洛谷P4841 城市规划(生成函数 多项式求逆)

题意

链接

Sol

Orz yyb

一开始想的是直接设\(f_i\)表示\(i\)个点的无向联通图个数，枚举最后一个联通块转移，发现有一种情况转移不到。。。

正解是先设\(g(n)\)表示\(n\)个点的无向图个数，这个方案是\(2^{\frac{i(i-1)}{2}}\)(也就是考虑每条边选不选)

考虑如何得到\(g\)

\[g(n) = \sum_{i=0}^n C_{n-1}^{i-1}f(i) g(n-i)
\]

直接将\(2^{\frac{n(n-1)}{2}}\)带入然后化简一下可以得到这个式子

\[\frac{2^{C_n^2}}{(n-1)!} = \sum_{i=1}^n \frac{f(i)}{(i-1)!} \frac{2^{C_{n-i}^2}}{(n-i)!}
\]

然后就可以多项式求逆啦。

#include<bits/stdc++.h>
#define Pair pair<int, int>
#define MP(x, y) make_pair(x, y)
#define fi first
#define se second
#define LL long long
#define ull unsigned long long
#define Fin(x) {freopen(#x".in","r",stdin);}
#define Fout(x) {freopen(#x".out","w",stdout);}
using namespace std;
const int MAXN = 2e6 + 10, INF = 1e9 + 1;
const double eps = 1e-9, pi = acos(-1);
inline int read() {
    char c = getchar(); int x = 0, f = 1;
    while(c < '0' || c > '9') {if(c == '-') f = -1; c = getchar();}
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar();
    return x * f;
}
int N, M, a[MAXN], b[MAXN], c[MAXN], d[MAXN], fac[MAXN], ifac[MAXN];
namespace Poly {
    int rev[MAXN], GPow[MAXN], A[MAXN], B[MAXN], C[MAXN], lim, INV2;
    const int G = 3, mod = 1004535809, mod2 = 1004535808;
    template <typename A, typename B> inline LL add(A x, B y) {if(x + y < 0) return x + y + mod; return x + y >= mod ? x + y - mod : x + y;}
    template <typename A, typename B> inline void add2(A &x, B y) {if(x + y < 0) x = x + y + mod; else x = (x + y >= mod ? x + y - mod : x + y);}
    template <typename A, typename B> inline LL mul(A x, B y) {return 1ll * x * y % mod;}
    template <typename A, typename B> inline void mul2(A &x, B y) {x = (1ll * x * y % mod + mod) % mod;}
    int fp(int a, int p, int P = mod) {
        int base = 1;
        for(; p > 0; p >>= 1, a = 1ll * a * a % P) if(p & 1) base = 1ll * base *  a % P;
        return base;
    }
    int inv(int x) {
    	return fp(x, mod - 2);
	}
    int GetLen(int x) {
        int lim = 1;
        while(lim <= x) lim <<= 1;
        return lim;
    }
    int GetOrigin(int x) {//¼ÆËãÔ­¸ù
        static int q[MAXN]; int tot = 0, tp = x - 1;
        for(int i = 2; i * i <= tp; i++) if(!(tp % i)) {q[++tot] = i;while(!(tp % i)) tp /= i;}
        if(tp > 1) q[++tot] = tp;
        for(int i = 2, j; i <= x - 1; i++) {
            for(j = 1; j <= tot; j++) if(fp(i, (x - 1) / q[j], x) == 1) break;
            if(j == tot + 1) return i;
        }
        return -1;
    }
    void Init(/*int P,*/ int Lim) {
        INV2 = fp(2, mod - 2);
        for(int i = 1; i <= Lim; i++) GPow[i] = fp(G, (mod - 1) / i);
    }
    void NTT(int *A, int lim, int opt) {
        int len = 0; for(int N = 1; N < lim; N <<= 1) ++len;
        for(int i = 1; i <= lim; i++) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (len - 1));
        for(int i = 0; i <= lim; i++) if(i < rev[i]) swap(A[i], A[rev[i]]);
        for(int mid = 1; mid < lim; mid <<= 1) {
            int Wn = GPow[mid << 1];
            for(int i = 0; i < lim; i += (mid << 1)) {
                for(int j = 0, w = 1; j < mid; j++, w = mul(w, Wn)) {
                    int x = A[i + j], y = mul(w, A[i + j + mid]);
                    A[i + j] = add(x, y), A[i + j + mid] = add(x, -y);
                }
            }
        }
        if(opt == -1) {
            reverse(A + 1, A + lim);
            int Inv = fp(lim, mod - 2);
            for(int i = 0; i <= lim; i++) mul2(A[i], Inv);
        }
    }
    void Mul(int *a, int *b, int N, int M) {
        memset(A, 0, sizeof(A)); memset(B, 0, sizeof(B));
        int lim = 1, len = 0;
        while(lim <= N + M) len++, lim <<= 1;
        for(int i = 0; i <= N; i++) A[i] = a[i];
        for(int i = 0; i <= M; i++) B[i] = b[i];
        NTT(A, lim, 1); NTT(B, lim, 1);
        for(int i = 0; i <= lim; i++) B[i] = mul(B[i], A[i]);
        NTT(B, lim, -1);
        for(int i = 0; i <= N + M; i++) b[i] = B[i];
        memset(A, 0, sizeof(A)); memset(B, 0, sizeof(B));
    }
    void Inv(int *a, int *b, int len) {//B1 = 2B - A1 * B^2
        if(len == 1) {b[0] = fp(a[0], mod - 2); return ;}
        Inv(a, b, len >> 1);
        for(int i = 0; i < len; i++) A[i] = a[i], B[i] = b[i];
        NTT(A, len << 1, 1); NTT(B, len << 1, 1);
        for(int i = 0; i < (len << 1); i++) mul2(A[i], mul(B[i], B[i]));
        NTT(A, len << 1, -1);
        for(int i = 0; i < len; i++) add2(b[i], add(b[i], -A[i]));
        for(int i = 0; i < (len << 1); i++) A[i] = B[i] = 0;
    }
    void Dao(int *a, int *b, int len) {
        for(int i = 1; i < len; i++) b[i - 1] = mul(i, a[i]); b[len - 1] = 0;
    }
    void Ji(int *a, int *b, int len) {
        for(int i = 1; i < len; i++) b[i] = mul(a[i - 1], fp(i, mod - 2)); b[0] = 0;
    }
    void Ln(int *a, int *b, int len) {//G(A) = \frac{A}{A'} qiudao zhihou jifen
        static int A[MAXN], B[MAXN];
        Dao(a, A, len);
        Inv(a, B, len);
        NTT(A, len << 1, 1); NTT(B, len << 1, 1);
        for(int i = 0; i < (len << 1); i++) B[i] = mul(A[i], B[i]);
        NTT(B, len << 1, -1);
        Ji(B, b, len << 1);
        memset(A, 0, sizeof(A)); memset(B, 0, sizeof(B));
    }
    void Exp(int *a, int *b, int len) {//F(x) = F_0 (1 - lnF_0 + A) but code ..why....
        if(len == 1) return (void) (b[0] = 1);
        Exp(a, b, len >> 1); Ln(b, C, len);
        C[0] = add(a[0] + 1, -C[0]);
        for(int i = 1; i < len; i++) C[i] = add(a[i], -C[i]);
        NTT(C, len << 1, 1); NTT(b, len << 1, 1);
        for(int i = 0; i < (len << 1); i++) mul2(b[i], C[i]);
        NTT(b, len << 1, -1);
        for(int i = len; i < (len << 1); i++) C[i] = b[i] = 0;
    }
    void Sqrt(int *a, int *b, int len) {
        static int B[MAXN];
        Ln(a, B, len);
        for(int i = 0; i < len; i++) B[i] = mul(B[i], INV2);
        Exp(B, b, len);
    }
};
using namespace Poly; 
signed main() {
    N = read(); int Lim = GetLen(N); Init(4 * Lim);
    fac[0] = 1;
    for(int i = 1; i <= N; i++) fac[i] = mul(i, fac[i - 1]);
    ifac[N] = fp(fac[N], mod - 2);
    for(int i = N; i >= 1; i--) ifac[i - 1] = mul(i, ifac[i]);
    for(int i = N; i >= 1; i--) {
    	int tmp;
		if(i & 1) tmp = fp(2, 1ll * ((i - 1) / 2) * i % mod2);
		else tmp = fp(2, 1ll * (i / 2) * (i - 1) % mod2);
		a[i] = mul(tmp, ifac[i - 1]),
    	b[i] = mul(tmp, ifac[i]);
	}
	b[0] = 1;
    Inv(b, c, Lim);
    Mul(a, c, Lim, Lim);
	cout << mul(c[N], fac[N - 1]);
    return 0;
}