简单计数 题解

最近在搞线代，拿了一道以前zzq的模拟题做了一下

前置技能：best定理

一个有向图的欧拉回路个数等于内向树个数乘上所有(deg[i]-1)!的乘积

我们考虑拆开算贡献，考虑经过(u,v),(v,w)的欧拉回路个数，我们把(u,v),(v,w)断开，连上(u,w)，如果v是孤立点，特判一下，否则我们发现我们改的矩阵的位置只会是4个，其中3个和v相关，直接删除掉即可，剩下的那个位置考虑拉普拉斯展开，所以我们要求的本质就是对于所有i，去掉第i行第i列后的矩阵的逆矩阵，分治消元即可

//waz
#include <bits/stdc++.h>

using namespace std;

#define mp make_pair
#define pb push_back
#define fi first
#define se second
#define ALL(x) (x).begin(), (x).end()
#define SZ(x) ((int)((x).size()))

typedef pair<int, int> PII;
typedef vector<int> VI;
typedef long long int64;
typedef unsigned int uint;
typedef unsigned long long uint64;

#define gi(x) ((x) = F())
#define gii(x, y) (gi(x), gi(y))
#define giii(x, y, z) (gii(x, y), gi(z))

int F()
{
    char ch;
    int x, a;
    while (ch = getchar(), (ch < '0' || ch > '9') && ch != '-');
    if (ch == '-') ch = getchar(), a = -1;
    else a = 1;
    x = ch - '0';
    while (ch = getchar(), ch >= '0' && ch <= '9')
        x = (x << 1) + (x << 3) + ch - '0';
    return a * x;
}

const int mod = 998244353;

int inc(int a, int b) { a += b; return a >= mod ? a - mod : a; }

int dec(int a, int b) { a -= b; return a < 0 ? a + mod : a; }

int fpow(int a, int x)
{
    int ret = 1;
    for (; x; x >>= 1)
    {
        if (x & 1) ret = 1LL * ret * a % mod;
        a = 1LL * a * a % mod;
    }
    return ret;
}

const int N = 310;

struct matrix
{
    int n;
    int a[N][N];
    matrix() { memset(a, 0, sizeof a); }
    int *operator[](int pos) { return a[pos]; }
    void swap(int i, int j) { return std::swap(a[i], a[j]); }
    void rev()
    {
        for (int i = 1; i <= n; ++i)
            for (int j = i + 1; j <= n; ++j)
                std::swap(a[i][j], a[j][i]);
    }
    friend matrix operator * (matrix x, int y)
    {
        for (int i = 1; i <= x.n; ++i)
            for (int j = 1; j <= x.n; ++j)
                x[i][j] = 1LL * x[i][j] * y % mod;
        return x;
    }
    matrix del(int t)
    {
        matrix c = *this;
        for (int i = 1; i <= c.n; ++i)
            for (int j = 1; j <= c.n; ++j)
            {
                int x = i, y = j;
                if (i >= t) ++x;
                if (j >= t) ++y;
                c[i][j] = c[x][y];
            }
        c.n--;
        return c;
    }
    void out()
    {
        printf("debug : %d
", n);
        for (int i = 1; i <= n; ++i)
            for (int j = 1; j <= n; ++j)
                printf("%d%c", a[i][j], ",
"[j == n]);
    }
} c[N], o;

int det(matrix a)
{
    int ans = 1;
    for (int i = 1; i <= a.n; ++i)
    {
        if (!a[i][i])
        {
            for (int j = i + 1; j <= a.n; ++j)
                if (a[j][i])
                {
                    a.swap(i, j);
                    ans = dec(mod, ans);
                    break;
                }
        }
        if (!a[i][i]) return 0;
        ans = 1LL * ans * a[i][i] % mod;
        int v = fpow(a[i][i], mod - 2);
        for (int j = i; j <= a.n; ++j) a[i][j] = 1LL * a[i][j] * v % mod;
        for (int j = i + 1; j <= a.n; ++j)
        {
            int t = a[j][i];
            for (int k = i; k <= a.n; ++k)
                a[j][k] = dec(a[j][k], 1LL * t * a[i][k] % mod);
        }
    }
    return ans;
}

matrix inv(matrix a)
{
    matrix b;
    b.n = a.n;
    for (int i = 1; i <= b.n; ++i) b[i][i] = 1;
    for (int i = 1; i <= a.n; ++i)
    {
        if (!a[i][i])
        {
            for (int j = i + 1; j <= a.n; ++j)
                if (a[j][i])
                {
                    a.swap(i, j);
                    b.swap(i, j);
                    break;
                }
        }
        int v = fpow(a[i][i], mod - 2);
        for (int j = 1; j <= a.n; ++j) a[i][j] = 1LL * a[i][j] * v % mod, b[i][j] = 1LL * b[i][j] * v % mod;
        for (int j = 1; j <= a.n; ++j)
        {
            if (j == i) continue;
            int t = a[j][i];
            for (int k = 1; k <= a.n; ++k)
                a[j][k] = dec(a[j][k], 1LL * t * a[i][k] % mod), b[j][k] = dec(b[j][k], 1LL * t * b[i][k] % mod);
        }
    }
    return b;
}

void guess(matrix &a, matrix &b, int l, int r)
{
    for (int i = l; i <= r; ++i)
    {
        if (!a[i][i])
        {
            for (int j = 1; j <= a.n; ++j)
                if (a[j][i])
                {
                    a.swap(i, j);
                    b.swap(i, j);
                    break;
                }
        }
        int v = fpow(a[i][i], mod - 2);
        for (int j = 1; j <= a.n; ++j) a[i][j] = 1LL * a[i][j] * v % mod, b[i][j] = 1LL * b[i][j] * v % mod;
        for (int j = 1; j <= a.n; ++j)
        {
            if (j == i) continue;
            int t = a[j][i];
            for (int k = 1; k <= a.n; ++k)
                a[j][k] = dec(a[j][k], 1LL * t * a[i][k] % mod), b[j][k] = dec(b[j][k], 1LL * t * b[i][k] % mod);
        }
    }
}

int n, m, x[N * N], u[N * N], v[N * N];

int w[N][N];

int g[N];

int deg[N];

int fac[N], rfac[N];

void fz(int l, int r, pair<matrix, matrix> now)
{
    if (l == r) 
    {
        c[l] = now.se.del(l);
        c[l] = c[l] * g[l];
        c[l].rev();
        return;
    }
    int mid = (l + r) >> 1;
    pair<matrix, matrix> t = now;
    guess(t.fi, t.se, l, mid);
    fz(mid + 1, r, t);
    guess(now.fi, now.se, mid + 1, r);
    fz(l, mid, now);
}

int main()
{
    freopen("count.in", "r", stdin);
    freopen("count.out", "w", stdout);
    gii(n, m); o.n = n;
    fac[0] = 1;
    for (int i = 1; i <= n; ++i) fac[i] = 1LL * fac[i - 1] * i % mod;
    rfac[n] = fpow(fac[n], mod - 2);
    for (int i = n; i; --i) rfac[i - 1] = 1LL * rfac[i] * i % mod;
    for (int i = 1; i <= m; ++i) 
        giii(x[i], u[i], v[i]), w[u[i]][v[i]] = x[i], ++deg[u[i]], 
        o[u[i]][u[i]] = inc(o[u[i]][u[i]], 1), o[u[i]][v[i]] = dec(o[u[i]][v[i]], 1);
    c[0] = o.del(1);
    g[0] = det(c[0]);
    for (int i = 1; i <= n; ++i) g[i] = g[i - 1];
    matrix I;
    I.n = n;
    for (int i = 1; i <= n; ++i) I[i][i] = 1;
    fz(1, n, mp(o, I));
    //for (int i = 1; i <= n; ++i) c[i] = o.del(i);
    //for (int i = 1; i <= n; ++i) g[i] = det(c[i]), c[i] = inv(c[i]), c[i] = c[i] * g[i], c[i].rev();
    int mul = 1, ans = 0;
    for (int i = 1; i <= n; ++i) mul = 1LL * mul * fac[deg[i] - 1] % mod;
    //c[3].out();
    for (int mid = 1; mid <= n; ++mid)
    {
        for (int from = 1; from <= n; ++from)
            if (w[from][mid])
            {
                for (int to = 1; to <= n; ++to)
                    if (w[mid][to] == w[from][mid])
                    {
                        int i = from > mid ? from - 1 : from;
                        int j = to > mid ? to - 1 : to;
                        int ret = g[mid];
                        if (deg[mid] == 1)
                        {
                            ans = (ans + 1LL * mul * ret) % mod;
                            //cerr << from << ", " << mid << ", " << to << ", " << ret << endl;
                        }
                        else
                        {
                            ret = dec(ret, c[mid][i][j]);
                            int gg = mul;
                            gg = 1LL * gg * rfac[deg[mid] - 1] % mod;
                            gg = 1LL * gg * fac[deg[mid] - 2] % mod;
                            ans = (ans + 1LL * gg * ret) % mod;
                            //cerr << from << ", " << mid << ", " << to << ", " << ret << endl;
                        }
                    }
            }
    }
    printf("%d
", ans);
}