「BZOJ 4228」Tibbar的后花园

「BZOJ 4228」Tibbar的后花园

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解题思路

可以证明最终的图中所有点的度数都 \(< 3\) ，且不存在环长是 \(3\) 的倍数的环。这是充分必要的，由于图不联通，其就是由若干个联通块组成的，每个联通块是一条链或者环长不是 \(3\) 的倍数的环，然后强上EGF就好了。

列出链的EGF和环的EGF

\[A(x)=x+\sum_{i\geq2}\dfrac{x^i}{2} \\ B(x)=\sum_{i>3,i\bmod3>0} \dfrac{x^i}{2i} \]

答案的EGF就是 \(\exp(A(x)+B(x))\) ，多项式 \(\exp\) 完再乘个阶乘，复杂度 \(\mathcal O(n\log n)\) ，需要板子比较快。

code

/*program by mangoyang*/
#include<bits/stdc++.h>
#define inf (0x7f7f7f7f)
#define Max(a, b) ((a) > (b) ? (a) : (b))
#define Min(a, b) ((a) < (b) ? (a) : (b))
typedef long long ll;
using namespace std;
template <class T>
inline void read(T &x){
    int ch = 0, f = 0; x = 0;
    for(; !isdigit(ch); ch = getchar()) if(ch == '-') f = 1;
    for(; isdigit(ch); ch = getchar()) x = x * 10 + ch - 48;
    if(f) x = -x;
}
const int N = (1 << 22) + 5, P = 1004535809, G = 3;
namespace poly{
	int rev[N], W[N], invW[N], len, lg;	
	inline int Pow(int a, int b){
		int ans = 1;
		for(; b; b >>= 1, a = 1ll * a * a % P)
			if(b & 1) ans = 1ll * ans * a % P;
		return ans;
	}
	inline void init(){
		for(int k = 2; k < N; k <<= 1)
			W[k] = Pow(G, (P - 1) / k), invW[k] = Pow(W[k], P - 2);
	}
	inline void timesinit(int lenth){
		for(len = 1, lg = 0; len <= lenth; len <<= 1, lg++);
		for(int i = 0; i < len; i++)
			rev[i] = (rev[i>>1] >> 1) | ((i & 1) << (lg - 1));
	}
	inline void DFT(int *a, int sgn){
		for(int i = 0; i < len; i++) if(i < rev[i]) swap(a[i], a[rev[i]]);
		for(int k = 2; k <= len; k <<= 1){
			int w = ~sgn ? W[k] : invW[k];
			for(int i = 0; i < len; i += k){
				int now = 1;
				for(int j = i; j < i + (k >> 1); j++){
					int x = a[j], y = 1ll * a[j+(k>>1)] * now % P;
					a[j] = (x + y) % P, a[j+(k>>1)] = (x - y + P) % P;
					now = 1ll * now * w % P;
				}
			}
		}
		if(sgn == -1){
			int Inv = Pow(len, P - 2);
			for(int i = 0; i < len; i++) a[i] = 1ll * a[i] * Inv % P;
		}
	}
	inline void getinv(int *a, int *b, int n){
		static int tmp[N];
		if(n == 1) return (void) (b[0] = Pow(a[0], P - 2));
		getinv(a, b, (n + 1) / 2);
		timesinit(n * 2 - 1);
		for(int i = 0; i < len; i++) tmp[i] = i < n ? a[i] : 0;
		DFT(tmp, 1), DFT(b, 1);
		for(int i = 0; i < len; i++) 
			b[i] = 1ll * (2 - 1ll * tmp[i] * b[i] % P + P) % P * b[i] % P;
		DFT(b, -1);
		for(int i = n; i < len; i++) b[i] = 0;
		for(int i = 0; i < len; i++) tmp[i] = 0;
	}
	inline void getsqrt(int *a, int *b, int n){
		static int tmp1[N], tmp2[N];
		if(n == 1) return (void) (b[0] = 1);
		getsqrt(a, b, (n + 1) / 2);
		for(int i = 0; i < n; i++) tmp1[i] = a[i];
		getinv(b, tmp2, n), timesinit(n * 2 - 1);
		DFT(tmp1, 1), DFT(tmp2, 1);
		for(int i = 0; i < len; i++) tmp1[i] = 1ll * tmp1[i] * tmp2[i] % P;
		DFT(tmp1, -1);
		for(int i = 0; i < len; i++) 
			b[i] = 1ll * (b[i] + tmp1[i]) % P * Pow(2, P - 2) % P; 
		for(int i = n; i < len; i++) b[i] = 0;
		for(int i = 0; i < len; i++) tmp1[i] = tmp2[i] = 0;
	}
	inline void getln(int *a, int *b, int n){
		static int tmp[N];
		getinv(a, b, n), timesinit(n * 2 - 1);
		for(int i = 1; i < n; i++) tmp[i-1] = 1ll * a[i] * i % P;
		DFT(tmp, 1), DFT(b, 1);
		for(int i = 0; i < len; i++) b[i] = 1ll * tmp[i] * b[i] % P;
		DFT(b, -1);
		for(int i = len - 1; i; i--) b[i] = 1ll * b[i-1] * Pow(i, P - 2) % P;
		b[0] = 0;
		for(int i = n; i < len; i++) b[i] = 0;
		for(int i = 0; i < len; i++) tmp[i] = 0;
	}
	inline void getexp(int *a, int *b, int n){
		static int tmp[N]; 
		if(n == 1) return (void) (b[0] = 1);
		getexp(a, b, (n + 1) / 2);
		getln(b, tmp, n), timesinit(n * 2 - 1);
		for(int i = 0; i < n; i++) tmp[i] = (!i - tmp[i] + a[i] + P) % P;
		DFT(tmp, 1), DFT(b, 1);
		for(int i = 0; i < len; i++) b[i] = 1ll * b[i] * tmp[i] % P;
		DFT(b, -1);
		for(int i = n; i < len; i++) b[i] = 0;
		for(int i = 0; i < len; i++) tmp[i] = 0;
	}
	inline void power(int *a, int *b, int n, int m, ll k){
		static int tmp[N];
		for(int i = 0; i < m; i++) b[i] = 0;
		int fir = -1;
		for(int i = 0; i < n; i++) if(a[i]){ fir = i; break; }
		if(fir && k >= m) return;
		if(fir == -1 || 1ll * fir * k >= m) return;
		for(int i = fir; i < n; i++) b[i-fir] = a[i];
		for(int i = 0; i < n - fir; i++) 
			b[i] = 1ll * b[i] * Pow(a[fir], P - 2) % P;
		getln(b, tmp, m);
		for(int i = 0; i < m; i++) 
			b[i] = 1ll * tmp[i] * (k % P) % P, tmp[i] = 0;
		getexp(b, tmp, m);
		for(int i = m; i >= fir * k; i--) 
			b[i] = 1ll * tmp[i-fir*k] * Pow(a[fir], k % (P - 1)) % P;
		for(int i = 0; i < fir * k; i++) b[i] = 0;
		for(int i = 0; i < m; i++) tmp[i] = 0;
	}
}
using poly::Pow;
using poly::DFT;
using poly::timesinit;
int a[N], b[N], A[N], f[N], g[N], n, m;
int main(){
	poly::init(), read(n);
	int inv2 = Pow(2, P - 2);
	a[1] = 1;
	for(int i = 2; i <= n; i++) a[i] = inv2;
	for(int i = 4; i <= n; i++) if(i % 3 > 0) 
		(a[i] += 1ll * inv2 * Pow(i, P - 2) % P) %= P;
	poly::getexp(a, b, n + 1);
	int fac = 1;
	for(int i = 1; i <= n; i++) fac = 1ll * fac * i % P;
	cout << 1ll * fac * b[n] % P << endl;
	return 0;
}