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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 ...
The size n of the orbit is called the length of the corresponding cycle; when n = 1, the single element in the orbit is called a fixed point of the permutation. A permutation is determined by giving an expression for each of its cycles, and one notation for permutations consist of writing such expressions one after another in some order.
Cycle notation describes the effect of repeatedly applying the permutation on the elements of the set S, with an orbit being called a cycle. The permutation is written as a list of cycles; since distinct cycles involve disjoint sets of elements, this is referred to as "decomposition into disjoint cycles".
The composition of permutations, when they are written in cycle notation, is obtained by juxtaposing the two permutations (with the second one written on the left) and then simplifying to a disjoint cycle form if desired. Thus, the above product would be given by: = () = ().
The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index. Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action.
The notation in parentheses is the cycle notation. ... In terms of permutations the two group elements of G / A 3 are the set of even permutations and the set of odd ...
Cycle index; Cycle notation; Cycles and fixed points; Cyclic order; Direct sum of permutations; Enumerations of specific permutation classes; Factorial. Falling factorial; Permutation matrix. Generalized permutation matrix; Inversion (discrete mathematics) Major index; Ménage problem; Permutation graph; Permutation pattern; Permutation ...
These identities may be derived by enumerating permutations directly. For example, a permutation of n elements with n − 3 cycles must have one of the following forms: n − 6 fixed points and three two-cycles; n − 5 fixed points, a three-cycle and a two-cycle, or; n − 4 fixed points and a four-cycle.