Find the Permutations
Sorting is one of the most used operations in real life, where Computer Science comes into act. It is well-known that the lower bound of swap based sorting is nlog(n). It means that the best possible sorting algorithm will take at least O(nlog(n)) swaps to sort a set of n integers. However, to sort a particular array of n integers, you can always find a swapping sequence of at most (n − 1) swaps, once you know the position of each element in the sorted sequence. For example consider four elements <1 2 3 4>. There are 24 possible permutations and for all elements you know the position in sorted sequence. If the permutation is <2 1 4 3>, it will take minimum 2 swaps to make it sorted. If the sequence is <2 3 4 1>, at least 3 swaps are required. The sequence <4 2 3 1> requires only 1 and the sequence <1 2 3 4> requires none. In this way, we can find the permutations of N distinct integers which will take at least K swaps to be sorted. Input Each input consists of two positive integers N (1 ≤ N ≤ 21) and K (0 ≤ K < N) in a single line. Input is terminated by two zeros. There can be at most 250 test cases. Output For each of the input, print in a line the number of permutations which will take at least K swaps.
Sample Input
3 1
3 0
3 2
0 0
Sample Output
3
1
2
【题意】
给出1~n的一个排列，可以通过一系列的交换变成{1,2,…,n}。比如{2,1,4,3}需要两次交换。给定n和k，统计有多少个排列至少需要k次交换才能变成{1,2,…,n}。
【分析】
先考虑一下怎么计算最少变换次数。
显然，如果把它弄成x个循环的乘积，最少变换次数为n-x。
问题变成了，给你n个数，分成n-x份的圆排列方案。这个方案刚好就是第一类斯特林数啊。
所以很简单，用第一类斯特林数的方程求方案就行了。
1 #include<cstdio> 2 #include<cstdlib> 3 #include<cstring> 4 #include<iostream> 5 #include<algorithm> 6 using namespace std; 7 #define LL unsigned long long 8 #define Maxn 9 10 LL s1[30][30]; 11 12 void init() 13 { 14 memset(s1,0,sizeof(0)); 15 s1[0][0]=1; 16 for(int i=1;i<=21;i++) 17 for(int j=1;j<=21;j++) 18 s1[i][j]=s1[i-1][j-1]+s1[i-1][j]*(i-1); 19 // printf("==%lld ",s1[2][0]); 20 } 21 22 int main() 23 { 24 init(); 25 int n,k; 26 while(1) 27 { 28 scanf("%d%d",&n,&k); 29 if(n==0&&k==0) break; 30 printf("%llu ",s1[n][n-k]); 31 } 32 return 0; 33 }
注意要用unsigned long long ，还是看了别人的代码才知道的。。。不然会WA、。。。。
其实这题只是用了小小的置换的思想而已。
2017-01-11 19:16:56