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:
- a0 = a1 = 1;
- an = an − 1 + (n − 1)an − 2, for n > 1.
The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in OEIS); these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells.
Read more about this topic: Involution (mathematics)
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