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Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). One of the most prominent combinatorialists of recent times was Gian-Carlo Rota, who helped formalize the subject beginning in the 1960s. The prolific problem-solver Paul Erdös worked mainly on extremal questions. The study of how to count objects is sometimes thought of separately as the field of enumeration.
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 is the product of all the natural numbers from one to fifty-two. It may seem surprising that this number, about 8.065817517094 × 1067, is so large. That is 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, "the number of atoms, molecules, etc., in a gram mole".
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 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 combinatorial problem. 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; 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 n+2 Fibonacci number, which can be expressed in closed form as:
where φ = (1 + √5) / 2, the Golden mean. 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
which may be more satisfactory (from a purely combinatorial view), since it more clearly shows why the result is as shown.
Another approach is to find an asymptotic formula
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
as n becomes large.
Finally, and most usefully, f(n) may be expressed by a formal power seriesAbstract algebra Ring theory Combinatorics In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of "convergence". They are also useful, called its generating functionIn mathematics a generating function is a formal power series whose coefficients encode information about a sequence a that is indexed by the natural numbers. There are various types of generating functions definitions and examples are given below. Every, which is most commonly either the ordinary generating function
or the exponential generating function
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.
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