bzoj4517 "SDOI2016" permutation count

4517: [Sdoi2016] permutation count time limit: Sec Memory Limit: MB
Submit: 576 Solved: 358
[Submit] [Status] [Discuss] Description How many sequence a of length n, satisfies the following conditions: 1 ~ n the number of n is present in the sequence once, if the number of I a[i] is the value of I, then I is said to be stable. Sequences that have exactly m number are stable enough to satisfy the condition of the sequence may be many, sequence number pairs 10^9+7 modulo. The first line of input is a number T, which indicates that there is a T group of data. Next T line, two integers per line n, M. T=500000,n≤1000000,m≤1000000output

Outputs T line, one number per line, indicating the number of sequences to be calculated

Sample Input5
1 0
1 1
5 2
100 50
10000Sample Output0
1
20
578028887
60695423HINT

Source

Acknowledgement Menci Upload

modulus of combination number + Wrong Row problem

First, it is obvious that the answer equals C (n,m) *f[n-m], where f[n-m] represents the number of scenarios where the number of (N-M) is all misplaced.

As long as the inverse of factorial and factorial can be preprocessed, the problem is how to find the F-array quickly.

For the F array, there is a recursive relationship f[i]= (f[i-1]+f[i-2]) * (I-1), the formula for this staggered scheme is still to be remembered ...

The proof is as follows:

1-n Total n number of all row wrong, assuming that 1 row to the position K (k≠1).

If K is ranked at position 1, then the number of remaining (n-2) is all wrong and the scheme count is f[n-2]. If K is not ranked to position 1, consider the position k to position 1 has a "portal", that is equal to the number of 2-k all the wrong, the scheme number is f[n-1]. K has a value of (n-1), then f[n]= (F[n-1]+f[n-2]) * (n-1).

 #include <iostream> #include <cstdio> #include <cstring> #include <cmath> #include <cstring># Include<algorithm> #define F (I,j,n) for (int. i=j;i<=n;i++) #define D (i,j,n) for (int i=j;i>=n;i--) #define LL Long long#define maxn 1000005#define mod 1000000007using namespace Std;int t,n,m,x,y;ll F[MAXN],FAC[MAXN],INV[MAXN]; inline int read () {int X=0,f=1;char ch=getchar (); while (ch< ' 0 ' | | Ch> ' 9 ') {if (ch== '-') F=-1;ch=getchar ();} while (ch>= ' 0 ' &&ch<= ' 9 ') {x=x*10+ch-' 0 '; Ch=getchar ();} return x*f;} int main () {fac[0]=1; F (i,1,1000000) fac[i]=fac[i-1]*i%mod;inv[0]=1;inv[1]=1; F (i,2,1000000) {x=mod/i+1;y=x*i-mod;inv[i]=inv[y]*x%mod;} F (i,1,1000000) inv[i]=inv[i-1]*inv[i]%mod;f[0]=1;f[1]=0; F (i,2,1000000) f[i]= (F[i-1]+f[i-2])%mod* (i-1)%mod;t=read (), while (t--) {n=read (); M=read ();p rintf ("%lld\n", fac[n]* INV[M]%MOD*INV[N-M]%MOD*F[N-M]%MOD);} return 0;} 

bzoj4517 "SDOI2016" permutation count