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In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration. Each permutation π of a finite set of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables ''a''1, ''a''2, … that describes the type of this partition (the cycle type of π): the exponent of ''a''''i'' is the number of cycles of π of size ''i''. 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. This is the main ingredient in the Pólya enumeration theorem. Performing formal algebraic and differential operations on these polynomials and then interpreting the results combinatorially lies at the core of species theory. ==Permutation groups and group actions== Let ''X'' be a set. A bijective map from ''X'' onto itself is called a permutation and the set of all permutations of ''X'' forms a group under the composition of mappings, called the Symmetric group of ''X'', Sym(''X''). Every subgroup of Sym(''X'') is called a permutation group of ''degree'' |''X''|. Let ''G'' be an abstract group with a group homomorphism,φ, from ''G'' into Sym(''X''). The image, φ(''G''), is a permutation group. The group homomorphism can be thought of as a means for permitting the group ''G'' to "act" on the set ''X'' (using the permutations associated with the elements of ''G''). Such a group homomorphism is formally called a group action and the image of the homomorphism is a ''permutation representation'' of ''G''. A given group can have many different permutation representations, corresponding to different actions. Suppose that group ''G'' ''acts'' on set ''X'' (that is, a group action exists). In combinatorial applications the interest is in the set ''X''; for instance, counting things in ''X'' and knowing what structures might be left invariant by ''G''. Little is lost by working with permutation groups in such a setting, so in these applications, when a group is considered, it is a permutation representation of the group which will be worked with, and thus, a group action must be specified. Algebraists, on the other hand, are more interested in the groups themselves and would be more concerned with the kernels of the group actions, which measure how much is lost in passing from the group to its permutation representation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cycle index」の詳細全文を読む スポンサード リンク
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