Inversion (discrete mathematics)

Under faulty state or inversion of a permutation is understood in combinatorics a pair of elements of an ordered set, the order will be reversed by the permutation. The number of failed objects is called a permutation wrong state number or inversion number of the permutation. About the erroneous state number can be the sign of a permutation determine where an even permutation is an even number and an odd failure state permutation has an odd number of erroneous state.

There are different ways to represent the missing objects of a permutation, for example via the inversion table, the Lehmer code or a Rothe diagram. Summing up the entries of the inversion table or the Lehmer codes as a number in a faculty- based number system, each permutation can be assigned a unique number. Next can be defined on the set of permutations of a partial order using the incorrect items.

After the failure state number of a permutation can be regarded as a measure of the disorder of the permuted by the permutation numbers, missing objects play an important role in the analysis of sorting algorithms.

• 3.1 mis stand number
• 3.2 distribution
• 3.3 Generating function
• 3.4 Expectation and Variance

• 4.1 Inversion Table
• 4.2 Lehmer code
• 4.3 Rothe diagram
• 4.4 Permutationsgraph

• 5.1 Enumeration of permutations
• 5.2 Arrangement of permutations

Definition

Is the symmetric group of all permutations of the set, then a failure state of a permutation is a pair for which

Applies. The number of missing objects of a permutation is then given by

Given. Occasionally, the pair is referred to as a false state in the literature instead of the couple.

General also permutations of arbitrary finite ordered sets can be considered for the mathematical analysis can, however, on the first natural numbers limit.

Examples

A concrete example

The number of missing items of the permutation

Is

You can identify these five false items in that up all the numbers you are looking for in the second row for each number of which are larger and are left of the number. In the example, the pairs and. The defect levels are then respectively associated number of pairs of the first line.

More general examples

The identity permutation is the only permutation without blemish stands, so

A Nachbarvertauschung generates exactly one incorrect state

A transposition with has the following faulty items:

Number

Incorrect level number

The number of failed objects is called a permutation wrong state number or inversion number of the permutation. The failure state number can be viewed as a measure of the disorder of the permuted by the permutation numbers. About the erroneous state number can be the sign of a permutation determine, because it is

If the MI is stand number is even, then one speaks of a straight permutation, otherwise by an odd permutation. The failure state of the inverse number of permutation is the same as the prior number of defects Ausgangspermutation, i.e.

Because the number of missing items of the inverse permutation has the representation

Distribution

The number of permutations with exactly mis - digit levels is defined as

After the identity permutation is the only permutation without blemish stands, applies to everyone. Since there are Nachbarvertauschungen with exactly one incorrect state, is on for all. The maximum number of a faulty stand - digit permutation

And is assumed accurate for that permutation which reverses the order of all the numbers. Furthermore, the symmetry applies

With the Convention and the numbers satisfy the recursion ( A008302 in OEIS sequence )

And the sum representation

Generating function

The generating function for the number of incorrect items has the relatively simple form

This result goes back to Olinde Rodrigues ( 1839).

Expected value and variance

The expected value of the missing status of a number ( uniformly distributed ) random permutation is

Why sorting algorithm bubblesort, the exact fix a faulty reading per step, not only in the worst case, but also in the average case have a square term. For the variance of the faulty state number of a random permutation applies accordingly

Whereby the standard deviation of the number of precipitates with a failure state value of about relatively large. The number of incorrect items a random permutation is asymptotically normally distributed for.

Representations

Inversion table

The inversion table or inversion vector of a permutation maps each number to the number of incorrect items that it produces. Refers to

The number of numbers that are in the Tupeldarstellung from left to and greater than, then the inversion table is a permutation of the vector

The number can generate a maximum of faulty items, and thus always applies. The failure state number of permutation is then obtained as the sum

From the inversion table is the underlying permutation can be inversely determined. To this end, we determined the sequence, the relative rankings of numbers, each indicating the position at which the number occurring within the already considered numbers. This is for the first digit, the second digit, and so on. This one-to -one correspondence of permutation and related inversion table is of great practical importance, as can be combinatorial problems related to permutations through consideration of inversion tables are often resolved more easily. The reason for this is that the entries of the inversion table within the specified limits can be chosen independently of each other, while the numbers must be different pairs.

Example

In the example above the inversion table

From the inversion table will be returned in the underlying permutation, by detecting the following measures in order:

Lehmer code

In a sense, the dual of the inversion table of the Lehmer code ( named after Derrick Lehmer ), which also summarizes the errors objects to a permutation. Refers to

The number of numbers that are in the Tupeldarstellung of the right of and smaller than, then the Lehmer code of a permutation of the vector

Also here and thus always. The failure state number of permutation results accordingly as the sum

From the Lehmer code also the underlying permutation can be determined. To this end, we first listed all the numbers up one behind the other. Below is removed from this list each in -th step, the - th number and they quoted it as an. Again, there is a one-to -one correspondence between the permutation and the associated Lehmer code.

Example

In the example above the Lehmer code

From the Lehmer code will be returned in the underlying permutation, by detecting the following measures in order:

Rothe diagram

Another way to represent the missing objects of a permutation is the Rothe diagram ( named after Heinrich August Rothe ). In a regimen consisting of fields that column is marked with a dot initially in each row applies. These fields correspond to just the entries with value of the corresponding permutation matrix. The incorrect readings of the permutation then correspond to those fields that both a point below in the same column, as also a point right in the same row. These fields are marked with a cross. In this way a field is marked with a cross if and only if a failure state of being.

From the Rothe diagram, both the inversion table, as well as the Lehmer code can be read. The number corresponds exactly to the number of crosses in the column and the number of number of crosses in the line. If you transpose the chart ( ie interchanging the rows and columns ), one obtains a representation of the false readings of the associated inverse permutation. Does the Rothe diagram of a permutation in the field a cross on, then this is true for the graph of the corresponding inverse permutation in the field. Therefore applies Due to the symmetry property of the Rothe diagram for the inverse permutation

For self- inverse permutations, ie, permutations, applies to the, therefore agree inversion table and Lehmer code.

Permutationsgraph

Each permutation can using the wrong items, a Permutationsgraph (not to be confused with the graph representation of a permutation ) are assigned. The Permutationsgraph a permutation is an undirected graph with vertex set

And the edge set

The edges of the Permutationsgraphen so connect those pairs of numbers that provide an incorrect status. Permutationsgraphen can also geometrically as a graph of routes

Be defined for. The end points of these lines lie on two parallel lines and two lines intersect if and only if the numbers at the endpoints generate an incorrect state. Permutationsgraphen may also be characterized in that both the graph, as well as its Komplementgraph are Vergleichbarkeitsgraphen. The Komplementgraph corresponds to the Permutationsgraphen the reverse permutation.

Example

For example, has the Permutationsgraph the permutation the edge set

Use

Enumeration of permutations

Summing up the inversion table or the Lehmer code as a number in a faculty- based number system, each permutation can be assigned a unique number in the crowd. From the inversion table the number is obtained as

And from the Lehmer code number

These two numbers are only correct match for even inverse permutations. Other variants for numbering the permutations exist by looking at the number of couples that take place in the wrong state definition and / or place to meet. These pairs of numbers then correspond to the Rothe diagram crosses the right instead of the left or below rather than above the points. The vectors consisting of the sums of crosses per row or column can be interpreted as numbers in a faculty- based number system then also.

Example

For the permutation obtained from the corresponding inversion table, the number

And from the associated Lehmer code number

Arrangement of permutations

Next can be specified by considering the faulty items on the set of digit permutations is a partial order. Such an order relation is for permutations by

Defined. Two permutations are doing in relation when the number of missing items of the first permutation is a subset of the faulty state amount of the second permutation. The minimum element with respect to this order is the identity permutation, while the maximal element is the one permutation which reverses the order of all the numbers.

Graphically, this can order relation illustrate using a Hasse diagram. Two permutations are connected by an edge if they proceed through a Nachbarvertauschung apart. The nodes and edges of the Hasse diagram form a Cayley graph which is isomorphic to the line graph of the corresponding Permutaeders.

Example

In the adjacent Hasse diagram of permutations of the symmetric group, the smallest permutation with respect to this order is at the bottom and the largest permutation at the top. Blue, green and red edges correspond to the Nachbarvertauschungen, and, seen from below upwards always exactly produce a false state.

History

The concept of mis -matter of a permutation was introduced in 1750 by Gabriel Cramer in his work Introduction à l' analyze the lignes courbes algébriques. As part of the eponymous cramer 's rule to specify the solution of linear systems of equations, he defined the determinant of a square matrix by

Wherein the sum runs over all digit permutations. Cramer's rule was the impetus for the development of a comprehensive theory of determinants.

For the concept of the false object different terms were used in the course of time. Cramer himself called faulty items as Derangement ( permutation ), Pierre -Simon Laplace in 1772 used the term variation ( change ) and Joseph Gergonne eventually led in 1813 the term inversion ( inversion) a, which is now mainly used in the English-speaking world. The German term " failure state " was popularized in the early 20th century by Gerhard Kowalewski.