1. Arrangement and Reverse Order
What is permutation: For
example: 1, 2, 3, 4, 5...n The
above permutation is also called n-level permutation, and the combination of n-level permutation has n factorial species.
Reverse order number:
"Reverse order" means that the larger number comes first.
"Reverse ordinal number" is the sum of how many numbers behind each number in the sequence are smaller than itself.
For example: 45231 = 3 + 3 +1 + 1 + 0 = 8
(2 3 1 after the first number 4 is smaller than it, so it is 3)
(The second number 5 after 2 3 1 is smaller than it, so it is 3. )
(The third number after 2 is smaller than it, so it is 1)
(The fourth number after 3 is smaller than 1, so it is 1)
(The fifth number after 1 is not smaller than it, so it is 0)
So it adds up to 3 + 3 + 1 + 1 + 0 = 8
Main theorem
swap: 1 and 4 make transposition
number reverse 2413 of 3 (odd arrangement), the number 2143 is 2 reverse (even order)
Theorem of: changing the number of columns of the transposed parity
theorems on: permutations n, there are n The factorial and odd-even of each account for half (n>1)
Theorem 3: Any arrangement can be exchanged with the natural arrangement through a series of exchanges. The parity is consistent with the number of swaps. (Odd for odd = even even for even = even)
Use of the sum sign
Second-order determinant
Third-order determinant
n-order determinant
The n-order determinant has n rows and n columns.
Definition:
Composition item: All rows are arranged, and the column is randomly selected from one column, which is taken from different rows and different columns.
Symbol: The column index arrangement is the negative of the odd arrangement, and the even arrangement is the positive number.
- It is a number or a formula.
- First-order determinant |a₁₁|=a₁₁
- D=|aᵢⱼ| (n rows and n columns)
- n! term (factorial term with n) positive and negative half (n>=2)
- Theoretical significance is greater than the computational significance (unless there are too many zeros)
Lower triangular determinant:
Upper triangular determinant:
Diagonal determinant
No name
