HDU 3664 Permutation Counting
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Permutation Counting
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 32768/32768 K (Java/Others)Total Submission(s): 1330 Accepted Submission(s): 669
Problem Description
Given a permutation a1, a2, … aN of {1, 2, …, N}, we define its E-value as the amount of elements where ai > i. For example, the E-value of permutation {1, 3, 2, 4} is 1, while the E-value of {4, 3, 2, 1} is 2. You are requested to find how many permutations of {1, 2, …, N} whose E-value is exactly k.
Input
There are several test cases, and one line for each case, which contains two integers, N and k. (1 <= N <= 1000, 0 <= k <= N).
Output
Output one line for each case. For the answer may be quite huge, you need to output the answer module 1,000,000,007.
Sample Input
3 03 1
Sample Output
14HintThere is only one permutation with E-value 0: {1,2,3}, and there are four permutations with E-value 1: {1,3,2}, {2,1,3}, {3,1,2}, {3,2,1}
Source
2010 Asia Regional Harbin
解题思路:给你1到n的n个数的全排列,求使得a[i]>i的数恰有k个方案的个数
令dp[i][j]表示i个数有j个a[i]>i的方案数,对于dp[i-1][j-1],最后一个数我们可以使它与a[i]<=i的任意一个数交换,共有(i-1-(j-1)) 种可能,对于dp[i-1][j],它要么直接放在最后一位,要么与a[i]>i的任意一个数交换,共有(j+1)种可能,所以
dp[i][j]=dp[i-1][j-1]*(i-j)+dp[i-1][j]*(j+1);
#include<cstdio>#include<iostream>using namespace std;const int MAXN=1001;long long dp[MAXN][MAXN];const long long MOD=1000000007;int main(){ int n,k; int i,j; for(i=1;i<=1000;i++) { dp[i][0]=1; for(j=1;j<i;j++) dp[i][j]=(dp[i-1][j]+dp[i-1][j]*j+dp[i-1][j-1]*(i-j))%MOD; } while(scanf("%d%d",&n,&k)!=EOF) printf("%I64d\n",dp[n][k]); return 0; }
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