"circle, wheel, any circular body," also "circular motion, cycle of events," = Example 1 =įor example, suppose we have an ordered set of objects $\)$$ The etymology of the word "cyclic" or "cycle" comes from the Greek "kyklos" which means With this terminology, a circular permutation is just exactly a permutation consisting of a single cycle that permutes all of the objects. The standard important result is that any permutation is the composition of disjoint cycles in a unique way. = Cycles =Īnother way to state this is that in an extended sense, a cycle is a permutation of $n$ objects that is a circular permutation on a subset of the objects and leaves the others fixed, whereas in a strict sense no object is fixed. The context should make it clear which meaning is being used. Which means it is any composition power of a circular permutation and is a different meaning than the Wikipedia definition because an ordering of elements is assumed and there is an offset which, if it is not one, may produce multiple disjoint cycles. The Mathworld article Cyclic permutation statesĪ permutation which shifts all elements of a set by a fixed offset, with the elements shifted off the end inserted back at the beginning. The Wikipedia article Cyclic permutation statesĪ cyclic permutation (or cycle) is a permutation of the elements of some set X which maps the elements of some subset $S$ of $X$ to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of $X$. In other words a circular permutation on a set makes it circularly ordered in the sense that the permutation maps each element to its immediate successor in the circular ordering. Since there are $n!$ possible linear orders, there are $(n−1)!$ possible cyclic ordersĪnd there is much more interesting information in the article. Choosing a linear order is equivalent to choosing a first element, so there are exactly $n$ linear orders that induce a given cyclic order. The Wikipedia article Cyclic order states:Ī set with a cyclic order is called a cyclically ordered set or simply a cycle.Ī cyclic order on a set $X$ can be determined by a linear order on $X$, but not in a unique way. MathWorld-A Wolfram Web Resource.In the context of an linear ordered set $n$ objects, a circular permutation takes the first object to the second, the second to the third, and so on until the last object gets taken to the first object which is precisely what makes the permutation circular. On Wolfram|Alpha Permutation Cite this as: Skiena,ĭiscrete Mathematics: Combinatorics and Graph Theory with Mathematica. "Permutations: Johnson's' Algorithm."įor Mathematicians. "Permutation Generation Methods." Comput. Knuth,Īrt of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. "Generation of Permutations byĪdjacent Transpositions." Math. "Permutations by Interchanges." Computer J. "Arrangement Numbers." In Theīook of Numbers. The permutation which switches elements 1 and 2 and fixes 3 would be written as (2)(143) all describe the same permutation.Īnother notation that explicitly identifies the positions occupied by elements before and after application of a permutation on elements uses a matrix, where the first row is and the second row is the new arrangement. There is a great deal of freedom in picking the representation of a cyclicĭecomposition since (1) the cycles are disjoint and can therefore be specified inĪny order, and (2) any rotation of a given cycle specifies the same cycle (Skienaġ990, p. 20). This is denoted, corresponding to the disjoint permutation cycles (2)Īnd (143). The unordered subsets containing elements are known as the k-subsetsĪ representation of a permutation as a product of permutation cycles is unique (up to the ordering of the cycles). (Uspensky 1937, p. 18), where is a factorial.
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