a repository of mathematical know-how

Revision of Techniques for counting from Sat, 18/04/2009 - 19:28

Quick description

Counting techniques loosely go under the name of combinatorics. Counting the Total Number of things is called enumerative combinatorics, while counting the maximum, minimum, or minimally sufficient number of things goes by the name of Extremal Combinatorics. Trivial examples are counting the number of ways to roll "18" on four 6-sided die, counting the number of ways to obtain a full house when playing poker with a standard deck, or counting the number of ways to distribute 17 doughnuts of different types to 5 people.


Algebra (see General Discussion)

Example 1

1. Let's say I roll 6 dice. What is the probability I roll the number k = 6, 7, \dots, 36?

2. Counting the number of ways to order 8 couples around a (round) table such that no couple is sitting together.

3. Counting the number of ways to write the integer n as the sum of k positive integers (e.g. 5 = 2 + 2 + 1 or 5 = 3 + 1 + 1).

4. Counting the number of strings of length 8 that can be made from the letters A, B, and C, such that no string has more than two repeating letters (in a row.)

5. Finding the largest integer not of the form 6x + 10y + 15z where x,y,z are non-negative integers.

6. Counting the number of ways to color a graph G with \lambda colors.

7. Counting the maximum number of edges in a graph that does not have a clique of order k.

General discussion

Combinatorics is a field of applied mathematics that can be approached in numerous ways. The standard introduction requires a basic knowledge of algebra, including the factorial function. For a more advanced introduction that may use recurrence relations, basic linear algebra could be useful. Advanced combinatorial tricks tend to use generating functions to transform a recurrence equation into an algebraic or differential equation, and then solve those.

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.


Combinations and Permutations

Binomial Theorem

List of Combinatorial Identities

Recurrence Relations

Generating Functions

Probabilistic Method

Sieve Method

Combinatorial Snake Oil Method

Wilf-Zeilberger Lemma


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!