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)
Famous quotes containing the words general and/or properties:
“In communist society, where nobody has one exclusive sphere of activity but each can become accomplished in any branch he wishes, society regulates the general production and thus makes it possible for me to do one thing today and another tomorrow, to hunt in the morning, fish in the afternoon, rear cattle in the evening, criticize after dinner, just as I have a mind, without ever becoming hunter, fisherman, shepherd or critic.”
—Karl Marx (18181883)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)