The first few terms of this sequence are
232 (sequence A000085 in the
OEIS); these numbers are called the
telephone numbers, and they also count the number of
Young tableaux with a given number of cells.
The number can also be expressed by non-recursive formulas, such as the sum
Some basic examples of involutions include the functions
and more generally the function
is an involution for constants and that satisfy
These are not the only pre-calculus involutions. Another one within the positive reals is
graph of an involution (on the real numbers) is
symmetric across the line . This is due to the fact that the inverse of any general function will be its reflection over the line . This can be seen by "swapping" with . If, in particular, the function is an involution, then its graph is its own reflection.
An involution is a
projectivity of period 2, that is, a projectivity that interchanges pairs of points.: 24
Any projectivity that interchanges two points is an involution.
The three pairs of opposite sides of a
complete quadrangle meet any line (not through a vertex) in three pairs of an involution. This theorem has been called
Desargues's Involution Theorem. Its origins can be seen in Lemma IV of the lemmas to the Porisms of Euclid in Volume VII of the Collection of
Pappus of Alexandria.
If an involution has one
fixed point, it has another, and consists of the correspondence between
harmonic conjugates with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called double points.: 53
Another type of involution occurring in projective geometry is a polarity which is a
correlation of period 2.
In linear algebra, an involution is a linear operator T on a vector space, such that . Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1s and −1s on the diagonal of the corresponding matrix. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.
For example, suppose that a basis for a vector space V is chosen, and that e1 and e2 are basis elements. There exists a linear transformation f which sends e1 to e2, and sends e2 to e1, and which is the identity on all other basis vectors. It can be checked that f(f(x)) = x for all x in V. That is, f is an involution of V.
For a specific basis, any linear operator can be represented by a
matrixT. Every matrix has a
transpose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices.
The definition of involution extends readily to
modules. Given a module M over a
ringR, an Rendomorphismf of M is called an involution if f2 is the identity homomorphism on M.
Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; i.e., group was taken to mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution.
Generally in non-classical logics, negation that satisfies the law of double negation is called involutive. In algebraic semantics, such a negation is realized as an involution on the algebra of
truth values. Examples of logics which have involutive negation are Kleene and Bochvar
Łukasiewicz many-valued logic,
fuzzy logic IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in
t-norm fuzzy logics.
XORbitwise operation with a given value for one parameter is an involution. XOR
masks were once used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state. The
NOT bitwise operation is also an involution, and is a special case of the XOR operation where one parameter has all bits set to 1.
Another example is a bit mask and shift function operating on color values stored as integers, say in the form RGB, that swaps R and B, resulting in the form BGR.
RC4 cryptographic cipher is an involution, as encryption and decryption operations use the same function.
Practically all mechanical cipher machines implement a
reciprocal cipher, an involution on each typed-in letter.
Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.