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A cyclic permutation consisting of a single 8-cycle. There is not widespread consensus about the precise definition of a cyclic permutation. Some authors define a permutation σ of a set X to be cyclic if "successive application would take each object of the permuted set successively through the positions of all the other objects", [1] or, equivalently, if its representation in cycle notation ...
An arrangement of distinct objects in a circular manner is called a circular permutation. [ 39 ] [ e ] These can be formally defined as equivalence classes of ordinary permutations of these objects, for the equivalence relation generated by moving the final element of the linear arrangement to its front.
There are a few equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes all of X into a single cycle, which is a special type of permutation - a circular permutation. Alternatively, a cycle with n elements is also a Z n-torsor: a set with a free transitive action by a finite cyclic group. [1]
In linear algebra, a circulant matrix is a square matrix in which all rows are composed of the same elements and each row is rotated one element to the right relative to the preceding row. It is a particular kind of Toeplitz matrix .
Furthermore, every permutation can be written as a product of adjacent transpositions, that is, transpositions of the form (a a+1). For instance, the permutation g from above can also be written as g = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5). The sorting algorithm bubble sort is an application of this fact. The representation of a permutation as a ...
A circular shift is a special kind of cyclic permutation, which in turn is a special kind of permutation. Formally, a circular shift is a permutation σ of the n entries in the tuple such that either
A rotation system specifies a circular ordering of the edges around each vertex, while a rotation map specifies a (non-circular) permutation of the edges at each vertex. In addition, rotation systems can be defined for any graph, while as Reingold et al. define them rotation maps are restricted to regular graphs.
A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]