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Techniques for counting

Quick description

Counting techniques are used predominantly in combinatorial fields, but are occasionally borrowed for a proofs in other branches of mathematics. This (unfinished) article is a quick resource to point the user towards some of the combinatorial techniques developed to count discrete objects, and specific examples of how counting techniques have been applied in different fields.

Discussion of Prerequisites

Certain counting techniques require almost no background knowledge. Basic combinatorial arguments, those that deal with finite arrangements of a finite number of objects, require a background in Algebra, and an understanding of the factorial function. More advanced combinatorics requires linear algebra and exposure to recurrence relations. The more advanced combinatorial techniques require a background in calculus and differential equations, as typical advanced techniques involve generating functions.

Combinatorics is the basis for counting techniques. If you are looking to learn how to solve certain problems, see the Combinatorics front page. Otherwise the techniques below should help to serve your purposes.


Elementary counting techniques

Combinations and Permutations

Binomial Theorem

List of Combinatorial Identities

Recurrence Relations

Generating Functions

Probabilistic Method

Sieve Method

Combinatorial Snake Oil Method

Wilf-Zeilberger Lemma

Counting Techniques in Other Fields

Counting Techniques in Set Theory

Counting Techniques in Graph Theory

Counting Techniques in Topology

Counting Techniques in Knot Theory

Counting Techniques in Geometry


Not sure where to link to (Counting)

If you know there's an article on any of the tricks linked to in the "Techniques" section, please edit the text so it goes to the right article. Thanks!

I can add a link for the

I can add a link for the probabilistic method, and will do so in a moment. We should perhaps think about whether this is the most appropriate title for the article – it would also be naturally described as "Enumerative Combinatorics Front Page," except perhaps for the brief mentions of probabilistic and extremal combinatorics. It's genuinely not obvious to me what the right way to organize things is here. One possibility might be to have a rather short navigation page that describes what the word "Counting" means to mathematicians (called "Techniques for counting") which could then link to the combinatorics front page, which is waiting for an enumerative combinatorics front page, which this could perhaps be.

Re: Counting Techniques vs. Combinatorics

I think you're right. I took what was the introduction and made it the Enumerative Combinatorics front page (still needs some tuning up) and re-wrote 'techniques for counting'. Presumably the distinction is that 'techniques for counting' should link not only to the combinatorial methods, but also to articles or proofs in non-combinatorial fields that make use of combinatorial techniques. That is, it should link to the answer to the question: what does [technique A] look like when applied in [field W]?

Hmm, that's not quite what I

Hmm, that's not quite what I meant. If you have a look at most subject-matter front pages (the combinatorics one is a bit of an exception) then you will see that they tend to consist of a lot of links and not much text. So what you now have as "Techniques for counting" is in that kind of Front Page style, and what you have as "Enumerative combinatorics front page" is not really in that style.

What I want to encourage is two routes to a typical article: one by means of gradual zeroing in on its subject matter, and the other by means of classifying the type of problem that the article can help you solve. So I would imagine an article entitled "Techniques for counting" as doing something like starting with a general description of what counting is, giving some kind of classification of counting problems into different types, trying to explain briefly which techniques are good for which types of problems, and giving links to fuller articles about those techniques. (For example, in the Princeton Companion to Mathematics, Doron Zeilberger gives a nice explanation of when you use straight generating functions and when you use exponential generating functions.) It seems to me that we'd get a better approximation to all this if we exchanged the titles of your two articles!

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