Combinatorics

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Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics), with deciding when the criteria can be met and then constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics).

Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century. Much of combinatorics is about graphs, to whose study all types of combinatorics can contribute.

An example of a combinatorial question is the following: What is the number of possible orderings of a deck of 52 playing cards? That number equals 52! (i.e., "fifty-two factorial"). It may seem surprising that this number, about 8.065817517094 × 1067, is so large —a little bit more than 8 followed by 67 zeros! Comparing that number to some other large numbers, it is greater than the square of Avogadro's number, 6.022 × 1023.

An example of another kind is this problem: Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people are in exactly one set together, every two sets have exactly one person in common, and no set contains all or all but one of the people? The answer depends on n and is only partially known to this day. See "Design theory" below for a partial answer.

Enumerative combinatorics came to prominence because counting configurations is essential to elementary probability, starting with the work of Pascal and others. Modern combinatorics began to develop in the late nineteenth century and became a distinguishable field of study in the twentieth century, partly through the publication of the systematic enumerative treatise Combinatory Analysis by Percy Alexander MacMahon in 1915 and the work of R.A. Fisher in design of experiments in the 1920s. Two of the most prominent combinatorialists of recent times were the prolific problem-raiser and problem-solver Paul Erdős, who worked mainly on extremal questions, and Gian-Carlo Rota, who helped to formalize the subject beginning in the 1960s, mostly in enumeration and algebraization. The study of how to count objects is sometimes thought of separately as the field of enumeration.

Enumerative combinatorics

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 does not 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 {Si} 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 Sn 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 Si 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 simple functions like factorials, powers, and so on. For instance, and 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 example, let f(n) be the number of distinct subsets of the integers in the interval [1,n] that do not contain two consecutive integers; e.g., with n = 4, we have the sets {}, {1}, {2}, {3}, {4}, {1,3}, {1,4}, {2,4}, so f(4) = 8. It turns out that f(n) is the n+2nd Fibonacci number, F(n+2), so it can be expressed in closed form as

f(n)={\frac  {\phi ^{{n+2}}}{{\sqrt  {5}}}}-{\frac  {(1-\phi )^{{n+2}}}{{\sqrt  {5}}}},

where Φ = (1 + √5) / 2, the golden ratio. However, given that we are looking at a counting function, the presence of the √5 in the result may be considered unaesthetic. As an alternative that shows more clearly why f(n) is a positive integer, f(n) may be expressed by the recurrence

f(n)=f(n-1)+f(n-2).


Another approach is to find an asymptotic formula

f(n)\sim 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

f(n)\sim {\frac  {\phi ^{{n+2}}}{{\sqrt  {5}}}}

as n becomes large.

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

\sum f(n)x^{n}

or the exponential generating function

\sum f(n){\frac  {x^{n}}{n!}},

where the sums are taken over the range 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; this allows one to extend results from one combinatorial problem in order to solve others.

Structural combinatorics

There are some very subtle combinatorial patterns and surprising theorems.

Design theory

A simple result in this area of combinatorics is that the problem of forming sets, described in the introduction, has a solution only if n has the form q2 + q + 1. It is less simple to prove that a solution exists if q is a prime power, may exist if q is a sum of two square numbers, but cannot exist for any other positive integer q. This last result, the Bruck-Ryser theorem, is proved by a combination of constructive methods based on finite fields and an application of quadratic forms.

When such a structure does exist, it is called a finite projective plane; thus showing how finite geometry and combinatorics intersect.

Ramsey theory

Frank P. Ramsey proved that, given any group of six people, it is always the case that one can find three people out of this group that either all know each other, or all do not know each other.

The proof is a short proof by contradiction: suppose the claim is false. This means that we can have a group of six people such that whenever we look at any three of the six, there are at least two people among these three that know each other and at least two who do not know each other. Consider now one person among the six; call this person "A." Now, among the remaining five people, there must be at least three who either all know A or all do not know A--this is clear since the negation of one condition immediately implies the other condition. Assume first former condition: that at least three of the remaining five know A. Among those three people, at least two of them must know each other, since otherwise we would have three people who all don't know each other, contrary to our hypothesis. But then we have two people who know each other, and know A, and so these two people, along with A, constitute a group of three people among the six who all know each other. This contradicts our initial hypothesis. Assuming that other condition--that three of the remaining five do not know A--results in a similar contradiction.

This is a special case of Ramsey's theorem.

The idea of finding order in random configurations gives rise to Ramsey theory. Essentially this theory says that any sufficiently large configuration will contain at least one instance of some other type of configuration.

Matroid theory

This part of combinatorics abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory.

For instance, given a set of n vectors in Euclidean space, what is the largest number of planes they can generate? (Answer: the binomial coefficient C(n,3).) Is there a set that generates exactly one less plane? (No, in almost all cases.) These are extremal questions in geometry.

Extremal combinatorics

Many extremal questions deal with set systems. A simple example is the following: what is the largest number of subsets of an n-element set one can have, if no two of the subsets are disjoint? Answer: half the total number of subsets. Proof: Call the n-element set S. Between any subset T and its complement ST, at most one can be chosen. This proves the maximum number of chosen subsets is not greater than half the number of subsets. To show one can attain half the number, pick one element x of S and choose all the subsets that contain x.

A more difficult problem is to characterize the extremal solutions; in this case, to show that no other choice of subsets can attain the maximum number while satisfying the requirement.

Often it is too hard even to find the extremal answer f(n) exactly and one can only give an asymptotic estimate.

See also

References

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