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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 have the same parity (odd or even) as n. [1] That is,
The factorial of is , or in symbols, ! =. There are several motivations for this definition: For n = 0 {\displaystyle n=0} , the definition of n ! {\displaystyle n!} as a product involves the product of no numbers at all, and so is an example of the broader convention that the empty product , a product of no factors, is equal to the ...
The following table lists many specialized symbols commonly used in modern mathematics, ordered by their introduction date. The table can also be ordered alphabetically by clicking on the relevant header title.
1. Factorial: if n is a positive integer, n! is the product of the first n positive integers, and is read as "n factorial". 2. Double factorial: if n is a positive integer, n!! is the product of all positive integers up to n with the same parity as n, and is read as "the double factorial of n". 3.
These symbols are collectively called factorial powers. [ 2 ] The Pochhammer symbol , introduced by Leo August Pochhammer , is the notation ( x ) n {\displaystyle (x)_{n}} , where n is a non-negative integer .
The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, [1] and the LaTeX symbol.
The ratio of the factorial!, that counts all permutations of an ordered set S with cardinality, and the subfactorial (a.k.a. the derangement function) !, which counts the amount of permutations where no element appears in its original position, tends to as grows.
The number of perfect matchings of the complete graph K n (with n even) is given by the double factorial (n – 1)!!. [12] The crossing numbers up to K 27 are known, with K 28 requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project. [13] Rectilinear Crossing numbers for K n are