Skew and direct sums of permutations

A sum of permutation is a combinatorial combination of two permutations, gives rise to a new permutation. The length of the Ergebnispermutation corresponds to the sum of the lengths of the two Ausgangspermutationen. There are two ways of summing, the direct sum and the sum crooked. In the direct sum the second permutation is appended moved to the top, moved the first permutation in the oblique sum of the second prefixed. Zuzugehörigen the permutation matrices have a similar block structure.

The formation of purely direct or purely slate sums of permutations is associative, mixed direct and oblique sums, however, the associative law does not hold in general. Sums of complementary or reverse permutations can be represented by sums of Ausgangspermutationen. The inverse of a sum of permutations is the sum of inverses. Direct and oblique sums of permutations play an important role in the decomposition of permutations into their basic building blocks and in the characterization of separable permutations.

Definition

If the symmetric group of permutations of length and and two permutations in Tupelschreibweise not necessarily the same length, then their direct sum (English direct sum) is

And their crooked sum (English skew sum) by

Defined. Tupelschreibweise in the second permutation is thus shifted to the first and attached at an incline, the first sum of permutation shifted to the second prefixed with a direct sum.

Examples

The direct sum of two identical permutations and is given by

While their sum by oblique

Is given.

Matrix representation

Is the related to permutation permutation, then the permutation matrix is the direct sum of two permutations and a block matrix of the form

And the permutation of the corresponding oblique sum of a block matrix of the form

Here, each stands for a zero matrix of appropriate size. For example, if and, then results

Properties

Associativity

The formation of purely direct and purely slate buzz is associative, that is, for permutations, and applies

And

However, the associative law generally does not apply to mixed direct and oblique sums, as the example

Shows. And the commutative law is not satisfied in general.

Symmetries

The complementary to a permutation permutation. For the complement of the sum of two permutations and

As well as

Accordingly, the reverse permutation to a permutation. For the reverse, and is the sum of two permutations

As well as

The corresponding permutation matrices are mirrored according to a horizontal or vertical axis.

Inverse

For the inverse of the sum of two permutations and results

As well as

The corresponding permutation matrices are each mirrored at the diagonal, ie transposed.

Use

Direct and oblique sums play an important role in the decomposition of permutations into their basic building blocks in combinatorics. Such a decomposition is not necessarily unique, however, due to the associativity of summation. Those permutations that can be fully represented as a direct sum of trivial or crooked permutations are called separable permutations. The number of separable permutations of length is given by the ( large ) Schröder numbers ( sequence A006318 in OEIS ). Separable permutations are characterized by a special recursive block structure of the corresponding permutation matrices. You are investigated, among others, in the sort theory.

Direct and oblique sums also appear in the study of Permutationsmustern (English permutation pattern ). The decomposition of permutations in not indivisible Teilpermutationen allowed the characterization and enumeration of certain pattern classes.