Factorial in the context of Exponential function

In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function ⁠${\displaystyle f:A\rightarrow A}$⁠, where A is a set; the function ⁠${\displaystyle f}$⁠ is a unary operation on A.

Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g. transpose A). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument.

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 S_i indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in S_n 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.

In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set.

A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function n! is defined by the rules

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ${\displaystyle \mathrm {S} _{n}}$ defined over a finite set of ${\displaystyle n}$ symbols consists of the permutations that can be performed on the ${\displaystyle n}$ symbols. Since there are ${\displaystyle n!}$ (${\displaystyle n}$ factorial) such permutation operations, the order (number of elements) of the symmetric group ${\displaystyle \mathrm {S} _{n}}$ is ${\displaystyle n!}$.

Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.

In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop is fixed before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions.

The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not particularly easy to devise a computable function that is not primitive recursive; some examples are shown in section § Limitations below.