Signs of
The order of the elements	+	-	+
The composition of the elements	-	+	+

Permutation

6 permutations of 3 balls

In combinatorics, transposition is an ordered set of numbers. usually interpreted as a bijection on a set The i- th element from the set corresponds to i . The number n is here called the permutation order . As a synonym for the word "permutation" in this sense, some authors use the word arrangement .

In group theory, a permutation of an arbitrary set means a bijection of this set onto itself. As a synonym for the word "permutation" in this sense, some authors use the word substitution . (Other authors refer to substitution as a visual way of recording a permutation.)

Content

• 1 Properties
• 2 Related definitions
  • 2.1 Special types of permutations
  • 2.2 Substitution
  • 2.3 Cycle products and permutation signs
• 3 Permutations with repetition
• 4 Random swap
• 5 See also
• 6 Notes
• 7 Literature
• 8 References

Properties

• The number of all permutations of order equal to the number of allocations from n to n , that is, factorial: [1] [2] [3] [4]

• The composition defines the operation of a product on permutations of one order: Regarding this operation, the set of permutations of order n forms a group, which is called symmetric and is usually denoted .
• Any group is a subgroup of the permutation group of the set of elements of this group (Cayley's theorem). In addition, each element matches the permutation given by identity where g is an arbitrary element of the group G , and - group operation.

Related definitions

• Permutation media Is a subset of the set defined as
• Fixed point permutation is any fixed point display that is, the element of the set The set of all fixed permutation points is an addition to its carrier in .
• Inversion permutation order n is called every pair of indices such that and . The parity of the number of inversions in a permutation determines the parity of the permutation .

Special types of permutations

• Identical permutation - permutation which is each item displays in itself:
• Involution - permutation which is the reverse of itself, that is
• Disorder is a permutation without fixed points.
• Cycle length called such a substitution which is identical on the whole set except a subset and Denoted by . The number of permutations containing k cycles is Stirling numbers of the first kind.
• Transposition - permutation of elements of the set which swaps two elements. Transposition is a cycle of length 2.

Substitution [edit]

Permutation sets can be written as a substitution , for example:

Where and

Cycle products and permutation sign [edit]

Any permutation can be decomposed into a product (composition) of disjoint length cycles and uniquely up to the order of the cycles in the product. For example:

Any cycle can be decomposed into a product of (not necessarily disjoint) transpositions. For arbitrary length cycle decomposition can be written like this: Loops of length 1 act as the identity permutation and can also be easily decomposed, since the square of any transposition is the identity permutation: Such a decomposition of cycles into a product of transpositions will not be the only one:

Thus, any permutation can be decomposed into a product of transpositions. Although this decomposition will not be unique, but the parity of the number of transpositions included in the decomposition is preserved. Let the permutation laid out in the product transpositions, then the permutation sign (otherwise: the permutation parity or permutation signature ) call the number wherein called an even permutation if and an odd permutation if

Permutation Sign can also be determined through the number of inversions in this permutation:

Comment. There are two conventions for multiplying permutations and cycles:

one) .

For example: .

2) .

For example: .

Repetition permutations

Consider n elements of m different types, and in each type all elements are the same. Then the permutations of all these elements up to the order of the sequence of similar elements are called permutations with repetition . If k i is the number of elements of the i -th type, then and the number of various permutations with repetitions is equal to the multinomial coefficient

Random permutation

Main article: Generalized layout

Random permutation is called a random vector. all elements of which take natural values ​​from 1 to and the probability of coincidence of any two elements is 0.

An independent random permutation is such a random permutation. , for which

for some such that

If at the same time do not depend on then the permutation called equally distributed . If there is no dependence on , i.e that called homogeneous .

Accommodation

In combinatorics, an arrangement (from n to k ) is an ordered set of k different elements from some set of different n elements.

Example 1: - This is a 4-element placement from a 6-element set. .

Example 2: some placements of the elements of a set on 2: ... ... ...

Unlike combinations, placements take into account the order of the objects. So, for example, sets and are different though they consist of the same elements. (i.e., match as combinations).

Content

• 1 Number of placements
• 2 Placement with repetitions
  • 2.1 Number of placements with repetitions
• 3 See also
• 4 References

Number of placements

The number of allocations of n for k , denoted by , is equal to decreasing factorial:

The last expression has a natural combinatorial interpretation: each placement from n to k uniquely corresponds to some combination of n to k and some rearrangement of the elements of this combination; the number of combinations of n in k is equal to the binomial coefficient , while permutations on k elements are exactly k! pieces

For k = n, the number of placements is equal to the number of permutations of order n :

Repetition posting

Posting or returning is the placement of “items” under the assumption that each “item” can participate in the placement several times.

Number of placements with repetitions

By the multiplication rule, the number of arrangements with repetitions of n by k , denoted by , equals:

For example, the number of variants of a 3-digit code, in which each character is a digit from 0 to 9 and can be repeated, is:

Another example: placements with repetitions of 4 elements a, b, c, d with 2 are these accommodations are as follows:

Combination

In combinatorics, the combination of by called a set elements selected from this set containing various items. Sets that differ only in the order of the elements (but not the composition) are considered to be the same, this combination differs from placements.

So, for example, sets (3-element combinations, subsets, ) {2, 1, 3} and {3, 2, 1} of the 6-element set {1, 2, 3, 4, 5, 6} ( ) are the same (while the placement would be different) and consist of the same elements {1,2,3}.

In general, a number indicating how many ways you can choose elements from the set containing different elements standing at the intersection th diagonal and Pascal's triangle line. [one]

Content

• 1 Number of combinations
• 2 Combinations with repetitions
• 3 See also
• 4 Notes
• 5 References

Number of combinations

Main article: Binomial coefficient

Combination of by equal to the binomial coefficient

With fixed generating function of a sequence of numbers of combinations , , , … is an:

The two-dimensional generating function of the numbers of combinations is

Combinations with repetitions

Combination with repetitions are called sets, in which each element can participate several times.

The number of combinations with repetitions from by equal to the binomial coefficient

Evidence

Let there be object types, and objects of the same type are indistinguishable. Suppose there is an unlimited (or large enough, in any case, no less than ) the number of objects of each type. From this range, choose objects; The sample can contain objects of the same type, the order of selection does not matter. Denote by number of selected objects of type i, , . Then .But the number of solutions of this equation is easily calculated with the help of “balls and partitions”: each solution corresponds to the arrangement of balls and partitions in a series so that exactly balls are located between the -th and -th partitions . But such arrangements are exactly what was required to prove. ■

With a fixed generating function of numbers of combinations with repetitions from by is an:

The two-dimensional generating function of numbers of combinations with repetitions is: