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Combination and placement permutations (with and without repetitions)

Lecture



Signs of Combination and placement permutations (with and without repetitions) Combination and placement permutations (with and without repetitions) Combination and placement permutations (with and without repetitions)
The order of the elements + - +
The composition of the elements - + +

Combination and placement permutations (with and without repetitions)

Permutation

Combination and placement permutations (with and without repetitions)

6 permutations of 3 balls

In combinatorics, transposition is an ordered set of numbers. Combination and placement permutations (with and without repetitions) usually interpreted as a bijection on a set Combination and placement permutations (with and without repetitions) 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 Combination and placement permutations (with and without repetitions) equal to the number of allocations from n to n , that is, factorial: [1] [2] [3] [4]

Combination and placement permutations (with and without repetitions)

  • The composition defines the operation of a product on permutations of one order: Combination and placement permutations (with and without repetitions) Regarding this operation, the set of permutations of order n forms a group, which is called symmetric and is usually denoted Combination and placement permutations (with and without repetitions) .
  • Any group is a subgroup of the permutation group of the set of elements of this group (Cayley's theorem). In addition, each element Combination and placement permutations (with and without repetitions) matches the permutation Combination and placement permutations (with and without repetitions) given by identity Combination and placement permutations (with and without repetitions) where g is an arbitrary element of the group G , and Combination and placement permutations (with and without repetitions) - group operation.

Related definitions

  • Permutation media Combination and placement permutations (with and without repetitions) Is a subset of the set Combination and placement permutations (with and without repetitions) defined as Combination and placement permutations (with and without repetitions)
  • Fixed point permutation Combination and placement permutations (with and without repetitions) is any fixed point display Combination and placement permutations (with and without repetitions) that is, the element of the set Combination and placement permutations (with and without repetitions) The set of all fixed permutation points Combination and placement permutations (with and without repetitions) is an addition to its carrier in Combination and placement permutations (with and without repetitions) .
  • Inversion permutation Combination and placement permutations (with and without repetitions) order n is called every pair of indices Combination and placement permutations (with and without repetitions) such that Combination and placement permutations (with and without repetitions) and Combination and placement permutations (with and without repetitions) . The parity of the number of inversions in a permutation determines the parity of the permutation .

Special types of permutations

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

Substitution [edit]

Permutation Combination and placement permutations (with and without repetitions) sets Combination and placement permutations (with and without repetitions) can be written as a substitution , for example:

Combination and placement permutations (with and without repetitions)

Where Combination and placement permutations (with and without repetitions) and Combination and placement permutations (with and without repetitions)

Cycle products and permutation sign [edit]

Any permutation Combination and placement permutations (with and without repetitions) can be decomposed into a product (composition) of disjoint length cycles Combination and placement permutations (with and without repetitions) and uniquely up to the order of the cycles in the product. For example:

Combination and placement permutations (with and without repetitions)

Any cycle can be decomposed into a product of (not necessarily disjoint) transpositions. For arbitrary length cycle Combination and placement permutations (with and without repetitions) decomposition can be written like this: Combination and placement permutations (with and without repetitions) 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: Combination and placement permutations (with and without repetitions) Such a decomposition of cycles into a product of transpositions will not be the only one: Combination and placement permutations (with and without repetitions)

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 Combination and placement permutations (with and without repetitions) laid out in the product Combination and placement permutations (with and without repetitions) transpositions, then the permutation sign (otherwise: the permutation parity or permutation signature ) Combination and placement permutations (with and without repetitions) call the number Combination and placement permutations (with and without repetitions) wherein Combination and placement permutations (with and without repetitions) called an even permutation if Combination and placement permutations (with and without repetitions) and an odd permutation if Combination and placement permutations (with and without repetitions)

Permutation Sign Combination and placement permutations (with and without repetitions) can also be determined through the number of inversions Combination and placement permutations (with and without repetitions) in this permutation: Combination and placement permutations (with and without repetitions)


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

one) Combination and placement permutations (with and without repetitions) .

For example: Combination and placement permutations (with and without repetitions) .

2) Combination and placement permutations (with and without repetitions) .

For example: Combination and placement permutations (with and without repetitions) .

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 Combination and placement permutations (with and without repetitions) and the number of various permutations with repetitions is equal to the multinomial coefficient Combination and placement permutations (with and without repetitions)

Random permutation

Main article: Generalized layout

Random permutation is called a random vector. Combination and placement permutations (with and without repetitions) all elements of which take natural values ​​from 1 to Combination and placement permutations (with and without repetitions) and the probability of coincidence of any two elements is 0.

An independent random permutation is such a random permutation. Combination and placement permutations (with and without repetitions) , for which

Combination and placement permutations (with and without repetitions)

for some Combination and placement permutations (with and without repetitions) such that

Combination and placement permutations (with and without repetitions)

Combination and placement permutations (with and without repetitions)

If at the same time Combination and placement permutations (with and without repetitions) do not depend on Combination and placement permutations (with and without repetitions) then the permutation Combination and placement permutations (with and without repetitions) called equally distributed . If there is no dependence on Combination and placement permutations (with and without repetitions) , i.e Combination and placement permutations (with and without repetitions) that Combination and placement permutations (with and without repetitions) 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: Combination and placement permutations (with and without repetitions) - This is a 4-element placement from a 6-element set. Combination and placement permutations (with and without repetitions) .

Example 2: some placements of the elements of a set Combination and placement permutations (with and without repetitions) on 2: Combination and placement permutations (with and without repetitions)Combination and placement permutations (with and without repetitions)Combination and placement permutations (with and without repetitions)Combination and placement permutations (with and without repetitions) ... Combination and placement permutations (with and without repetitions)Combination and placement permutations (with and without repetitions)Combination and placement permutations (with and without repetitions) ... Combination and placement permutations (with and without repetitions) ...

Unlike combinations, placements take into account the order of the objects. So, for example, sets Combination and placement permutations (with and without repetitions) and Combination and placement permutations (with and without repetitions) are different though they consist of the same elements. Combination and placement permutations (with and without repetitions) (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 Combination and placement permutations (with and without repetitions) , is equal to decreasing factorial:

Combination and placement permutations (with and without repetitions)

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 Combination and placement permutations (with and without repetitions) , 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 :

Combination and placement permutations (with and without repetitions)

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 Combination and placement permutations (with and without repetitions) , equals:

Combination and placement permutations (with and without repetitions)

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:

Combination and placement permutations (with and without repetitions)

Another example: placements with repetitions of 4 elements a, b, c, d with 2 are Combination and placement permutations (with and without repetitions) these accommodations are as follows:

Combination and placement permutations (with and without repetitions)

Combination

In combinatorics, the combination of Combination and placement permutations (with and without repetitions) by Combination and placement permutations (with and without repetitions) called a set Combination and placement permutations (with and without repetitions) elements selected from this set containing Combination and placement permutations (with and without repetitions) 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, Combination and placement permutations (with and without repetitions) ) {2, 1, 3} and {3, 2, 1} of the 6-element set {1, 2, 3, 4, 5, 6} ( Combination and placement permutations (with and without repetitions) ) 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 Combination and placement permutations (with and without repetitions) elements from the set containing Combination and placement permutations (with and without repetitions) different elements standing at the intersection Combination and placement permutations (with and without repetitions) th diagonal and Combination and placement permutations (with and without repetitions) 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 Combination and placement permutations (with and without repetitions) by Combination and placement permutations (with and without repetitions) equal to the binomial coefficient

Combination and placement permutations (with and without repetitions)

With fixed Combination and placement permutations (with and without repetitions) generating function of a sequence of numbers of combinations Combination and placement permutations (with and without repetitions) , Combination and placement permutations (with and without repetitions) , Combination and placement permutations (with and without repetitions) , … is an:

Combination and placement permutations (with and without repetitions)

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

Combination and placement permutations (with and without repetitions)

Combinations with repetitions

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

The number of combinations with repetitions from Combination and placement permutations (with and without repetitions) by Combination and placement permutations (with and without repetitions) equal to the binomial coefficient

Combination and placement permutations (with and without repetitions)

Evidence

Let there be Combination and placement permutations (with and without repetitions) object types, and objects of the same type are indistinguishable. Suppose there is an unlimited (or large enough, in any case, no less than Combination and placement permutations (with and without repetitions) ) the number of objects of each type. From this range, choose Combination and placement permutations (with and without repetitions) objects; The sample can contain objects of the same type, the order of selection does not matter. Denote by Combination and placement permutations (with and without repetitions)number of selected objects of Combination and placement permutations (with and without repetitions)type i,Combination and placement permutations (with and without repetitions) , Combination and placement permutations (with and without repetitions) . Then Combination and placement permutations (with and without repetitions) .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 Combination and placement permutations (with and without repetitions)balls and Combination and placement permutations (with and without repetitions)partitions in a series so that exactly balls are located between the Combination and placement permutations (with and without repetitions)-th and Combination and placement permutations (with and without repetitions)-th partitions Combination and placement permutations (with and without repetitions). But such arrangements are exactly Combination and placement permutations (with and without repetitions)what was required to prove. ■

With a fixed Combination and placement permutations (with and without repetitions)generating function of numbers of combinations with repetitions fromCombination and placement permutations (with and without repetitions) by Combination and placement permutations (with and without repetitions) is an:

Combination and placement permutations (with and without repetitions)

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

Combination and placement permutations (with and without repetitions)

See also

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19.5.2020 23:23

еще систематизировать информацию можно в этой статье
https://intellect.icu/formuly-dlya-vsekh-vidov-soedinenij-v-kombinatorike-perestanovki-i-razmeshheniya-s-povtoreniyami-i-bez-povtorenij-s-primerami-4270


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Discrete Math. Set theory. Graph theory. Combinatorics.

Terms: Discrete Math. Set theory. Graph theory. Combinatorics.