View source for Enumerative combinatorics

In [[mathematics]], '''enumerative combinatorics''' is an area of [[combinatorics]] that deals with the number of ways that certain patterns can be formed.

== Description ==

Two examples of this type of problem are counting [[Combination|combinations]] and counting [[Permutation|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.

== See also ==

* [[Algebraic combinatorics]]
* [[Asymptotic combinatorics]]
* [[Burnside's lemma]]
* [[Combination]]
* [[Combinatorial explosion]]
* [[Combinatorial principles]]
* [[Combinatorial species]]
* [[Combinatorics]]
* [[Inclusion-exclusion principle]]
* [[Method of distinguished element]]
* [[Pólya enumeration theorem]]
* [[Sieve theory]]
* [[Twelvefold way]]

== External links ==

* [https://en.wikipedia.org/wiki/Enumerative_combinatorics Enumerative combinatorics] @ Wikipedia.org 

[[Category:Combinatorics]]
[[Category:Mathematics]]