「新年的追逐战」题解

我们研究\(G_1 \times G_2\)的连通块的性质。不妨设\(G_1, G_2\)连通，且都不是孤立点。(若不连通，可以拆成若干连通块。若\(G_1\)是孤立点，则答案就是\(\lvert V(G_2) \rvert\)）

考虑\(G_1 \times G_2\)的两个点\((u_1, u_2), (v_1, v_2)\)连通的条件是：
存在两条长度相同的可以经过重复点的路径，使得第一条路径的边在\(G_1\)上，起点为\(u_1\)，终点为\(v_1\)，第二条路径在\(G_2\)上，起点为\(u_2\)，终点为\(v_2\)。

注意到路径长度（此处均可以经过重复点）在充分大的时候只与图是否是二分图，以及两个点是否在同一区域里有关。通过对路径奇偶性的分析容易得出下列结论：

当\(G_1, G_2\)都不是二分图的时候，形成一个非二分图的连通块。
当\(G_1, G_2\)有恰好一个二分图的时候，形成一个二分图的连通块。
当\(G_1, G_2\)全是二分图的时候，形成两个二分图的连通块。

于是，我们发现对于一张图，只需要关心孤立点、二分图的连通块、非孤立点的连通块的性质！

假设这\(n\)个图中有\(a_1, a_2, ..., a_n\)的非孤立点，那么孤立点对连通块的贡献为：\(\prod_{i = 1}^{n} m_i - \prod_{i = 1}^{n} a_i\)。
这一部分我们只需计算出\(a_1, ..., a_n\)的期望。计算每个点是孤立点的概率，容易得到\(E(a_i) = [1 - (\frac{1}{2})^{m_i - 1})]m_i\)。

接下来处理非孤立点的贡献。现在设这\(n\)个图都没有孤立点，我们记二元组 \((A_i, B_i)\)，\(A_i\)表示\(G_i\)的连通块个数，\(B_i\)表示\(G_i\)的二分图的连通块的个数。

那么根据前面的结论，\(G_1 \times G_2\)对应的二元组是\((A_1B_1 + A_2B_2, A_1B_2 + A_2B_1)\)。这正好是异或卷积的形式!所以我们只需要求出\(A_i + B_i\)，\(A_i - B_i\)的期望，再乘起来做\(IFWT\)即可。

记\(M = \max_{i = 1}^{n} m_i\)，\(F(x) = \sum_{i = 0}^{M} \frac{2^{\frac{i(i - 1)}{2}}}{i!} x^i\)。则\(\ln F(x)\)的每项系数就是\(i\)阶连通图的个数的EGF。\(F(x) \ln F(x)\)就是\(n\)阶图总连通块数的EGF。于是\(A_i\)就求出来了。

记\(G(x) = \sum_{i = 0}^{M} \sum_{j = 0}^{i} \frac{2^{j(i - j)}}{j!(i - j)!}\)，它的每一项意义表示把\(i\)阶图的每个点二染色以及连边的情况数。那么\(\frac{\ln G(x)}{2}\)就是\(i\)阶连通二分图的个数的EGF。那么\(\frac{F(x) \ln G(x)}{2}\)就是\(n\)阶图的总的二分图的连通块数的EGF。

现在唯一的问题就是求出\(G(x)\)了。这只需要把\(j(i - j)\)看成\(\frac{i(i - 1)}{2} - \frac{j(j - 1)}{2} - \frac{(i - j)(i - j - 1)}{2}\)，然后做一次卷积即可！

时间复杂度\(O(M \log M)\)，可以获得\(100\)分。
代码如下：

#include <bits/stdc++.h>
#define debug(x) cerr << #x << " " << (x) << endl
using namespace std;

const int N = 100005, M = 100005, K = 21;
const long long mod = 998244353ll, R = 3ll;

template <class T>
void read (T &x) {
	int sgn = 1;
	char ch;
	x = 0;
	for (ch = getchar(); (ch < '0' || ch > '9') && ch != '-'; ch = getchar()) ;
	if (ch == '-') ch = getchar(), sgn = -1;
	for (; '0' <= ch && ch <= '9'; ch = getchar()) x = x * 10 + ch - '0';
	x *= sgn;
}
template <class T>
void write (T x) {
	if (x < 0) putchar('-'), write(-x);
	else if (x < 10) putchar(x + '0');
	else write(x / 10), putchar(x % 10 + '0');
}

int n, a[N], m = 0;
long long w[K][M << 2], inv[M << 2], fac[M << 2], inv_fac[M << 2];
long long qpow (long long x, long long y) {
	long long res = 1ll;
	for (; y; y >>= 1, x = x * x % mod) {
		if (y & 1) res = res * x % mod;
	}
	return res;
}
void init () {
	inv[1] = 1ll;
	for (int i = 2; i <= (m + 2 << 2); i++) inv[i] = (mod / i) * (mod - inv[mod % i]) % mod;
	fac[0] = 1ll;
	for (int i = 1; i <= (m + 2 << 2); i++) fac[i] = fac[i - 1] * i % mod;
	inv_fac[0] = 1ll;
	for (int i = 1; i <= (m + 2 << 2); i++) inv_fac[i] = inv_fac[i - 1] * inv[i] % mod;
	
	for (int i = 1, j = 0; i <= (m + 2 << 2); i <<= 1, j++) {
		w[j][0] = 1ll, w[j][1] = qpow(R, (mod - 1) >> j);
		for (int k = 2; k <= i; k++) w[j][k] = w[j][k - 1] * w[j][1] % mod;
	}
}

int rev[M << 2];
void ntt (long long *f, int len, int ty) {
	rev[0] = 0;
	for (int i = 1; i < len; i++) {
		rev[i] = rev[i >> 1] >> 1;
		if (i & 1) rev[i] |= (len >> 1);
	}
	for (int i = 0; i < len; i++) {
		int j = rev[i];
		if (j > i) swap(f[i], f[j]);
	}
	for (int i = 1, j = 1; i < len; i <<= 1, j++) {
		for (int k = 0; k < len; k += (i << 1)) {
			for (int l = 0; l < i; l++) {
				long long u = f[k + l], v = f[k + l + i];
				long long rt = w[j][ty > 0 ? l : (i << 1) - l];
				f[k + l] = (u + rt * v) % mod;
				f[k + l + i] = (u + rt * (mod - v)) % mod;
			}
		}
	}
	if (ty < 0) {
		for (int i = 0; i < len; i++) f[i] = f[i] * inv[len] % mod;
	}
}

long long poly[M << 2], poly_inv[M << 2], tmp1[M << 2], tmp2[M << 2];
void get_inv (int len) {
	for (int i = 0; i < (len << 2); i++) poly_inv[i] = 0ll;
	poly_inv[0] = 1ll;
	for (int i = 1; i < len; i <<= 1) {
		for (int j = 0; j < (i << 2); j++) tmp1[j] = tmp2[j] = 0ll;
		for (int j = 0; j < (i << 1); j++) tmp1[j] = poly[j], tmp2[j] = poly_inv[j];
		ntt(tmp1, i << 2, 1), ntt(tmp2, i << 2, 1);
		for (int j = 0; j < (i << 2); j++) tmp1[j] = (2ll * tmp2[j] + mod - tmp1[j] * tmp2[j] % mod * tmp2[j] % mod) % mod;
		ntt(tmp1, i << 2, -1);
		for (int j = 0; j < (i << 1); j++) poly_inv[j] = tmp1[j];
	}
}

long long poly_ln[M << 2];
void get_ln (int len) {
	get_inv(len + 1);
	for (int i = 0; i < (len << 2); i++) tmp1[i] = tmp2[i] = poly_ln[i] = 0ll;
	for (int i = 1; i < len; i++) tmp1[i - 1] = poly[i] * i % mod;
	for (int i = 0; i < len; i++) tmp2[i] = poly_inv[i];
	int L = 1;
	while (L < (len << 1)) L <<= 1;
	ntt(tmp1, L, 1), ntt(tmp2, L, 1);
	for (int i = 0; i < L; i++) tmp1[i] = tmp1[i] * tmp2[i] % mod;
	ntt(tmp1, L, -1);
	for (int i = 0; i < len; i++) poly_ln[i + 1] = tmp1[i] * inv[i + 1] % mod;
}

long long total_g[M], ans1[M], ans2[M], ans3[M], mul = 1ll, ans = 0ll;
int main () {
	read(n);
	for (int i = 0; i < n; i++) read(a[i]), m = max(m, a[i]);
	init();
	for (int i = 0; i <= m; i++) total_g[i] = qpow(2ll, 1ll * i * (i - 1) / 2);
	for (int i = 0; i <= m; i++) poly[i] = total_g[i] * inv_fac[i] % mod;	
	get_ln(m + 1);
	
	int L = 1;
	while (L <= (m + 1 << 1)) L <<= 1;
	for (int i = 0; i < L; i++) tmp1[i] = tmp2[i] = 0ll;
	for (int i = 0; i <= m; i++) tmp1[i] = poly[i], tmp2[i] = poly_ln[i];
	ntt(tmp1, L, 1), ntt(tmp2, L, 1);
	for (int i = 0; i < L; i++) tmp1[i] = tmp1[i] * tmp2[i] % mod;
	ntt(tmp1, L, -1);
	for (int i = 1; i <= m; i++) ans1[i] = tmp1[i] * fac[i] % mod;
	
	for (int i = 0; i <= m; i++) poly[i] = qpow(inv[2], 1ll * i * (i - 1) / 2) * inv_fac[i] % mod;
	for (int i = 0; i < L; i++) tmp1[i] = 0ll;
	for (int i = 0; i <= m; i++) tmp1[i] = poly[i];
	ntt(tmp1, L, 1);
	for (int i = 0; i < L; i++) tmp1[i] = tmp1[i] * tmp1[i] % mod;
	ntt(tmp1, L, -1);
	for (int i = 0; i <= m; i++) poly[i] = tmp1[i] * total_g[i] % mod;
	get_ln(m + 1);
	
	for (int i = 0; i < L; i++) tmp1[i] = tmp2[i] = 0ll;
	for (int i = 0; i <= m; i++) {
		tmp1[i] = poly_ln[i] * inv[2] % mod;
		tmp2[i] = total_g[i] * inv_fac[i] % mod;
	}
	ntt(tmp1, L, 1), ntt(tmp2, L, 1);
	for (int i = 0; i < L; i++) tmp1[i] = tmp1[i] * tmp2[i] % mod;
	ntt(tmp1, L, -1);
	for (int i = 0; i <= m; i++) {
		ans2[i] = tmp1[i] * fac[i] % mod;
		poly[i] = total_g[i];
	}
	for (int i = 1; i <= m; i++) {
		ans1[i] = (ans1[i] + mod - poly[i - 1] * i % mod) % mod;
		ans2[i] = (ans2[i] + mod - poly[i - 1] * i % mod) % mod;
		ans3[i] = (poly[i] + mod - poly[i - 1]) * i % mod;
	}
	for (int i = 0; i < n; i++) mul = mul * poly[a[i]] % mod * a[i] % mod;
	ans = mul, mul = 1ll;
	for (int i = 0; i < n; i++) mul = mul * ans3[a[i]] % mod;
	ans = (ans + mod - mul) % mod, mul = 1ll;
	for (int i = 0; i < n; i++) mul = mul * (ans1[a[i]] + ans2[a[i]]) % mod;
	ans = (ans + mul * inv[2]) % mod, mul = 1ll;
	for (int i = 0; i < n; i++) mul = mul * (ans1[a[i]] + mod - ans2[a[i]]) % mod;
	ans = (ans + mul * inv[2]) % mod;
	write(ans), putchar('\n');
	return 0;
}