Description
题库链接
求有多少种长度为 \(n\) 的序列 \(A\) , 满足以下条件:
\(1 \sim n\) 这 \(n\) 个数在序列中各出现了一次
若第 \(i\) 个数 \(A[i]\) 的值为 \(i\) , 则称 \(i\) 是稳定的序列恰好有 \(m\) 个数是稳定的
满足条件的序列可能很多, 序列数对 \(10^9+7\) 取模
- \(T\leq 500000\) , \(n\leq 1000000\) , \(m\leq 1000000\)
- Solution
需要用到组合数学和错排公式
\(ans=C_n^m\cdot D_{n-m}\)
其中 \(D_n=n!\left(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!}\right)\)
- Code
- //It is made by Awson on 2018.2.14
- #include <bits/stdc++.h>
- #define LL long long
- #define dob complex<double>
- #define Abs(a) ((a) < 0 ? (-(a)) : (a))
- #define Max(a, b) ((a) > (b) ? (a) : (b))
- #define Min(a, b) ((a) < (b) ? (a) : (b))
- #define Swap(a, b) ((a) ^= (b), (b) ^= (a), (a) ^= (b))
- #define writeln(x) (write(x), putchar('\n'))
- #define lowbit(x) ((x)&(-(x)))
- using namespace std;
- const int N = 1000000;
- const int MOD = 1e9+7;
- void read(int &x) {
- char ch; bool flag = 0;
- for (ch = getchar(); !isdigit(ch) && ((flag |= (ch == '-')) || 1); ch = getchar());
- for (x = 0; isdigit(ch); x = (x<<1)+(x<<3)+ch-48, ch = getchar());
- x *= 1-2*flag;
- }
- void print(int x) {if (x > 9) print(x/10); putchar(x%10+48); }
- void write(int x) {if (x < 0) putchar('-'); print(Abs(x)); }
- int A[N+5], inv[N+5], D[N+5], n, m, t;
- void work() {
- inv[0] = inv[1] = A[1] = A[0] = D[0] = 1;
- for (int i = 2; i <= N; i++) inv[i] = -1ll*(MOD/i)*inv[MOD%i]%MOD;
- for (int i = 2; i <= N; i++) A[i] = 1ll*A[i-1]*i%MOD, inv[i] = 1ll*inv[i]*inv[i-1]%MOD;
- for (int i = 1; i <= N; i++)
- if (i%2 == 1) D[i] = (D[i-1]-inv[i])%MOD;
- else D[i] = (D[i-1]+inv[i])%MOD;
- for (int i = 0; i <= N; i++) D[i] = 1ll*D[i]*A[i]%MOD;
- read(t);
- while (t--) {
- read(n), read(m);
- if (n < m) puts("0");
- else writeln(int((1ll*A[n]*inv[m]%MOD*inv[n-m]%MOD*D[n-m]%MOD+MOD)%MOD));
- }
- }
- int main() {
- work(); return 0;
- }
来源: http://www.bubuko.com/infodetail-2498740.html