What is "permutation"

Permutation \Per`mu*ta"tion\, n. [L. permutatio: cf. F. permutation. See Permute.]

1. The act of permuting; exchange of the thing for another; mutual transference; interchange.

   The violent convulsions and permutations that have been made in property.
   --Burke.

2. (Math.)

   1. The arrangement of any determinate number of things, as units, objects, letters, etc., in all possible orders, one after the other; -- called also alternation. Cf. Combination, n., 4.

   2. Any one of such possible arrangements.

3. (Law) Barter; exchange.

   Permutation lock, a lock in which the parts can be transposed or shifted, so as to require different arrangements of the tumblers on different occasions of unlocking.

mid-14c., from Old French permutacion "change, shift" (14c.), from Latin permutationem (nominative permutatio) "a change, alteration, revolution," noun of action from past participle stem of permutare "change thoroughly, exchange," from per- "thoroughly" (see per) + mutare "to change" (see mutable).

n. 1 (context mathematics English) A one-to-one mapping from a finite set to itself. 2 (context mathematics combinatorics English) An ordering of a finite set of distinct elements. 3 (context music English) A transformation of a set's prime form, by applying one or more of certain operations, specifically, transposition, inversion, and retrograde.

1. n. an event in which one thing is substituted for another; "the replacement of lost blood by a transfusion of donor blood" [syn: substitution, transposition, replacement, switch]

2. the act of changing the arrangement of a given number of elements

3. complete change in character or condition; "the permutations...taking place in the physical world"- Henry Miller

4. act of changing the lineal order of objects in a group

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting. These differ from combinations, which are selections of some members of a 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). These are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. In this example, the letters are already ordered in the original word and the anagram is a reordering of the 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 distinct objects is factorial, usually written as , which means the product of all positive integers less than or equal to .

In algebra and particularly in group theory, a permutation of a set is defined as a bijection from to itself. That is, it is a function from to for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of in which each element is replaced by the corresponding . The collection of such permutations form a group called the symmetric group of . 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 -permutations, or partial permutations, are the ordered arrangements of distinct elements selected from a set. When is equal to the size of the set, these are the permutations of the set.

In music, a permutation (order) of a set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation (also called rotation), or multiplicative operations (such as the cycle of fourths and cycle of fifths transforms). These may produce reorderings of the members of the set, or may simply map the set onto itself.

Order is particularly important in the theories of compositional techniques originating in the 20th century such as the twelve-tone technique and serialism. Analytical techniques such as set theory take care to distinguish between ordered and unordered collections. In traditional theory concepts such voicing and form include ordering. For example, many musical forms, such as rondo, are defined by the order of their sections.

The permutations resulting from applying the inversion or retrograde operations are categorized as the prime form's inversions and retrogrades, respectively. Likewise, applying both inversion and retrograde to a prime form produces its retrograde-inversions, which are considered a distinct type of permutation.

Permutation may be applied to smaller sets as well. However, the use of transformation operations to such smaller sets do not necessarily result in permutation of the original set. Here is an example of non-permutation of trichords, using the operations of retrogradation, inversion, and retrograde-inversion, combined in each case with transposition, as found within in the tone row (or twelve tone series) from Anton Webern's Concerto:

B, B, D, E, G, F, G, E, F, C, C, A

If the first three notes are regarded as the "original" cell, then the next three are its transposed retrograde inversion (backwards and upside down), the next three are the transposed retrograde (backwards), and the last three are its transposed inversion (upside down).

Not all prime series have the same number of variations because the transposed and inverse transformations of a tone row may be identical to each other, a quite rare phenomenon: less than 0.06% of all series admit 24 forms instead of 48.

One technique facilitating twelve-tone permutation is the use of number values corresponding with musical letter names. The first note of the first of the primes, actually prime zero (commonly mistaken for prime one), is represented by 0. The rest of the numbers are counted half-step-wise such that: B = 0, C = 1, C/D = 2, D = 3, D/E = 4, E = 5, F = 6, F/G = 7, G = 8, G/A = 9, A = 10, and A/B = 11.

Prime zero is retrieved entirely by choice of the composer. To receive the retrograde of any given prime, the numbers are simply rewritten backwards. To receive the inversion of any prime, each number value is subtracted from 12 and the resulting number placed in the corresponding matrix cell (see twelve-tone technique). The retrograde inversion is the values of the inversion numbers read backwards.

Therefore:

A given prime zero (derived from the notes of Anton Webern's Concerto):

0, 11, 3, 4, 8, 7, 9, 5, 6, 1, 2, 10

The retrograde:

10, 2, 1, 6, 5, 9, 7, 8, 4, 3, 11, 0

'''The inversion: '''

0, 1, 9, 8, 4, 5, 3, 7, 6, 11, 10, 2

The retrograde inversion:

2, 10, 11, 6, 7, 3, 5, 4, 8, 9, 1, 0

More generally, a musical permutation is any reordering of the prime form of an ordered set of pitch classes or, with respect to twelve-tone rows, any ordering at all of the set consisting of the integers modulo 12. In that regard, a musical permutation is a combinatorial permutation from mathematics as it applies to music. Permutations are in no way limited to the twelve-tone serial and atonal musics, but are just as well utilized in tonal melodies especially during the 20th and 21st centuries, notably in Rachmaninoff's "Variations on the Theme of Paganini" for orchestra and piano.

Cyclical permutation is the maintenance of the original order of the tone row with the only change being that of the initial pitch-class, with the original order following after. This is also called rotation. A secondary set may be considered a cyclical permutation beginning on the sixth member of a hexachordally combinatorial row. The tone row from Berg's Lyric Suite, for example, is realized thematically and then cyclically permuted (0 is bolded for reference):

5 4 0 9 7 2 8 1 3 6 t e
3 6 t e 5 4 0 9 7 2 8 1

Permutation is a mathematical concept.

Permutation may also refer to:

• Permutation (music), a concept in musical set theory
• Permutation (policy debate), a type of argument in policy debate
• Permutation (album), an album by Brazilian electronic artist Amon Tobin
• "Permutation", an instrumental song by the Red Hot Chili Peppers
• Permutation test, in statistics

Permutation is the tenth solo album by American composer Bill Laswell, released on November 30, 1999 by ION Records.

Permutation is the third studio album by Brazilian electronic music producer Amon Tobin and the second under his own name. It was released in 1998, just over a year after Bricolage. The album was a success for Tobin and found him playing sold-out shows at the Montreal Jazz Festival, the Knitting Factory in New York and the Coachella Valley Music and Arts Festival. He went on to release Supermodified in 2000.

Tobin makes references to David Lynch films a number of times on Permutation. The song "Like Regular Chickens" contains a line of dialogue spoken in Eraserhead, while the title of the song "People Like Frank" is a line of dialogue from Blue Velvet. The song also samples a part of Angelo Badalamenti's score to that film. "Fast Eddie" is probably a reference to Lynch's Lost Highway character Mr. Eddie. On the LP edition of the album, "Melody Infringement" is added as a bonus track.

In policy debate (although sometimes used in Lincoln-Douglas debate, especially on the national circuit), a permutation is an argument made by the 2AC (in Lincoln-Douglas debate the 1AR) to test the competition of a counterplan or kritik testing the comparative desirability of the plan and all or part of the counterplan or kritik against the counterplan or kritik by itself. Most permutations are tests rather than advocacies and thus do not change the fiat of the affirmative plan in the world where the negative does not advocate the counterplan or the kritik.

The easiest way to describe the function of a permutation perm is in the context of counterplan theory. A counterplan functions to test the opportunity cost of a plan. The negative proposes a counterplan that is competitive with the affirmative's plan. For example, if the Aff plan is to grant amnesty to all illegal immigrants within the US, a counterplan could be to declare all illegal immigrants felons. The neg would argue that their counterplan, made impossible by the aff's plan, will garner more benefits than plan. A perm is a way to test whether or not the counterplan and plan are mutually exclusive. An example of a perm would be this: Aff plan is to send a mission to the moon. Counterplan is to invest in renewable energy. The Aff can run a perm, i.e., claim that sending a mission to the moon does not make it impossible to invest in renewable energy. The perm demonstrates that the counterplan is not an opportunity cost to plan, and therefore does not garner any benefits for neg.

The same is basically true for perms in Kritiks. A kritik "is generally a type of argument that challenges a certain mindset, assumption, or discursive element that exists within the advocacy of the opposing team" ( Kritik). A simple example of Kritik is that capitalism is bad (to put it simply). The team running the K will argue that the nature of capitalism is bad, and has horrible implications for society. Part of the K is an alternative. If you kritik capitalism, a simple alternative might be to endorse Marxism, or "reject, and rethink" (meaning, vote the other team down, and have a good long think about how to replace capitalism). A permutation, again, is a way of showing a lack of competition between the opposing sides of the debate. The side having the K run on them could, in our example, say "we need to do our plan, but capitalism is also bad. Vote for our plan, but while you're at it, rethink capitalism."