题目链接：
题意：
求 \(n\) 个点的无向简单(无重边无自环)连通图的个数。\((n<=50)\)
题解：
这题你甚至能OEIS。
但不支持这样做。TAT
间接做。
总方案数减去不合法方案。
因为\(n\)个点的完全图有 \(C(n,2)={n(n-1) \over 2}\) 条边，显然就有 \(2^{C(n,2)}\) 种子图，即枚举每条边是否选择。
设$ f[i]$ 表示每个点都和点 \(1\) 相连的连通图的个数 。
假设\(1\) 号点所在的连通块大小为 \(i\)。那么和 \(1\) 连通的这 \(i−1\) 个点就有 \(C(n-1,i-1)\) ，方案数则为：\(C(n-1,i-1)*f[i]\)。
然后其它 \(n−i\) 个点间任意连边即可，这时方案数为 \(2^{C(n-i,2)}\)。
综上，则存在递推式：
$$f[n]=2{C(n,2)}-\sum_{i=1}f[i]C(n-1,i-1)2^{C(n-i,2)}$$
答案就是\(f[n]\)。
然后就套个大数板子。额...过世OJ好像不支持c++11。懒得改了，虽然不过，但代码是正确的。QAQ
代码：
#include <iostream>
#include <vector>
#include <cmath>
#include <complex>
#include <cstring>
#include <stdlib.h>
#include <string.h>
using namespace std;
typedef long long LL;
const double PI = acos(-1);
void rader(vector<complex<double> >& y) {
int len = y.size();
int i, j, k;
for (i = 1, j = len / 2; i < len - 1; i++) {
if (i < j) swap(y[i], y[j]);
k = len / 2;
while (j >= k) {
j -= k;
k /= 2;
}
if (j < k) j += k;
}
}
void fft(vector<complex<double> >& y, int on) {
int len = y.size();
rader(y);
for (int h = 2; h <= len; h <<= 1) {
complex<double> wn(cos(-on * 2 * PI / h), sin(-on * 2 * PI / h));
for (int j = 0; j < len; j += h) {
complex<double> w(1, 0);
for (int k = j; k < j + h / 2; k++) {
complex<double> u = y[k];
complex<double> t = w * y[k + h / 2];
y[k] = u + t;
y[k + h / 2] = u - t;
w = w * wn;
}
}
}
if (on == -1) for (auto& i : y) i.real(i.real() / len);
}
class BigInt {
private:
string num;
string sign;
public:
const string to_string() const {
if (this->sign == "-") return this->sign + this->num;
else return this->num;
}
const LL toll() { return stoll(this->to_string()); }
const int toi() { return stoi(this->to_string()); }
BigInt() : num("0"), sign("+") {}
BigInt(const int t) {
if (t < 0) {
this->num = std::to_string(-t);
this->sign = "-";
} else {
this->num = std::to_string(t);
this->sign = "+";
}
}
BigInt(const LL t) {
if (t < 0) {
this->num = std::to_string(-t);
this->sign = "-";
} else {
this->num = std::to_string(t);
this->sign = "+";
}
}
BigInt(const string& t) {
if (t[0] == '-') {
this->num = t.substr(1);
this->sign = "-";
} else {
this->num = t;
this->sign = "+";
}
int flag = 0;
while (flag < (int)this->num.length() - 1 && this->num[flag] == '0') flag++;
this->num = this->num.substr(flag);
}
BigInt(char* const t) : BigInt(string(t)) {}
friend bool operator< (const BigInt& t, const BigInt& s) {
if (t.sign != s.sign) {
if (t.sign == "-") return true;
else return false;
} else {
if (t.sign == "-") {
if (t.num.length() == s.num.length()) {
return t.num > s.num;
} else {
return t.num.length() > s.num.length();
}
} else {
if (t.num.length() == s.num.length()) {
return t.num < s.num;
} else {
return t.num.length() < s.num.length();
}
}
}
}
friend bool operator> (const BigInt& t, const BigInt& s) {
return s < t;
}
friend bool operator== (const BigInt& t, const BigInt& s) {
return t.num == s.num && t.sign == s.sign;
}
friend bool operator!= (const BigInt& t, const BigInt& s) {
return !(t == s);
}
friend bool operator<= (const BigInt& t, const BigInt& s) {
return t == s || t < s;
}
friend bool operator>= (const BigInt& t, const BigInt& s) {
return t == s || t > s;
}
friend const BigInt abs(const BigInt& t) {
BigInt ans = t;
if (ans.sign == "-") ans.sign = "+";
return ans;
}
friend const BigInt operator- (const BigInt& t) {
BigInt ans = t;
if (ans.sign == "-") ans.sign = "+";
else ans.sign = "-";
return ans;
}
friend istream& operator>> (istream& in, BigInt& t) {
string s;
in >> s;
t = s;
return in;
}
friend ostream& operator<< (ostream& out, const BigInt& t) {
out << t.to_string();
return out;
}
friend const BigInt operator+ (const BigInt& t, const BigInt& s) {
BigInt ans, sub;
if (t.num.length() < s.num.length()) {
ans = s;
sub = t;
} else if (t.num.length() == s.num.length()) {
if (t.num < s.num) {
ans = s;
sub = t;
} else {
ans = t;
sub = s;
}
} else {
ans = t;
sub = s;
}
int sub_l = sub.num.length();
int ans_l = ans.num.length();
if (t.sign == s.sign) {
for (int i = 1; i <= sub_l; i++) {
ans.num[ans_l - i] += sub.num[sub_l - i] - '0';
}
int flag = 0;
for (int i = 1; i <= ans_l; i++) {
if (ans.num[ans_l - i] > '9') {
ans.num[ans_l - i] -= 10;
if (i == ans_l) {
flag = 1;
} else {
ans.num[ans_l - i - 1] += 1;
}
} else if (i >= sub_l) {
break;
}
}
if (flag) ans.num = "1" + ans.num;
} else {
for (int i = 1; i <= sub_l; i++) {
ans.num[ans_l - i] -= sub.num[sub_l - i] - '0';
}
for (int i = 1; i <= ans_l; i++) {
if (ans.num[ans_l - i] < '0') {
ans.num[ans_l - i] += 10;
ans.num[ans_l - i - 1] -= 1;
} else if (i >= sub_l) {
break;
}
}
int flag = 0;
while (flag < ans_l - 1 && ans.num[flag] == '0') flag++;
ans.num = ans.num.substr(flag);
if (ans.num == "0") ans.sign = "+";
}
return ans;
}
friend const BigInt operator- (const BigInt& t, const BigInt& s) {
BigInt sub = s;
if (sub.sign == "+") sub.sign = "-";
else sub.sign = "+";
return t + sub;
}
friend const BigInt operator* (const BigInt& t, const BigInt& s) {
BigInt res;
if (s.sign == t.sign) res.sign = "+";
else res.sign = "-";
vector<complex<double> > x1, x2;
vector<int> sum;
string str1 = t.num, str2 = s.num;
int len1 = str1.length();
int len2 = str2.length();
int len = 1;
while (len < len1 * 2 || len < len2 * 2) len <<= 1;
for (int i = 0; i < len1; i++) {
x1.push_back(complex<double>(str1[len1 - 1 - i] - '0', 0));
}
for (int i = len1; i < len; i++) {
x1.push_back(complex<double>(0, 0));
}
for (int i = 0; i < len2; i++) {
x2.push_back(complex<double>(str2[len2 - 1 -i] - '0', 0));
}
for (int i = len2; i < len; i++) {
x2.push_back(complex<double>(0, 0));
}
fft(x1, 1);
fft(x2, 1);
for (int i = 0; i < len; i++) x1[i] = x1[i] * x2[i];
fft(x1, -1);
for (auto& i : x1) sum.push_back((int)(i.real() + 0.5));
for (int i = 0; i < len; i++) {
sum[i + 1] += sum[i] / 10;
sum[i] %= 10;
}
len = len1 + len2 - 1;
while (sum[len] <= 0 && len > 0) len--;
res.num = "";
for (int i = len; i >= 0; i--) res.num += sum[i] + '0';
if (res.num == "0") res.sign = "+";
return res;
}
friend const BigInt operator/ (const BigInt& t, const BigInt& s) {
if (s == 0) throw;
BigInt res;
if (s.sign == t.sign) res.sign = "+";
else res.sign = "-";
BigInt sub = abs(t), ans = abs(s);
int w = sub.num.length() - ans.num.length();
for (int i = 0; i < w; i++) ans.num += "0";
while (w >= 0) {
int d = 0;
while (ans <= sub) {
sub -= ans;
d++;
}
res.num += d + '0';
ans.num = ans.num.substr(0, ans.num.length() - 1);
w--;
}
int flag = 0;
while (flag < (int)res.num.length() - 1 && res.num[flag] == '0') flag++;
res.num = res.num.substr(flag);
if (res.num == "0") res.sign = "+";
return res;
}
friend const BigInt operator% (const BigInt& t, const BigInt& s) {
if (s == 0) throw;
BigInt sub = abs(t), ans = abs(s);
int w = sub.num.length() - ans.num.length();
for (int i = 0; i < w; i++) ans.num += "0";
while (w >= 0) {
int d = 0;
while (ans <= sub) {
sub -= ans;
d++;
}
w--;
ans.num = ans.num.substr(0, ans.num.length() - 1);
}
sub.sign = t.sign;
if (sub.num == "0") sub.sign = "+";
return sub;
}
friend BigInt& operator+= (BigInt& t, const BigInt& s) {
return t = t + s;
}
friend BigInt& operator-= (BigInt& t, const BigInt& s) {
return t = t - s;
}
friend BigInt& operator*= (BigInt& t, const BigInt& s) {
return t = t * s;
}
friend BigInt& operator/= (BigInt& t, const BigInt& s) {
return t = t / s;
}
friend BigInt& operator%= (BigInt& t, const BigInt& s) {
return t = t % s;
}
const BigInt subnum(int r, int l) {
BigInt ans = this->num.substr(this->num.length() - l, l - r);
ans.sign = this->sign;
return ans;
}
const BigInt subnum(int l) {
return this->subnum(0, l);
}
};
BigInt dp[110][110];
void init()
{
dp[0][0] = 1;
for(int i = 1; i <= 51; i++) {
dp[i][0] = 1;
for(int j = 1; j <= i; j++) {
dp[i][j] = dp[i - 1][j] + dp[i - 1][j - 1];
}
}
}
BigInt expo[2000];
BigInt f[52];
int main(int argc, char const *argv[]) {
init();
int n;
expo[0] = 1;
for(long long i = 1; i <= 1250; i++) {
expo[i] = 2LL * expo[i - 1];
}
f[1] = 1;
f[2] = 1;
while(std::cin >> n) {
if(n == 0) break;
if(n <= 2) {
std::cout << "1" << '\n';
continue;
}
BigInt tmp = 0;
for(int i = 3; i <= n; i++) {
BigInt tot = expo[(i * (i - 1) >> 1)];
// std::cout << "tot = " << tot << '\n';
for(int j = 1; j <= i - 1; j++) {
tot = tot - f[j] * dp[i - 1][j - 1] * expo[((i - j) * ((i - j) - 1)) >> 1];
}
f[i] = tot;
}
std::cout << f[n] << '\n';
}
return 0;
}
