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Concept
Enumerative combinatorics
Summary
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.
The simplest such functions are closed formulas, which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck of n cards is f(n) = n!. The problem of finding a closed formula is known as algebraic enumeration, and frequently involves deriving a recurrence relation or generating function and using this to arrive at the desired closed form.
Often, a complicated closed formula yields little insight into the behavior of the counting function as the number of counted objects grows.
In these cases, a simple asymptotic approximation may be preferable. A function is an asymptotic approximation to if as . In this case, we write
Generating functions are used to describe families of combinatorial objects. Let denote the family of objects and let F(x) be its generating function. Then
where denotes the number of combinatorial objects of size n. The number of combinatorial objects of size n is therefore given by the coefficient of . Some common operation on families of combinatorial objects and its effect on the generating function will now be developed.
The exponential generating function is also sometimes used. In this case it would have the form
Once determined, the generating function yields the information given by the previous approaches.
Official source
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