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Although a Mathematician, this article was hard for me to comprehend the first time I read it. My suggestion is to make it understandable for everyone by inserting the following practical but enlightening example in Definition 1: Suppose you have four objects labeled A,B,C,D and imagine that you arrange them in a circle side by side, each object being represented by a point on the circle. (A small scheme would be nice here). The arclengths between the points are not of significance. Choose a fixed orientation, say clockwise, and a point other than A, say C. Starting with C, read all the objects on the circle clockwise until each occurs exactly once. The outcome would be C,D,A,B and this is called a cyclic permutation of A,B,C,D. Following the same procedure for B and D we obtain all cyclic permutations of A,B,C,D.
Important Notice: As far I have searched, the concepts cyclic permutation and cycle are not identical. The only connection between them is that a cyclic permutation can be uniquely represented as a product of disjoint cycles, as can every permutation. —Preceding unsigned comment added by Ilknow (talk • contribs) 13:51, 21 January 2008 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.
The result was No consensus to merge -- Jreferee(Talk) 05:22, 17 August 2007 (UTC)
Do not merge. Rather, I would like to see the cyclic permutation article reference strictly to it's practice in music, and move mathematical information to the cycle article. I was directed to cyclic permutation from twelve-tone technique; this article would be best as an expansion of its use in that context. If there is that much overlap, maybe a disambiguation page is in order. count_lawrence 14:15, 23 August 2006 (UTC)
But at present there is nothing about music here: only three different definitions of what could be a cyclic permutation. Also, the article is completely ill written concerning formatting, language, punctuation, and missing references. — MFH:Talk 20:01, 10 November 2006 (UTC)
I've temporarily reverted the merge of Cycle (mathematics) into this article, since I'm unconvinced that this is a good idea. I would like to have more discussion about this. Please see the discussion [Wikipedia_talk:WikiProject_Mathematics/Archive_27#Merge_of_Cycle_.28mathematics.29.2C_into_Cyclic_permutation here]. Paul August☎ 22:30, 18 July 2007 (UTC)
Recently Cycle (mathematics) was merged into Cyclic permutation. Since I'm unconvinced that this is a good idea, I've temporarily reverted that merge until we could have a discussion here. So what does everyone think? Paul August☎ 22:29, 18 July 2007 (UTC)
Well, you should have told us what's wrong here in the first place, to explain your actions. Anyway, I see multiple problems with all three articles and the way how the merge was performed by a well-meaning but obviously under qualified person. So I guess I support your move. I will think a bit what can be done here. For starters, IMO the title Cycle (mathematics) must be made into a "subdisambig" page made of Cycle#Mathematics, since the qualifier is way too general. At the moment I am not familiar with English math terminology, so I am quite suspicious with seeing three quite distinct things supposedly called Cyclic permutation. It particular, it seems to confuse permutation cycle and cyclic permutation, not to say it has no references. Also, "i -> (i+k)(mod n)" is called shift permutation, cyclic shift permutationrotation permutation or simply rotation. Tangentially: in my olden days of Russian mathematics, a methodological distinction was drawn between permutations and substitutions (in the same way as graphs are not the same as (0,1)-matrices, although there is a 1-1 correspondence). I will think for more.`'Míkka 23:33, 18 July 2007 (UTC)
I agree with the unmerge and agree that Cycle (mathematics) should be a disambig. As for maths terminology, I am not surprised at all that three different things are given the same name, but I think that in the postscript, Míkka has been confused by the notation - it is definitely meant to be an acyclic permutation. JPD (talk) 11:14, 19 July 2007 (UTC)
The way the term cycle, in the sense of cycle of a function, is used by mathematicians, it is not restricted to bijections, unlike what is suggested by Cycle (mathematics). Let f : S → S be any function. Then a cycle of f, with period p, is an infinite sequence a0, a1, ..., such that ai+1 = f(ai) for i = 0, 1, ..., ak = a0 for some k > 0, where p is the least such value of k. See, for example, the use of the term in the Collatz conjecture article. If f is viewed as the state transition function of an automaton, this is the same as an infinite loop in programmers' parlance. --Lambiam 14:59, 19 July 2007 (UTC)
JPD wrote: "am not surprised at all that three different things are given the same name". Actually I am surprised. Maths used to be distinguished from all other domains by being least ambiguous and vague. Yes, some different things are called the same word, but usually they are in different branches of math, and the same word usually indicates the analogy. In this case we talking about a single very narrow domain of permutation theory/theory of permutations/permutation group theory (which even does not a Wikipedia article). So I smell that the naming mess is actually created by non-mathematicians. Therefore I suggest that whoever undertakes the job of cleaning the act here, please use definitions from reputable books, which are specifically devoted to permutations as a basis, not from random websites and arbitrary by-math articles. And of course, if some (mis)-usage is widespread, then we have to report it as well, but duly and clearly noting in which domain this alternative terminology is used. `'Míkka 16:22, 19 July 2007 (UTC)
P.S. Usually I don't waste my time in talk pages and put my words into deeds right away, but here I am slowing down. I used to work with permutations about 30 years ago (and even had a couple articles published), but since then I radically changed my interests, so I cannot call myself an expert here, and therefore I do not want to start rewriting myself (especially not having books in perms handy), but I will take part in this job. `'Míkka 17:12, 19 July 2007 (UTC)
The above discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.
I have reverted the last edit for several reasons.
1. Carmichael's definition of circular permutation (A permutation such as ... is called a circular permutation or a cyclic permutation. [and later] A circular permutation on two letters, such as (ab), is called a transposition. [and his first theorem is] Any given permutation is a product of circular permutations no two of which have a letter in common.) is what we call a cyclic permutation, with fixed points allowed - there is no question about this.
2. Expanding on the difference between cyclic and circular permutations (in the modern sense) was a good edit, but the improvement on this just muddied the waters. No cycle has a distinguished starting element, so to say that a circular permutation is a permutation without a distinguished starting element does not differentiate the two ideas. You have to bring in the circular arrangement versus the linear arrangement of the elements to see the difference.
3. I can not parse the statement about leaving circular permutations invariant by composing with cycles in any meaningful way. Perhaps the intent was to rotate the circular permutation?
I think the problems are coming from trying to mix the passive (= arrangement) and active (= mapping = substitution) forms of permutations. Cycles are active while circular permutations are passive. Bill Cherowitzo (talk) 04:57, 8 June 2014 (UTC)
Word not used in article. Equinox (talk) 18:54, 8 September 2015 (UTC)
As far as I can tell this is related to the above discussion on circular permutations. A circular permutation (or if you prefer a cyclic permutation with no fixed points) can have one of two "orientations", clockwise or counterclockwise. If clockwise is the preferred orientation for cyclic permutations then counterclockwise would be the other orientation, that is, an anti-cyclic permutation. This terminology seems to be used only in applications and almost always in the situation where only three things are permuted. The circular permutations on three elements are either cyclic or anti-cyclic. If there are more elements involved then a circular permutation does not have to be either cyclic or anti-cyclic. Why the redirect? - I think it stems from a basic confusion between the terms circular and cyclic that is to be found in the literature. What should we do about this? - Not sure, suggestions welcome. Bill Cherowitzo (talk) 21:29, 8 September 2015 (UTC)
A good starting-point is to look on Google Books or Google Scholar, and see whether there is any overwhelming similarity of usage; unfortunately I'm not mathematically competent enough to understand most papers of that kind. Equinox (talk) 02:24, 9 September 2015 (UTC)