Examining the Rooms

A murder happened in the hotel. As the best detective in the town, you should examine all the N rooms of the hotel immediately. However, all the doors of the rooms are locked, and the keys are just locked in the rooms, what a trap! You know that there is exactly one key in each room, and all the possible distributions are of equal possibility. For example, if N = 3, there are 6 possible distributions, the possibility of each is 1/6. For convenience, we number the rooms from 1 to N, and the key for Room 1 is numbered Key 1, the key for Room 2 is Key 2, etc.
To examine all the rooms, you have to destroy some doors by force. But you don’t want to destroy too many, so you take the following strategy: At first, you have no keys in hand, so you randomly destroy a locked door, get into the room, examine it and fetch the key in it. Then maybe you can open another room with the new key, examine it and get the second key. Repeat this until you can’t open any new rooms. If there are still rooms un-examined, you have to randomly pick another unopened door to destroy by force, then repeat the procedure above, until all the rooms are examined.
Now you are only allowed to destroy at most K doors by force. What’s more, there lives a Very Important Person in Room 1. You are not allowed to destroy the doors of Room 1, that is, the only way to examine Room 1 is opening it with the corresponding key. You want to know what is the possibility of that you can examine all the rooms finally.

 

The first line of the input contains an integer T (T ≤ 200), indicating the number of test cases. Then T cases follow. Each case contains a line with two numbers N and K. (1 < N ≤ 20, 1 ≤ K < N)

 

Output one line for each case, indicating the corresponding possibility. Four digits after decimal point are preserved by rounding.

 

 

 

Source

2010 Asia Regional Tianjin Site —— Online Contest

 

第一类Stirling数 s(p,k)

　　  

s(p,k)的一个的组合学解释是：将p个物体排成k个非空循环排列的方法数。

 

s(p,k)的递推公式： s(p,k)=(p-1)*s(p-1,k)+s(p-1,k-1) ,1<=k<=p-1

边界条件：s(p,0)=0 ,p>=1  s(p,p)=1  ,p>=0

 

递推关系的说明：

考虑第p个物品，p可以单独构成一个非空循环排列，这样前p-1种物品构成k-1个非空循环排列，方法数为s(p-1,k-1)；

也可以前p-1种物品构成k个非空循环排列，而第p个物品插入第i个物品的左边，这有(p-1)*s(p-1,k)种方法。

 

 1 //2017-08-05
 2 #include <cstdio>
 3 #include <cstring>
 4 #include <iostream>
 5 #include <algorithm>
 6 #define ll long long 
 7 
 8 using namespace std;
 9 
10 const int N = 22;
11 ll factorial[N], stir1[N][N];//factorial[n]存n的阶乘，stir1为第一类斯特林数
12 
13 int main()
14 {
15     int T, n, k;
16     cin >> T;
17     factorial[1] = 1;
18     for(int i = 2; i < N; i++)
19           factorial[i] = factorial[i-1]*i;
20     memset(stir1, 0, sizeof(stir1));
21     stir1[1][1] = 1;
22     stir1[0][0] = 0;
23     for(int i = 2; i < N; i++){
24           for(int j = 1; j <= i; j++)
25               stir1[i][j] = stir1[i-1][j-1] + (i-1)*stir1[i-1][j];
26     }
27     while(T--){
28         cin>>n>>k;
29         ll tmp = 0;
30         for(int i = 1; i <= k; i++)
31               tmp += stir1[n][i] - stir1[n-1][i-1];
32         printf("%.4lf\n", (double)tmp*1.0/factorial[n]);
33     }
34 
35     return 0;
36 }