Permutation Subset

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A Permutation Subset is a string produced by a permutation operation on some set (for an ordering of a set).

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/permutation Retrieved:2015-2-9.
• In mathematics, the notion of 'permutation relates to the act of rearranging, or permuting, all the members of a set into some sequence or order (unlike combinations, which are selections of some members of the set where order is disregarded). For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. The study of permutations of finite sets is a topic in the field of combinatorics.

Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science.

The number of permutations of n distinct objects is n factorial usually written as n!, which means the product of all positive integers less than or equal to n.

In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). The collection of such permutations form a group called the symmetric group of S. The key to this group's structure is the fact that the composition of two permutations (performing two given rearrangements in succession) results in another rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements (substitutions) of symbols.

In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.

2009

• (Wordnet, 2009) ⇒ http://wordnet.princeton.edu/perl/webwn
• substitution: an event in which one thing is substituted for another; "the replacement of lost blood by a transfusion of donor blood"
• the act of changing the arrangement of a given number of elements
• complete change in character or condition; "the permutations...taking place in the physical world"- Henry Miller
• act of changing the lineal order of objects in a group
• http://en.wiktionary.org/wiki/permutation
• A one-to-one mapping from a finite set to itself; An ordering of a finite set of distinct elements; Any reordering of an ordered set of pitch ...