[linear algebra] 1.2 total permutation and commutation

Catalog

One 、 Permutation and its reverse order number

1. array

2. Reverse order number

Two 、 exchange

One 、 Permutation and its reverse order number

1. array

hold n A row of different elements , It's called this n An element of Full Permutation ( Also abbreviated as array ).

from 1,2,3,...,n An ordered number of components is called n Level arrangement .

Arrangement is from 1,2,3,...,n, Is a continuous , There must be no missing number in the middle .

Such as 3145 It's not a permutation

n The number of all permutations of different elements , Usually use Express , The calculation is as follows ：

2. Reverse order number

  about n A different element , First, there is a standard order between the elements ( for example n Different natural numbers , Specify the standard order from small to large ), So here it is n In any arrangement of elements , When the order of a pair of elements is different from the standard order , It makes up a The reverse .

Reverse order number ： The total number of all reverse orders in an arrangement .

Odd permutation ： An arrangement in which the reverse order number is odd .

Even permutation ： An arrangement in which the reverse order number is even .

A simple way to find the number in reverse order ：

such as seek 32514 In reverse order

Just count the numbers one by one. After each number, there are several numbers smaller than itself

3 Back 2 and 1 Smaller than it ,2 An inverse ordinal number

2 Back 1 Smaller than it ,1 An inverse ordinal number

5 Back 1 and 4 Smaller than it ,2 An inverse ordinal number

1 There is no number smaller than it

4 Back There's no more

So the reverse order of this permutation is ：2+1+2=5

Two 、 exchange

In the arrangement , Swap any two elements , The rest of the elements don't understand , The procedure of making a new arrangement is called exchange .

Swap two adjacent elements , It's called Adjacent exchange .

Theorem 1： Any two elements in an arrangement are swapped , Permutation changes parity

It means an odd arrangement , The two elements are swapped , It becomes even permutation .

such as 32514,2 and 5 exchange , become 35214( The reverse order number is 6), It becomes even permutation .

Swap once , Change parity once .

Do an odd number of conversions , Change parity ; Do an even number of conversions , Do not change parity .

such as 32514, It was an odd arrangement , Swap once , Become even arrangement , Switch again , It becomes an odd arrangement again . Again , exchange 4、6、8... Time , Odd arrangement .

n In the order of levels , The number of odd permutations and even permutations are equal , Each half .

  inference ： The number of permutations from odd permutations to standard permutations is odd , Even permutations are replaced by standard permutations an even number of times .

such as 32514, Exchange for 12345 The number of conversions of is odd . 

Reference material ：

Engineering mathematics linear algebra sixth edition