desirabel to have a bit wider though still good definition of combinatorics and please go on and extend it. But I would object on simplistic and untrue definition at wikipedia. Not all the study of finite discrete objects is combinatorics as wikipedia suggests.
In informal language, “combinatorics” is used in such a broad way that it’s really hard to give a satisfying and encompassing idea of the scope of use. I agree that considering the theory of finite groups as “combinatorics” would be overly broad (and a tad reductionistic), but on the other hand there are aspects of say proof theory or game theory that definitely feel “combinatorial” to me (as do human games like chess and go), and it’s hard to describe the qualitative dividing line.
Maybe one solution is just to admit that fact up front, saying it’s a kind of catch-all term, and say that in practice the term could refer to any of the following (enumerative combinatorics, algebraic combinatorics, etc. etc.), with pages for each if desired. I’m leery of trying to give a good definition, because I’m skeptical that there is one.
I agree that it is not definable in general, though I do not see it incompatible with the picture of several levels of generality. Why don’t you add to the entry more impressions, and examples or links.
added pointer to:
Dmitry N. Kozlov, Trends in Topological Combinatorics [arXiv:math/0507390]
Dmitry Kozlov, Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics 21, Springer (2008) [doi:10.1007/978-3-540-71962-5]
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