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In mathematics, an (anti-)involution, or an involutory function, is a function that is its own inverse, : for all in the domain of . ==General properties== Any involution is a bijection. The identity map is a trivial example of an involution. Common examples in mathematics of more detailed involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other examples include circle inversion, rotation by a half-turn, and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The number of involutions, including the identity involution, on a set with ''n'' = 0, 1, 2, … elements is given by a recurrence relation found by Heinrich August Rothe in 1800: :''a''0 = ''a''1 = 1; :''a''''n'' = ''a''''n'' − 1 + (''n'' − 1)''a''''n'' − 2, for ''n'' > 1. The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 ; these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells.〔.〕 The composition of two involutions f and g is an involution if and only if they commute: .〔.〕 Every involution on an odd number of elements has at least one fixed point. More generally, for an involution on a finite set of elements, the number of elements and the number of fixed points have the same parity.〔.〕 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「Involution (mathematics)」の詳細全文を読む スポンサード リンク
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