复（学）习化学时突然的一个 idea
期中考试成功探底。。。但是某些化学问题还是很有信息学价值的。。。
n 烷同分异构体计数。
这个题 fanhq666 出过，就是一个 dp。
设 f[i] 表示含有 i 个节点的无标号不同构的度数限制为 4 的有根树的个数。那么转移时枚举最多 3 个子树的大小，如果大小相同时用组合数，否则乘法原理就好了。
最后统计答案时找到中心为根，然后最多 4 个子树，每个大小不超过 [n / 2] - 1（除法取上整），同样的累计方法（组合数/乘法原理）统计一下就好了；双重心时需要把两个重心同时提上去统计一遍和刚才的答案累加。
没事闲的写了个比较全的高精度，先贴高精度模板。
#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cctype>
#include <algorithm>
using namespace std;
int read() {
int x = 0, f = 1; char c = getchar();
while(!isdigit(c)){ if(c == '-') f = -1; c = getchar(); }
while(isdigit(c)){ x = x * 10 + c - '0'; c = getchar(); }
return x * f;
}
struct LL {
int len, A[510];
LL() { len = -1; }
LL(int x) {
A[1] = x; len = 1;
if(x) while(A[len] > 9) A[len+1] = A[len] / 10, A[len] %= 10, len++;
else len = -1;
}
LL operator = (const int& t) {
A[1] = t; len = 1;
if(t) while(A[len] > 9) A[len+1] = A[len] / 10, A[len] %= 10, len++;
else len = -1;
return *this;
}
LL operator + (const LL& t) const {
LL ans; ans.len = max(len, t.len);
for(int i = 1; i <= ans.len; i++) ans.A[i] = (i <= len ? A[i] : 0) + (i <= t.len ? t.A[i] : 0);
for(int i = 1; i < ans.len; i++) if(ans.A[i] > 9) ans.A[i+1] += ans.A[i] / 10, ans.A[i] %= 10;
while(ans.A[ans.len] > 9) ans.A[ans.len+1] = ans.A[ans.len] / 10, ans.A[ans.len] %= 10, ans.len++;
return ans;
}
LL operator += (const LL& t) {
*this = *this + t;
return *this;
}
LL operator + (const int& t) const {
LL ans; ans.len = max(len, 1);
for(int i = 1; i <= ans.len; i++) ans.A[i] = A[i]; ans.A[1] += t;
for(int i = 1; i < ans.len; i++) if(ans.A[i] > 9) ans.A[i+1] += ans.A[i] / 10, ans.A[i] %= 10;
while(ans.A[ans.len] > 9) ans.A[ans.len+1] = ans.A[ans.len] / 10, ans.A[ans.len] %= 10, ans.len++;
return ans;
}
LL operator += (const int& t) {
*this = *this + t;
return *this;
}
LL operator ++ () { // ++this;
*this = *this + 1;
return *this;
}
LL operator ++ (int x) { // this++;
*this = *this + 1;
return *this - 1;
}
LL operator - (const LL& t) const {
LL ans; ans.len = max(len, t.len);
for(int i = 1; i <= ans.len; i++) ans.A[i] = A[i] - t.A[i];
for(int i = 1; i < ans.len; i++) if(ans.A[i] < 0) {
int tmp = (-ans.A[i] + 9) / 10;
ans.A[i+1] -= tmp; ans.A[i] += tmp * 10;
}
while(!ans.A[ans.len]) ans.len--;
return ans;
}
LL operator -= (const LL& t) {
*this = *this - t;
return *this;
}
LL operator - (const int& t) const {
LL ans; ans.len = len;
for(int i = 1; i <= ans.len; i++) ans.A[i] = A[i]; ans.A[1] -= t;
for(int i = 1; i < ans.len; i++) if(ans.A[i] < 0) {
int tmp = (-ans.A[i] + 9) / 10;
ans.A[i+1] -= tmp; ans.A[i] += tmp * 10;
}
while(!ans.A[ans.len]) ans.len--;
return ans;
}
LL operator -= (const int& t) {
*this = *this - t;
return *this;
}
LL operator -- () { // --this;
*this = *this - 1;
return *this;
}
LL operator -- (int x) { // this--;
*this = *this - 1;
return *this + 1;
}
LL operator * (const LL& t) const {
LL ans; ans.len = len + t.len - 1;
if(len < 0 || t.len < 0) return ans.len = -1, ans;
for(int i = 1; i <= ans.len; i++) ans.A[i] = 0;
for(int i = 1; i <= len; i++)
for(int j = 1; j <= t.len; j++) ans.A[i+j-1] += A[i] * t.A[j];
for(int i = 1; i < ans.len; i++) if(ans.A[i] > 9) ans.A[i+1] += ans.A[i] / 10, ans.A[i] %= 10;
while(ans.A[ans.len] > 9) ans.A[ans.len+1] = ans.A[ans.len] / 10, ans.A[ans.len] %= 10, ans.len++;
return ans;
}
LL operator *= (const LL& t) {
*this = *this * t;
return *this;
}
LL operator * (const int& t) const {
LL ans = t;
return ans * (*this);
}
LL operator *= (const int& t) {
*this = *this * t;
return *this;
}
bool operator < (const LL& t) const {
if(len != t.len) return len < t.len;
for(int i = len; i > 0; i--) if(A[i] != t.A[i]) return A[i] < t.A[i];
return 0;
}
bool operator > (const LL& t) const {
if(len != t.len) return len > t.len;
for(int i = len; i > 0; i--) if(A[i] != t.A[i]) return A[i] > t.A[i];
return 0;
}
bool operator == (const LL& t) const {
if(len != t.len) return 0;
for(int i = len; i > 0; i--) if(A[i] != t.A[i]) return 0;
return 1;
}
bool operator != (const LL& t) const {
return !((*this) == t);
}
bool operator <= (const LL& t) const {
return *this < t || *this == t;
}
bool operator >= (const LL& t) const {
return *this > t || *this == t;
}
LL operator / (const LL& t) const {
LL ans, f = 0; ans.len = -1;
for(int i = 1; i <= len; i++) ans.A[i] = 0;
for(int i = len; i > 0; i--) {
f = f * 10 + A[i];
while(f >= t) {
f -= t; ans.A[i]++;
ans.len = max(ans.len, i);
}
}
return ans;
}
LL operator /= (const LL& t) {
*this = *this / t;
return *this;
}
LL operator / (const int& t) const {
LL ans; int f = 0; ans.len = -1;
for(int i = 1; i <= len; i++) ans.A[i] = 0;
for(int i = len; i > 0; i--) {
f = f * 10 + A[i];
while(f >= t) {
f -= t; ans.A[i]++;
ans.len = max(ans.len, i);
}
}
return ans;
}
LL operator /= (const int& t) {
*this = *this / t;
return *this;
}
void print() {
if(len < 0){ putchar('0'); return ; }
for(int i = len; i > 0; i--) putchar(A[i] + '0'); putchar('\n');
return ;
}
} A, B;
int main() {
while(1) {
A = read(); B = read();
(A + B).print();
(A - B).print();
(A * B).print();
(A / B).print();
(A++).print();
(++A).print();
(B--).print();
(--B).print();
}
return 0;
}
/*
8997622 2333
*/
再贴这道题的程序。
#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cctype>
#include <algorithm>
#include <string>
using namespace std;
int read() {
int x = 0, f = 1; char c = getchar();
while(!isdigit(c)){ if(c == '-') f = -1; c = getchar(); }
while(isdigit(c)){ x = x * 10 + c - '0'; c = getchar(); }
return x * f;
}
#define maxn 1010
struct LL {
int len, A[510];
LL() { len = -1; }
LL(int x) {
A[1] = x; len = 1;
if(x) while(A[len] > 9) A[len+1] = A[len] / 10, A[len] %= 10, len++;
else len = -1;
}
LL operator = (const int& t) {
A[1] = t; len = 1;
if(t) while(A[len] > 9) A[len+1] = A[len] / 10, A[len] %= 10, len++;
else len = -1;
return *this;
}
LL operator + (const LL& t) const {
LL ans; ans.len = max(len, t.len);
for(int i = 1; i <= ans.len; i++) ans.A[i] = (i <= len ? A[i] : 0) + (i <= t.len ? t.A[i] : 0);
for(int i = 1; i < ans.len; i++) if(ans.A[i] > 9) ans.A[i+1] += ans.A[i] / 10, ans.A[i] %= 10;
while(ans.A[ans.len] > 9) ans.A[ans.len+1] = ans.A[ans.len] / 10, ans.A[ans.len] %= 10, ans.len++;
return ans;
}
LL operator += (const LL& t) {
*this = *this + t;
return *this;
}
LL operator + (const int& t) const {
LL ans; ans.len = max(len, 1);
for(int i = 1; i <= ans.len; i++) ans.A[i] = A[i]; ans.A[1] += t;
for(int i = 1; i < ans.len; i++) if(ans.A[i] > 9) ans.A[i+1] += ans.A[i] / 10, ans.A[i] %= 10;
while(ans.A[ans.len] > 9) ans.A[ans.len+1] = ans.A[ans.len] / 10, ans.A[ans.len] %= 10, ans.len++;
return ans;
}
LL operator += (const int& t) {
*this = *this + t;
return *this;
}
LL operator ++ () { // ++this;
*this = *this + 1;
return *this;
}
LL operator ++ (int x) { // this++;
*this = *this + 1;
return *this - 1;
}
LL operator - (const LL& t) const {
LL ans; ans.len = max(len, t.len);
for(int i = 1; i <= ans.len; i++) ans.A[i] = A[i] - t.A[i];
for(int i = 1; i < ans.len; i++) if(ans.A[i] < 0) {
int tmp = (-ans.A[i] + 9) / 10;
ans.A[i+1] -= tmp; ans.A[i] += tmp * 10;
}
while(!ans.A[ans.len]) ans.len--;
return ans;
}
LL operator -= (const LL& t) {
*this = *this - t;
return *this;
}
LL operator - (const int& t) const {
LL ans; ans.len = len;
for(int i = 1; i <= ans.len; i++) ans.A[i] = A[i]; ans.A[1] -= t;
for(int i = 1; i < ans.len; i++) if(ans.A[i] < 0) {
int tmp = (-ans.A[i] + 9) / 10;
ans.A[i+1] -= tmp; ans.A[i] += tmp * 10;
}
while(!ans.A[ans.len]) ans.len--;
return ans;
}
LL operator -= (const int& t) {
*this = *this - t;
return *this;
}
LL operator -- () { // --this;
*this = *this - 1;
return *this;
}
LL operator -- (int x) { // this--;
*this = *this - 1;
return *this + 1;
}
LL operator * (const LL& t) const {
LL ans; ans.len = len + t.len - 1;
if(len < 0 || t.len < 0) return ans.len = -1, ans;
for(int i = 1; i <= ans.len; i++) ans.A[i] = 0;
for(int i = 1; i <= len; i++)
for(int j = 1; j <= t.len; j++) ans.A[i+j-1] += A[i] * t.A[j];
for(int i = 1; i < ans.len; i++) if(ans.A[i] > 9) ans.A[i+1] += ans.A[i] / 10, ans.A[i] %= 10;
while(ans.A[ans.len] > 9) ans.A[ans.len+1] = ans.A[ans.len] / 10, ans.A[ans.len] %= 10, ans.len++;
return ans;
}
LL operator *= (const LL& t) {
*this = *this * t;
return *this;
}
LL operator * (const int& t) const {
LL ans = t;
return ans * (*this);
}
LL operator *= (const int& t) {
*this = *this * t;
return *this;
}
bool operator < (const LL& t) const {
if(len != t.len) return len < t.len;
for(int i = len; i > 0; i--) if(A[i] != t.A[i]) return A[i] < t.A[i];
return 0;
}
bool operator > (const LL& t) const {
if(len != t.len) return len > t.len;
for(int i = len; i > 0; i--) if(A[i] != t.A[i]) return A[i] > t.A[i];
return 0;
}
bool operator == (const LL& t) const {
if(len != t.len) return 0;
for(int i = len; i > 0; i--) if(A[i] != t.A[i]) return 0;
return 1;
}
bool operator != (const LL& t) const {
return !((*this) == t);
}
bool operator <= (const LL& t) const {
return *this < t || *this == t;
}
bool operator >= (const LL& t) const {
return *this > t || *this == t;
}
LL operator / (const LL& t) const {
LL ans, f = 0; ans.len = -1;
for(int i = 1; i <= len; i++) ans.A[i] = 0;
for(int i = len; i > 0; i--) {
f = f * 10 + A[i];
while(f >= t) {
f -= t; ans.A[i]++;
ans.len = max(ans.len, i);
}
}
return ans;
}
LL operator /= (const LL& t) {
*this = *this / t;
return *this;
}
LL operator / (const int& t) const {
LL ans; int f = 0; ans.len = -1;
for(int i = 1; i <= len; i++) ans.A[i] = 0;
for(int i = len; i > 0; i--) {
f = f * 10 + A[i];
while(f >= t) {
f -= t; ans.A[i]++;
ans.len = max(ans.len, i);
}
}
return ans;
}
LL operator /= (const int& t) {
*this = *this / t;
return *this;
}
void print() {
if(len < 0){ putchar('0'); return ; }
for(int i = len; i > 0; i--) putchar(A[i] + '0');
return ;
}
};
LL f[maxn];
LL C(LL n, int m) {
LL sum = 1;
for(LL i = n; i >= n - m + 1; i--) sum *= i;
for(int i = 1; i <= m; i++) sum /= i;
return sum;
}
LL dp(int n) {
LL& ans = f[n];
if(ans > 0) return ans;
if(!n) return ans = 1;
// printf("dp(%d)\n", n);
for(int s1 = n - 1; s1 >= 0; s1--)
for(int s2 = s1; s2 >= 0; s2--) {
int s3 = n - s1 - s2 - 1;
if(s3 < 0) continue;
if(s2 < s3) break;
// printf("%d %d %d\n", s1, s2, s3);
if(s1 == s3) ans += C(dp(s1) + 2, 3);
else if(s1 == s2)
ans += C(dp(s1) + 1, 2) * dp(s3);
else if(s2 == s3)
ans += C(dp(s2) + 1, 2) * dp(s1);
else ans += dp(s1) * dp(s2) * dp(s3);
// printf("dp(%d): %lld\n", n, ans);
}
// printf("%d: %d\n", n, ans);
return ans;
}
string Names[15] = {"¼×", "ÒÒ", "±û", "¶¡", "Îì", "¼º", "¸ý", "ÐÁ", "ÈÉ", "¹ï"};
string getname(int n) {
if(n <= 10) return Names[n-1] + string("Íé");
char Name[maxn]; memset(Name, 0, sizeof(Name));
sprintf(Name, "%dÍé", n);
return string(Name);
}
int main() {
while(1) {
int n = read();
LL ans = 0, getrid = 0;
if(n & 1) {
for(int s1 = (n >> 1); s1 >= 0; s1--)
for(int s2 = s1; s2 >= 0; s2--)
for(int s3 = s2; s3 >= 0; s3--) {
int s4 = n - s1 - s2 - s3 - 1;
if(s4 < 0) continue;
if(s3 < s4) break;
// printf("%d %d %d %d\n", s1, s2, s3, s4);
if(s1 == s4) ans += C(dp(s1) + 3, 4);
else if(s1 == s3) ans += C(dp(s1) + 2, 3) * dp(s4);
else if(s2 == s4) ans += C(dp(s2) + 2, 3) * dp(s1);
else {
int S[4] = {s1, s2, s3, s4}, l = 0, r = 0;
LL tmp = 1;
while(r < 4) {
while(r < 3 && S[l] == S[r+1]) r++;
tmp *= C(dp(S[l]) + r - l, r - l + 1);
l = r = r + 1;
}
ans += tmp;
}
}
}
else {
for(int s1 = (n >> 1) - 1; s1 >= 0; s1--)
for(int s2 = s1; s2 >= 0; s2--)
for(int s3 = s2; s3 >= 0; s3--) {
int s4 = n - s1 - s2 - s3 - 1;
if(s4 < 0) continue;
if(s3 < s4) break;
// printf("%d %d %d %d\n", s1, s2, s3, s4);
if(s1 == s4) ans += C(dp(s1) + 3, 4);
else if(s1 == s3) ans += C(dp(s1) + 2, 3) * dp(s4);
else if(s2 == s4) ans += C(dp(s2) + 2, 3) * dp(s1);
else {
int S[4] = {s1, s2, s3, s4}, l = 0, r = 0;
LL tmp = 1;
while(r < 4) {
while(r < 3 && S[l] == S[r+1]) r++;
tmp *= C(dp(S[l]) + r - l, r - l + 1);
l = r = r + 1;
}
ans += tmp;
}
}
ans += C(dp(n >> 1) + 1, 2);
}
cout << getname(n); printf("ÓÐ "); ans.print(); puts(" ÖÖͬ·ÖÒì¹¹Ìå");
}
return 0;
}
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