7.12模拟T2（套路容斥+多项式求逆）

Description：

\(n<=10,max(w)<=1e6\)

题解：

考虑暴力，相当于走多维格子图，不能走有些点。

套路就是设\(f[i]\)表示第一次走到i的方案数
\(f[i]=起点到点i的方案数-\sum_{j在i前面}f[j]*j到i的方案数\)

不妨把前缀和后缀的分开，设为f和g。

f上的点形如(i,i,…)

设\(m=min(w)\),\(w-=m\)

则g上的点形如(w+i,……)

这样就顺序了。

且\(i->j的方案数\)之和坐标差有关，那就可以分治NTT了。

还可以优化，设\(f->f的转移多项式是A,f->g是B，g->f是C\)

则有
\(F=A-F*A-G*C\)
\(G=B-G*A-F*B\)

最后答案就是\(G[m]\)

消元可得：
\(G={B \over (1+A)^2-BC}\)

所以求逆即可。

Code：

#include<bits/stdc++.h>
#define fo(i, x, y) for(int i = x, B = y; i <= B; i ++)
#define ff(i, x, y) for(int i = x, B = y; i <  B; i ++)
#define fd(i, x, y) for(int i = x, B = y; i >= B; i --)
#define ll long long
#define pp printf
#define hh pp("\n")
using namespace std;

const int mo = 998244353;

ll ksm(ll x, ll y) {
    ll s = 1;
    for(; y; y /= 2, x = x * x % mo)
        if(y & 1) s = s * x % mo;
    return s;
}

typedef vector<ll> V;
#define pb push_back
#define si size()

const int nm = 1 << 21;
namespace ntt {
    ll w[nm], a[nm], b[nm]; int r[nm];
    void build() {
        for(int i = 1; i < nm; i *= 2) {
            ll v = ksm(3, (mo - 1) / 2 / i);
            w[i] = 1; ff(j, 1, i) w[i + j] = w[i + j - 1] * v % mo;
        }
    }
    void dft(ll *a, int n, int f) {
        ff(i, 0, n) {
            r[i] = r[i / 2] / 2 + (i & 1) * (n / 2);
            if(i < r[i]) swap(a[i], a[r[i]]);
        } ll b;
        for(int i = 1; i < n; i *= 2) for(int j = 0; j < n; j += 2 * i)
            ff(k, 0, i) b = a[i + j + k] * w[i + k], a[i + j + k] = (a[j + k] - b) % mo, a[j + k] = (a[j + k] + b) % mo;
        if(f == -1) {
            reverse(a + 1, a + n);
            b = ksm(n, mo - 2);
            ff(i, 0, n) a[i] = (a[i] + mo) * b % mo;
        }
    }
    void fft(V &p, V &q) {
        int p0 = p.si + q.si - 1, n = 1;
        for(; n < p0; n *= 2);
        ff(i, 0, n) a[i] = b[i] = 0;
        ff(i, 0, p.si) a[i] = p[i];
        ff(i, 0, q.si) b[i] = q[i];
        dft(a, n, 1); dft(b, n, 1);
        ff(i, 0, n) a[i] = a[i] * b[i] % mo;
        dft(a, n, -1);
        p.resize(p0);
        ff(i, 0, p0) p[i] = a[i];
    }
}

V operator * (V p, V q) {
    ntt :: fft(p, q);
    return p;
}
void dft(V &p, int f) {
    ff(i, 0, p.si) ntt :: a[i] = p[i];
    ntt :: dft(ntt :: a, p.si, f);
    ff(i, 0, p.si) p[i] = ntt :: a[i];
}
V qni(V a) {
    int a0 = a.si, n0 = 1;
    while(n0 < a0) n0 *= 2;
    V b; b.resize(1); b[0] = ksm(a[0], mo - 2);
    for(int n = 2; n <= n0; n *= 2) {
        V d = b; d.resize(n); b.resize(2 * n);
        V c = a; c.resize(n); c.resize(2 * n);
        dft(c, 1); dft(b, 1);
        ff(i, 0, b.si) b[i] = c[i] * b[i] % mo * b[i] % mo;
        dft(b, -1); b.resize(n);
        ff(i, 0, b.si) b[i] = (2 * d[i] - b[i] + mo) % mo;
    }
    b.resize(a0);
    return b;
}

const int N = 1e7 + 5;

ll fac[N], nf[N];
V a, b, c;

void build(int n) {
    fac[0] = 1; fo(i, 1, n) fac[i] = fac[i - 1] * i % mo;
    nf[n] = ksm(fac[n], mo - 2); fd(i, n, 1) nf[i - 1] = nf[i] * i % mo;
}

int n, m, w[11];

void build2() {
    a.resize(m + 1); b.resize(m + 1); c.resize(m + 1);
    fo(i, 1, m) a[i] = fac[n * i] * ksm(nf[i], n) % mo;
    fo(i, 0, m) {
        int s = 0; b[i] = 1;
        fo(j, 1, n) s += w[j] + i, b[i] = b[i] * nf[w[j] + i] % mo;
        b[i] = b[i] * fac[s] % mo;
    }
    fo(i, 0, m) {
        int s = 0; c[i] = 1;
        fo(j, 1, n) {
            if(-w[j] + i < 0) c[i] = 0; else
            s += -w[j] + i, c[i] = c[i] * nf[-w[j] + i] % mo;
        }
        c[i] = c[i] * fac[s] % mo;
    }
}

ll f[N], g[N];

int main() {
    freopen("queue.in", "r", stdin);
    freopen("queue.out", "w", stdout);
    ntt :: build();
    scanf("%d", &n); m = 1e6;
    fo(i, 1, n) scanf("%d", &w[i]), m = min(m, w[i]);
    fo(i, 1, n) w[i] -= m;
    {
        int ye = 1;
        fo(i, 1, n) if(w[i]) ye = 0;
        if(ye) {
            pp("0\n"); return 0;
        }
    }
    build(1e7);
    build2();
    a[0] ++; a = a * a; a.resize(m + 1);
    c = c * b; c.resize(m + 1);
    ff(i, 0, a.si) a[i] = (a[i] - c[i] + mo) % mo;
    a = qni(a); b = b * a;
    pp("%lld\n", b[m]);
}

7.12模拟T2（套路容斥+多项式求逆）

原文：https://www.cnblogs.com/coldchair/p/11178794.html