## 【日常小测】最长路径

### 解题报告

1. 任意一个竞赛图一定存在哈密尔顿路径
2. 一个竞赛图存在哈密尔顿回路当且仅当这个竞赛图强连通

$ans_{x} = \sum\limits_{i = 1}^{x}{{{n – 1}\choose{i – 1}} g_i {{n – i}\choose{x – i}} f_{x – i} f_{n – x}}$

### Code

#include<bits/stdc++.h>
#define LL long long
using namespace std;

const int N = 2009;

int n, MOD, f[N], g[N], pw[N * N], C[N][N];

char c = getchar();
int ret = 0, f = 1;
while (c < '0' || c > '9') {
f = c == '-'? -1: 1;
c = getchar();
}
while ('0' <= c && c <= '9') {
ret = ret * 10 + c - '0';
c = getchar();
}
return ret * f;
}

int main() {
freopen("path.in", "r", stdin);
freopen("path.out", "w", stdout);
pw[0] = 1;
for (int i = 1; i < n * n; i++) {
pw[i] = (pw[i - 1] << 1) % MOD;
}
C[0][0] = 1;
for (int i = 1; i <= n; ++i) {
C[i][0] = 1;
for (int j = 1; j <= n; j++) {
C[i][j] = (C[i - 1][j - 1] + C[i - 1][j]) % MOD;
}
}
f[0] = g[0] = 1;
for (int i = 1; i <= n; i++) {
f[i] = g[i] = pw[i * (i - 1) >> 1];
for (int j = 1; j < i; j++) {
g[i] = (g[i] - (LL)C[i][j] * g[j] % MOD * f[i - j]) % MOD;
}
}
for (int x = 1; x <= n; x++) {
int ans = 0;
for (int i = 1; i <= x; i++) {
ans = (ans + (LL)C[n - 1][i - 1] * g[i] % MOD * C[n - i][x - i] % MOD * f[x - i] % MOD * f[n - x]) % MOD;
}
printf("%d\n", ans > 0? ans: ans + MOD);
}
return 0;
}