Combinatorics : Combinatorial 

Counting functions

Calculating the number of ways that certain patterns can be formed is the beginning of combinatorics. Let S be a set with n objects. Combinations of k objects from this set S are subsets of S having k elements each (where the order of listing the elements doesn't distinguish two subsets). Permutations of k objects from this set S refer to sequences of k different elements of S (where two sequences are considered different if they contain the same elements but in a different order). Formulas for the number of permutations and combinations are readily available and important throughout combinatorics.

More generally, given an infinite collection of finite sets {S_i} typically indexed by the natural numbers, enumerative combinatorics seeks a variety of ways of describing a counting function, f(n), which counts the number of objects in S_n for any n. Although the activity of counting the number of elements in a set is a rather broad mathematical problem, in a combinatorial problem the elements S_i will usually have a relatively simple combinatorial description, and little additional structure.

The simplest such functions are closed formulas, which can be expressed as a composition of elementary functions such as factorials, powers, and so on. As noted above, the number of possible different orderings of a deck of n cards is f(n) = n!.

This approach may not always be entirely satisfactory (or practical) for every combinatoric problem. For example, let f(n) be the number of distinct subsets of the the integers in the interval [1,n] that do not contain two consecutive integers; thus for example, with n = 4, we have {}, {1}, {2}, {3}, {4}, {1,3}, {1,4}, {2,4}, so f(4) = 8. It turns out that f(n) is the the nth Fibonacci number, which can be expressed in closed form as:

<math>f(n) = \frac{\phi^n}{\sqrt{5}} - \frac{(1-\phi)^n}{\sqrt{5}}</math>

where φ = (1 + √5) / 2. However, given that we are looking at sets of integers, the presence of the √5 in the result may be considered as "unaesthetic" from a combinatoric viewpoint. Alternatively, f(n) may be expressed as the recurrence

<math>f(n) = f(n-1) + f(n-2)</math>

which may be more satisfactory (from a purely combinatoric view), since it more clearly shows why the result is as shown.

Another approach is to find an asymptotic formula f(n) ~ g(n) where g(n) is a "familiar" function, and where f(n) approaches g(n) as n approaches infinity. In some cases, a simple asymptotic function may be preferable to a horribly complicated closed formula that yields no insight to the behaviour of the counted objects. In the above example, an asymptotic formula would be

<math>f(n) \sim \frac{\phi^n}{\sqrt{5}}</math>

as n becomes large.

Finally, and most usefully, f(n) may be expressed by a formal power series, called its generating function, which is most commonly either the ordinary generating function

<math>\sum f(n) x^n</math>

or the exponential generating function

<math>\sum f(n) \frac{x^n}{n!}</math>

where the sums are taken for n ≥ 0. Once determined, the generating function may allow one to extract all the information given by the previous approaches. In addition, the various natural operations on generating functions such as addition, multiplication, differentiation, etc., have a combinatorial significance; and this allows one to extend results from one combinatorial problem in order to solve others.

Results

Some very subtle patterns can be developed and some surprising theorems proved. One example of a surprising theorem is of Frank P. Ramsey:

Suppose 6 people meet each other at a party. Some of those already know each other, some of them do not. It is always the case that one can find 3 people out of the 6 such that they either all know each other or that they are all strangers to each other.

The proof is a short proof by contradiction: suppose that there aren't 3 people who either all know each other or all don't know each other. Then consider any one person at the party, hereafter called person A: among the remaining 5 people, there must be at least three who either all know or all do not know A. Without loss of generality[?], assume three such people all know A. But then among those three people, at least two of them must know each other (otherwise we would have 3 people who all don't know each other). But then those two also know A, so we have 3 people who all know each other. (This is a special case of Ramsey's theorem)

The idea of finding order in random configuration gives rise to Ramsey theory. Essentially this theory says (in mathematical language) that any random configuration will, if it is large enough, contain smaller configuration of a given type. For example if you try hard enough any pattern of stars can be found in the sky. It has been used to debunk claims that some patterns are especially meaningful.

See also: Finite Mathematics

External links

• Frank P. Ramsey (http://cepa.newschool.edu/het/profiles/ramsey.htm)
• Gian-Carlo Rota (http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Rota.html)

References

• Handbook of Combinatorics, Volumes 1 and 2, R.L. Graham, M. Groetschel and L. Lovasz (Eds.), MIT Press, 1996. ISBN 026207169X
• Enumerative Combinatorics, Volumes 1 and 2 (http://www-math.mit.edu/~rstan/ec/), Richard P. Stanley, Cambridge University Press, 1997 and 1999, ISBN 0-521-55309-1n

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