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In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product. [1]
Definition. The factorial number system is a mixed radix numeral system: the i -th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i− 1)! (its place value). 7! 6!
Double factorial. The fifteen different chord diagrams on six points, or equivalently the fifteen different perfect matchings on a six-vertex complete graph. These are counted by the double factorial 15 = (6 − 1)‼. In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that ...
In this article, the symbol () is used to represent the falling factorial, and the symbol () is used for the rising factorial. These conventions are used in combinatorics , [ 4 ] although Knuth 's underline and overline notations x n _ {\displaystyle x^{\underline {n}}} and x n ¯ {\displaystyle x^{\overline {n}}} are increasingly popular.
Factorion. In number theory, a factorion in a given number base is a natural number that equals the sum of the factorials of its digits. [1][2][3] The name factorion was coined by the author Clifford A. Pickover. [4]
Roughly speaking, the simplest version of Stirling's formula can be quickly obtained by approximating the sum with an integral: The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating , one considers its natural logarithm, as this is a slowly varying function:
In mathematics, the gamma function (represented by Γ, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function is defined for all complex numbers except non-positive integers, and for every positive integer , The gamma function can be defined via a ...
The sum of the numbers in the factorial number system representation gives the number of inversions of the permutation, and the parity of that sum gives the signature of the permutation. Moreover, the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the ...