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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Even and odd permutations ： ウィキペディア英語版 Parity of a permutation In mathematics, when ''X'' is a finite set of at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of ''X'' is fixed, the parity (oddness or evenness) of a permutation $\backslash sigma$ of ''X'' can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements of ''X'' such that The sign or signature of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group ''S''''n''. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol ($\backslash epsilon\_\backslash sigma$), which is defined for all maps from ''X'' to ''X'', and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as : where ''N''(σ) is the number of inversions in σ. Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as : where is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well-defined.〔Jacobson (2009), p. 50.〕 == Example == Consider the permutation σ of the set } which turns the initial arrangement 12345 into 34521. It can be obtained by three transpositions: first exchange the places of 1 and 3, then exchange the places of 2 and 4, and finally exchange the places of 1 and 5. This shows that the given permutation σ is odd. Using the notation explained in the Permutation article, we can write : 3&4&5&2&1\end = \begin1&3&5\end \begin2&4\end = \begin1&5\end \begin1&3\end \begin2&4\end. There are many other ways of writing σ as a composition of transpositions, for instance :, but it is impossible to write it as a product of an even number of transpositions. 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「Parity of a permutation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.