题解 有标号DAG计数

题目传送门

题目大意

给出\(n\)，求出对于任意\(t\in[1,n]\)，点数为\(t\)的弱联通\(\texttt{DAG}\)个数。答案对\(998244353\)取模。

\(n\le 10^5\)

思路

看到\(\texttt{Karry5307}\)的题解里面有很多小问题（但这并不影响\(\texttt {Karry AK IOI}\)），这里给一篇可能没有什么错误的题解。

我们发现直接求似乎不是很好求，我们发现弱连通图组合在一起的话，就相当于一个不保证联通的\(\texttt{DAG}\)，于是我们的目标就是如何求出不保证联通性的\(\texttt{DAG}\)的个数。

我们设\(g_n\)为有\(n\)个点的不保证联通的\(\texttt{DAG}\)的个数，我们可以得到转移式:

\[g_n=\sum_{i=1}^{n} \binom{n}{i}2^{i(n-i)}g_{n-i}(-1)^{i-1}
\]

这个式子的意思就是，我们可以先选\(i\)个点入入度为\(0\)，然后其余的点构成\(\texttt{DAG}\)，这两部分之间的边随便连不连都能满足条件。但是我们并不能恰好有\(i\)个点入度为\(0\)，我们只能保证至少有\(i\)个点入度为\(0\)，所以我们就需要容斥一下。

我们发现这个式子中最难看的就是\(2^{i(n-i)}\)，而我们发现这个可以用\(\texttt{Bluestein}\)拆成\(\frac{(\sqrt{2})^{n^2}}{(\sqrt{2})^{i^2}(\sqrt{2})^{(n-i)^2}}\)，当然如果你喜欢的话你也可以拆成:\(\frac{2^{\binom{n}{2}}}{2^{\binom{i}{2}}2^{\binom{n-i}{2}}}\)。我用的是第一种拆成平方的方法。

于是，式子就变成了:

\[g_n=\sum_{i=1}^{n} \binom{n}{i}\frac{(\sqrt{2})^{n^2}}{(\sqrt{2})^{i^2}(\sqrt{2})^{(n-i)^2}}g_{n-i}(-1)^{i-1}
\]

而我们又有:

\[\sqrt{2}\equiv 116195171\pmod {998244353}
\]

所以我们暂时还没有碰到什么困难。

我们发现我们如果设:

\[G(x)=\sum_{i=0}^{n} \frac{g_i}{(\sqrt{2})^{i^2}i!}x^i
\]

\[H(x)=\sum_{i=1}^{n} \frac{(-1)^{i-1}}{(\sqrt{2})^{i^2}i!}x^i
\]

我们就可以得到:

\[G(x)=G(x)H(x)+1
\]

这个代一下就可以得到了。

于是，我们就得到:

\[G(x)=\frac{1}{1-H(x)}
\]

于是，我们就可以求到\(g_i\)了。我们考虑弱连通图和不保证连通性的\(\texttt{DAG}\)之间的关系。我们发现，其实不保证连通性的\(\texttt{DAG}\)就是把一堆弱连通图揉成一坨，于是，如果我们设\(F(x)\)为\(\{g_{0,1,2,...,n}\}\)的指数生成函数，那么，弱连通图的指数型生成函数就是:

\[\ln F(x)
\]

于是，我们就在\(\Theta(n\log n)\)的时间复杂度内解决了这个问题。

\(\text {Code}\)

#include <bits/stdc++.h>
using namespace std;

#define SZ(x) ((int)x.size())
#define Int register int
#define sqr2 116195171
#define mod 998244353
#define MAXN 1000005

int mul (int a,int b){return 1ll * a * b % mod;}
int dec (int a,int b){return a >= b ? a - b : a + mod - b;}
int add (int a,int b){return a + b >= mod ? a + b - mod : a + b;}
int qkpow (int a,int k){
	int res = 1;for (;k;k >>= 1,a = 1ll * a * a % mod) if (k & 1) res = 1ll * res * a % mod;
	return res;
}
int inv (int x){return qkpow (x,mod - 2);}

typedef vector <int> poly;

int rev[MAXN];

void ntt (poly &f,int lim,int type){
#define G 3
#define Gi 332748118
	int l = log2 (lim);
	for (Int i = 0;i < lim;++ i) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << l - 1);
	for (Int i = 0;i < lim;++ i) if (i < rev[i]) swap (f[i],f[rev[i]]);
	for (Int i = 1;i < lim;i <<= 1){
		int Wn = qkpow (type == 1 ? G : Gi,(mod - 1) / (i << 1));
		for (Int j = 0;j < lim;j += i << 1)
			for (Int k = 0,w = 1;k < i;++ k,w = 1ll * w * Wn % mod){
				int x = f[j + k],y = 1ll * w * f[i + j + k] % mod;
				f[j + k] = (x + y) % mod,f[i + j + k] = (x + mod - y) % mod;
			}
	}
	if (type == 1) return ;
	for (Int i = 0,Inv = inv (lim);i < lim;++ i) f[i] = 1ll * f[i] * Inv % mod;
#undef G
#undef Gi
}

poly operator + (poly a,poly b){
	a.resize (max (SZ (a),SZ (b)));
	for (Int i = 0;i < SZ (b);++ i) a[i] = add (a[i],b[i]);
	return a;
}

poly operator - (poly a,poly b){
	a.resize (max (SZ (a),SZ (b)));
	for (Int i = 0;i < SZ (b);++ i) a[i] = dec (a[i],b[i]);
	return a;
}

poly operator * (poly a,int b){
	for (Int i = 0;i < SZ (a);++ i) a[i] = mul (a[i],b);
	return a;
}

poly operator * (poly a,poly b){
	int d = SZ (a) + SZ (b) - 1,lim = 1;while (lim < d) lim <<= 1;
	a.resize (lim),b.resize (lim);
	ntt (a,lim,1),ntt (b,lim,1);
	for (Int i = 0;i < lim;++ i) a[i] = mul (a[i],b[i]);
	ntt (a,lim,-1),a.resize (d);
	return a;
}

poly operator << (poly a,int n){
	a.resize (SZ (a) + n);
	for (Int i = SZ (a) - 1;~i;-- i) a[i] = (i >= n ? a[i - n] : 0);
	return a;
}

poly inv (poly a,int n){
	poly b(1,inv (a[0])),c;
	for (Int l = 4;(l >> 2) < n;l <<= 1){
		c.resize (l >> 1);
		for (Int i = 0;i < (l >> 1);++ i) c[i] = i < n ? a[i] : 0;
		c.resize (l),b.resize (l);
		ntt (c,l,1),ntt (b,l,1);
		for (Int i = 0;i < l;++ i) b[i] = mul (b[i],dec (2,mul (b[i],c[i])));
		ntt (b,l,-1),b.resize (l >> 1);
	}
	b.resize (n);
	return b;
}

poly inv (poly a){return inv (a,SZ (a));}
poly der (poly a){
	for (Int i = 0;i < SZ (a) - 1;++ i) a[i] = mul (a[i + 1],i + 1);
	a.pop_back ();return a;
}
poly ine (poly a){
	a.push_back (0);
	for (Int i = SZ (a) - 1;i;-- i) a[i] = mul (a[i - 1],inv (i));
	a[0] = 0;return a;
}

poly ln (poly a,int n){
	a = ine (der (a) * inv (a));
	a.resize (n);
	return a;
}
poly ln (poly a){return ln (a,SZ (a));}

poly exp (poly a,int n){
	poly b (1,1),c;
	for (Int l = 2;(l >> 1) < n;l <<= 1){
		b.resize (l),c = ln (b);
		for (Int i = 0;i < l;++ i) c[i] = dec (i < n ? a[i] : 0,c[i]);
		c[0] = add (c[0],1);
		b = b * c,b.resize (l);
	}
	b.resize (n);
	return b;
}
poly exp (poly a){return exp (a,SZ (a));}

template <typename T> inline void read (T &t){t = 0;char c = getchar();int f = 1;while (c < '0' || c > '9'){if (c == '-') f = -f;c = getchar();}while (c >= '0' && c <= '9'){t = (t << 3) + (t << 1) + c - '0';c = getchar();} t *= f;}
template <typename T,typename ... Args> inline void read (T &t,Args&... args){read (t);read (args...);}
template <typename T> inline void write (T x){if (x < 0){x = -x;putchar ('-');}if (x > 9) write (x / 10);putchar (x % 10 + '0');}

poly H;
int n,fac[MAXN],ifac[MAXN];

signed main(){
	read (n);
	fac[0] = 1;for (Int i = 1;i <= n;++ i) fac[i] = mul (fac[i - 1],i);
	ifac[n] = inv (fac[n]);for (Int i = n;i;-- i) ifac[i - 1] = mul (ifac[i],i);
	H.resize (n + 1);for (Int i = 1;i <= n;++ i) H[i] = inv (mul (fac[i],qkpow (sqr2,1ll * i * i % (mod - 1)))),H[i] = i & 1 ? mod - H[i] : H[i];
	H[0] = 1,H = inv (H);for (Int i = 0;i <= n;++ i) H[i] = mul (H[i],qkpow (sqr2,1ll * i * i % (mod - 1)));H = ln (H);
	for (Int i = 1;i <= n;++ i) write (mul (H[i],fac[i])),putchar ('\n');
	return 0;
}

一个小小的总结

其实做了几道多项式与图计数的题目之后可以发现，对于一些不是很好求到的答案我们采用的办法就是把所求多项式与更好求的图的多项式建立关系，从而反推出该图的生成函数。反推的方法就不计其数了。