Talk:Cyclic permutation

message: I added a reference to stirling numbers. -  21:57, 22 June 2005 (UTC)
thread_title: Stirling numbers

message: Maybe I'm just being stupid but I don't understand what Offset means from this article as it isn't very clear... - Eraserhead1 14:30, 31 December 2005 (UTC)
thread_title: Offset

message: 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)
thread_title: Definition in terms of an example

message: :The result was No consensus to merge -- Jreferee (Talk) 05:22, 17 August 2007 (UTC)
replies: text: I transcribed the discussion below.--Jorfer 20:41, 30 July 2007 (UTC)
thread_title:  Merge tag 

message: 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)
thread_title:  I've reverted Merge 

message: : I transcribed the discussion below.--Jorfer 20:41, 30 July 2007 (UTC)
thread_title:  From Wikipedia talk:WikiProject Mathematics 

message: 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)
thread_title:  Merge of [[Cycle (mathematics)]], into [[Cyclic permutation]] 

message: 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)
thread_title:  circular permutation 

message: Word not used in article. Equinox (talk) 18:54, 8 September 2015 (UTC)
replies: text: 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)
replies: text:  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)
replies: text: Fitzpatrick uses the term to denote a shift of two elements.  See p. 18 of "Maxwell's equations and the principles of electromagnetism"  — Preceding unsigned comment added by 192.249.3.208 (talk) 14:56, 8 November 2019 (UTC)
thread_title:  Why does "anticyclic" redirect here? 

message: The above section is getting pretty hard to follow, so I'm starting a new section.  Summarizing what I and others have written above, I propose that we define the terms in terms of orbits.
* A cyclic permutation of a set  is a permutation of  for which every element of  is in the same orbit.  (In particular, there are  cyclic permutations of a set  with .  Furthermore, the identity permutation is a cyclic permutation only when .)
* A cycle of a set  is a permutation of  that has exactly one non-trivial orbit, where a non-trivial orbit is an orbit with size at least 2.  (In particular, the identity permutation is never a cycle, which is good because adding it to any cycle decomposition for a permutation would give a distinct cycle decomposition for that permutation.)
* A cycle decomposition is a representation of a permutation as the product of disjoint cycles.  Cycles are disjoint if the elements in their non-trivial orbits are disjoint.  (In particular, the cycle decomposition of the identity permutation is the (empty) product of zero cycles.  Furthermore, a cycle decomposition is unique up to rearrangement of the disjoint cycles.)
* I know that circular shift hasn't gotten much attention in the discussion here.  Perhaps we could leave it out.  Alternatively, those of you with access to relevant publications might give a summary of what is out there in the literature.  Possibilities include:
:# A circular shift of an ordered set  is a permutation of  of the form  for some }.  (In particular, it is a cyclic permutation only when  and  are relatively prime.)
:# A circular shift of a set  is a permutation of  that is some integer power of a cyclic permutation.  (In particular, the identity permutation for any  with  is a circular shift because it is the power 0 of any cyclic permutation.  Furthermore, every orbit of a circular shift is of the same size (at least when  is finite).)
* What is a circular permutation, if anything?
—Quantling (talk | contribs) 19:31, 12 July 2023 (UTC)
thread_title:  In terms of orbits