Permutation

In terms of Latin squares, an isotopy (α, β, γ) is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ. An autotopy is an isotopy from a quasigroup to itself.

Thus the totally antisymmetric Levi-Civita symbol extends the signature of a permutation, by setting for any permutation σ of n, and when no permutation σ exists such that for (or equivalently, whenever some pair of indices are equal).

Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant.

Any such homomorphism is called a (permutation) representation of G on M. For any permutation group, the action that sends (g, x) → g(x) is called the natural action of G on M. This is the action that is assumed unless otherwise indicated.

A permutation box (P-box) is a permutation of all the bits: it takes the outputs of all the S-boxes of one round, permutes the bits, and feeds them into the S-boxes of the next round.

During the course of a permutation fugue, it is quite uncommon, actually, for every single possible voice-combination (or "permutation") of the themes to be heard.

In these applications, the ordered arrangement view of a permutation is needed to talk about the positions in a permutation.

In two-line notation, the product of two permutations is obtained by rearranging the columns of the second (leftmost) permutation so that its first row is identical with the second row of the first (rightmost) permutation.

More generally, any permutation of the rows or columns multiplies the determinant by the sign of the permutation.

One advantage of this method is that the small amount of change from one permutation to the next allows the method to be implemented in constant time per permutation.

Other uses of the term permutation The concept of a permutation as an ordered arrangement admits several generalizations that are not permutations but have been called permutations in the literature.

Substitution-permutation networks A sketch of a Substitution-Permutation Network with 3 rounds, encrypting a plaintext block of 16 bits into a ciphertext block of 16 bits.

The method goes back to Narayana Pandita in 14th century India, and has been frequently rediscovered ever since. citation The following algorithm generates the next permutation lexicographically after a given permutation.

The multiplication is easily expressed in terms of the representation ring : the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module.

The product can then be written as the first row of the first permutation over the second row of the modified second permutation.

The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd.

The same can also easily generate the subset of even permutations, again in constant time per permutation, by skipping every other output permutation.

This convention is commonly used in the permutation group literature, but this article uses the convention where the rightmost permutation is applied first.

This permutation could also be written as: : Permutations are also often written in cyclic notation (cyclic form) especially when the algebraic properties of the permutation are of interest.

To use it, one starts by sorting the sequence in (weakly) increasing order (which gives its lexicographically minimal permutation), and then repeats advancing to the next permutation as long as one is found.