# Difference between revisions of "Factorial"

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(Created page with '==Definition== Let <math>n</math> be a nonnegative integer. The '''factorial''' of <math>n</math>, denoted <math>n!</math>, and read as ''n factorial'', is defined as the produc...') |
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Let <math>n</math> be a nonnegative integer. The '''factorial''' of <math>n</math>, denoted <math>n!</math>, and read as ''n factorial'', is defined as the product of all the natural numbers from <math>1</math> to <math>n</math>. Note that <math>0!</math> is defined as <math>1</math>. | Let <math>n</math> be a nonnegative integer. The '''factorial''' of <math>n</math>, denoted <math>n!</math>, and read as ''n factorial'', is defined as the product of all the natural numbers from <math>1</math> to <math>n</math>. Note that <math>0!</math> is defined as <math>1</math>. | ||

+ | |||

+ | <math>n!</math> is also the order of the [[groupprops:symmetric group|symmetric group]], or the group of ''all'' permutations, on a set of size <math>n</math>. | ||

==Behavior== | ==Behavior== | ||

+ | |||

+ | {{oeis|A000142}} | ||

===Initial values=== | ===Initial values=== | ||

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==Related notions== | ==Related notions== | ||

+ | * [[lcm of all numbers so far]]: This is the exponential of the [[second Chebyshev function]] and ''also'' equals the [[groupprops:exponent of a group|exponent]] of the symmetric group of degree <math>n</math>. | ||

+ | * [[Maximum product over additive partitions]] | ||

+ | * [[Maximum lcm over additive partitions]] | ||

* [[Primorial]] is the product of the first few primes. | * [[Primorial]] is the product of the first few primes. |

## Revision as of 20:18, 30 April 2009

## Definition

Let be a nonnegative integer. The **factorial** of , denoted , and read as *n factorial*, is defined as the product of all the natural numbers from to . Note that is defined as .

is also the order of the symmetric group, or the group of *all* permutations, on a set of size .

## Behavior

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A000142

### Initial values

Here are the values of for small : .

## Related notions

- lcm of all numbers so far: This is the exponential of the second Chebyshev function and
*also*equals the exponent of the symmetric group of degree . - Maximum product over additive partitions
- Maximum lcm over additive partitions
- Primorial is the product of the first few primes.