Specific mathematical Stirling number-----salute Kunth

Note that this is about the Stirling number, not the Sterling formula.

There are two types of sterling: the first class of Stirling and the second class of Stirling. be recorded separately.

First, the second class of Stirling numbers is described.

Description: The number of methods that divide a collection of n items into K non-empty subsets.

For example, the collection {1,2,3,4} has the following divisions:

{A-i} U{4} {1,2,4}u{3} {1,3,4}u{2} {2,3,4}u{1} {1,2}u{3,4} {1,3}u{2,4} {1,4}u{2,3}.

7 such a division.

Recorded as.

Then there is a second class of sterling numbers to give you calculation.

By definition, the easy answers are: 0,1,n,0.

Then consider a formula like this.

Divides a collection into two subsets.

Ideas:

Take the last element out.

There are two possible ways to do this. 1: The last element becomes a collection. 2: A subset of the last element and the preceding set is an element.

。

This idea is not only used at the time of k = 2. Extended to all K.

Principle does not speak. On the one hand, you can think for yourself when you look at it in the future. The other aspect is actually the same as the method of derivation.

List:

Tabular data is really important. Sometimes stiring number may be hidden in the title you can not find. And you can make a bold guess by playing the table.

1 7 6 1

1 15 25) 10 1

1 31 90 65 15 1

Worth remembering.

And then describe the first class of sterling.

Meaning: rotation. That is, n elements can be divided into k rotations.

Rotation: The new rotation cannot be obtained by using the old rotation for array shift.

[A,b,c,d] = [B,c,d,a] = [c,d,a,b] = [D,a,b,c].

All of the above represent the same rotation.

[A,b,c] and [a,c,b] are two different rotations.

For example, n = 4. When k=2.

There are 11 rotations:

[A] [4] [1,2,4][3] [1,3,4][2] [2,3,4][1]

[1,3,2] [4] [1,4,2][3] [1,4,3][2] [2,4,3][1]

[Up] [3,4] [1,3][2,4] [1,4][2,3]

Recorded as

Also for n>0

This formula is easy to get. n elements are all arranged. For one arrangement there is another n-1 which can be obtained by the array shift. So it is classified into 1 kinds.

That is n!/n = (n-1)!.

In addition, the following properties are also available.

Note the relationship between the two types of Stirling numbers.

In the same way, the first class of Stirling numbers also has a recursive type.

The point is the understanding of that (n-1).

Let me give you an example.

[4] added [1,2,3,4]

[2,3,1] Add 4 composition [2,3,1,4]

[3,1,2] Add 4 composition [3,1,2,4]

And the previous rotations are all part of a rotation. and constitute 3 different rotations. So it must not be the same as the second type of Stirling number is K.

And this situation can be found. Each number in the preceding collection appears once in the first position of the collection. So it's a n-1.

List:

2 3 1

6 11 6 1

24 50 35) 10 1

Worth remembering in the mind.

Also for rotation. We can correspond with the arrangement.

1 2 3 4 5 6 7 8 9

3 8 4 7 2 9 1 5 6

1->3 3->4 4->7 7->1 for a rotation [1,3,4,7]

2->8 8->5 5->2 for a rotation [2,8,5]

6->9 9->6 for a rotation [6,9]

There is always a rotation for any permutation that corresponds. Thus we can list.

In addition, in the back of the specific mathematics there is a formula that links the first class and the second class of Stirling numbers

There are, of course, many wonderful proofs and derivations in the midst of concrete mathematics. There are formulas. Specifically, it is not listed.

Just an introduction.

Specific mathematical Stirling number-----salute Kunth