Luogu4233 射命丸文的笔记 DP、多项式求逆

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注意到总共有\(\frac{n!}{n}\)条本质不同的哈密顿回路，每一条哈密顿回路恰好会出现在\(2^{\binom{n}{2} - n}\)个图中，所以我们实际上要算的是强连通有向竞赛图的数量。

设\(f_i\)表示点数为\(i\)的强连通竞赛图数，转移考虑用总数\(2^\binom{i}{2}\)减去不强连通的图数量。如果竞赛图不强连通，我们可以枚举拓扑序最靠后的一个强连通子图，如果它的大小为\(j\)，那么剩下\(i-j\)个点之间的边可以任意连，但是这\(i-j\)个和这\(j\)个点之间的边的方向是确定的，可以得到转移\(f_i = 2^\binom{i}{2} - \sum\limits_{j=1}^{i-1} \binom{i}{j} 2^\binom{i-j}{2} f_j\)，直接做复杂度\(O(n^2)\)

然后考虑优化。把\(\binom{i}{j}\)拆开然后左右两边各除以\(i!\)得到\(\frac{f_i}{i!} = \frac{2^\binom{i}{2}}{i!} - \sum\limits_{j=1}^{i-1} \frac{f_j}{j!} \frac{2^\binom{i-j}{2}}{i-j!}\)。设多项式\(F = \sum\limits_{i=1}^n \frac{f_i}{i!}x^i\)，\(G = \sum\limits_{i=1}^n \frac{2^\binom{i}{2}}{i!} x^i\)，可以得到\(F = G - F * G\)，即\(F = \frac{G}{G + 1}\)，多项式求逆即可。

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<vector>
#include<queue>
#include<cstring>
#include<iomanip>
#include<cmath>
#include<cassert>
//This code is written by Itst
using namespace std;

#define int long long
const int MOD = 998244353 , _ = (1 << 18) + 3;
int jc[_] , inv[_] , G[_] , H[_] , N;

#define ch2(x) ((x) * ((x) - 1) / 2)

int poww(int a , int b){
    int tms = 1;
    while(b){
        if(b & 1) tms = tms * a % MOD;
        a = a * a % MOD;
        b >>= 1;
    }
    return tms;
}

namespace poly{
    const int G = 3 , INV = 332748118;
    int dir[_] , need , invnd;

    void init(int len){
        need = 1;
        while(need < len) need <<= 1;
        invnd = poww(need , MOD - 2);
        for(int i = 1 ; i < need ; ++i)
            dir[i] = (dir[i >> 1] >> 1) | (i & 1 ? need >> 1 : 0);
    }

    void NTT(int *arr , int tp){
        for(int i = 1 ; i < need ; ++i)
            if(i < dir[i])
                arr[i] ^= arr[dir[i]] ^= arr[i] ^= arr[dir[i]];
        for(int i = 1 ; i < need ; i <<= 1){
            int wn = poww(tp == 1 ? G : INV , MOD / i / 2);
            for(int j = 0 ; j < need ; j += i << 1){
                int w = 1;
                for(int k = 0 ; k < i ; ++k , w = w * wn % MOD){
                    int x = arr[j + k] , y = arr[i + j + k] * w % MOD;
                    arr[j + k] = x + y >= MOD ? x + y - MOD : x + y;
                    arr[i + j + k] = x < y ? x + MOD - y : x - y;
                }
            }
        }
        if(tp != 1)
            for(int i = 0 ; i < need ; ++i)
                arr[i] = arr[i] * invnd % MOD;
    }

#define clr(x) memset(x , 0 , sizeof(int) * need)
    int A[_] , B[_];
    void getInv(int *a , int *b , int len){
        if(len == 1) return (void)(b[0] = poww(a[0] , MOD - 2));
        getInv(a , b , (len + 1) >> 1);
        memcpy(A , a , sizeof(int) * len);
        memcpy(B , b , sizeof(int) * len);
        init(len * 2 + 3); NTT(A , 1); NTT(B , 1);
        for(int i = 0 ; i < need ; ++i)
            A[i] = A[i] * B[i] % MOD * B[i] % MOD;
        NTT(A , -1);
        for(int i = 0 ; i < len ; ++i)
            b[i] = (2 * b[i] - A[i] + MOD) % MOD;
        clr(A); clr(B);
    }
}

signed main(){
    cin >> N;
    jc[0] = 1;
    for(int i = 1 ; i <= N ; ++i) jc[i] = jc[i - 1] * i % MOD;
    inv[N] = poww(jc[N] , MOD - 2);
    for(int i = N - 1 ; i >= 0 ; --i) inv[i] = inv[i + 1] * (i + 1) % MOD;
    for(int i = 1 ; i <= N ; ++i) G[i] = 1ll * poww(2 , ch2(i)) * inv[i] % MOD;
    G[0] = 1; poly::getInv(G , H , N + 1); G[0] = 0;
    poly::init(2 * N + 2); poly::NTT(G , 1); poly::NTT(H , 1);
    for(int i = 0 ; i < poly::need ; ++i)
        G[i] = 1ll * G[i] * H[i] % MOD;
    poly::NTT(G , -1);
    for(int i = 1 ; i <= N ; ++i)
        printf("%d\n" , i == 1 ? 1 : (i == 2 ? -1 : poww(G[i] , MOD - 2) * inv[i] % MOD * jc[i - 1] % MOD * poww(2 , ch2(i) - i) % MOD));
    return 0;
}