[2022 牛客多校2 E] Falfa with Substring (二项式反演 NTT)

题意

定义 $F_{n,k}$ 为所有长度为 $n$ 的字符串 $S$ 中恰好出现了 $k$ 次 bit 的个数。

求 $F_{n,0},F_{n,1},\cdots ,F_{n,n}$ 对 $998244353$ 取模。

$(1\le n\le 10^6)$

分析：

看到求恰好出现 $k$ 次，首先想到求出大于等于 $k$ 次再进行容斥。

考虑钦定出现 $k$ 次 bit 的字符串，将 $k$ 个 bit 进行捆绑，那么有 $n-3k+k=n-2k$ 个位置，并且剩下 $n-3k$ 个字母任意取值，从 $n-2k$ 个位置选出 $k$ 个放 bit，方案数为 $\left(\genfrac{}{}{0ex}{}{n-2k}{k}\right)\times 26^{n-3k}$，记为 $f(k)$

那么 $f(k)$ 由所有恰好出现 $k,k+1,\cdots ,n$ 次的方案数加起来，还要乘上对应次数选出 $k$ 个的方案数，记恰好出现 $k$ 次的方案数为 $g(k)$
$$f(k)=\sum _{i=k}^n\left(\genfrac{}{}{0ex}{}{i}{k}\right)g(i)$$
根据二项式反演公式 $f(n)=\sum _{i=n}^m\left(\genfrac{}{}{0ex}{}{i}{n}\right)g(i)\iff g(n)=\sum _{i=n}^m(-1{)}^{i-n}(\genfrac{}{}{0ex}{}{i}{n})g(i)$
$$g(k)=\sum _{i=k}^n(-1{)}^{i-k}(\genfrac{}{}{0ex}{}{i}{k})f(i)$$
展开组合数 $\left(\genfrac{}{}{0ex}{}{i}{k}\right)=\frac{i!}{k!\times (i-k)!}$
$$\begin{gathered} g(k)=\sum _{i=k}^{n}i!f(i)\frac{(-1{)}^{i-k}}{k!\times (i-k)!} \\ \iff k!g(k)=\sum _{i=k}^{n}i!f(i)\frac{(-1{)}^{i-k}}{(i-k)!} \end{gathered}$$
设 $P(i)=i!g(i),F(i)=i!f(i),G(i)=\frac{(-1{)}^i}{i!}$，则
$$P(i)=\sum _{i=k}^{n}F(i)\times G(i-k)$$
令 $(i-k)\to i$
$$P(i)=\sum _{i=0}^{n-k}G(i)\times F(i+k)$$
考虑多项式加速。我们知道常规的多项式卷积是 $F(i)=\sum _{i=0}^{n}f(i)\times g(n-i)$，所以上式中 $F$ 对应的下标应为 $n-k-i$，发现 $n-k-i+i+k=n$ 恰好是对称关系，所以可以对 $F$ 函数做翻转进行多项式卷积。

答案为 $\frac{P(i)}{i!}$ $(0\le i\le n)$

代码：

#include 
using namespace std;
using i64 = long long;
constexpr int mod = 998244353;
int norm(int x) {
    if (x < 0) {
        x += mod;
    }
    if (x >= mod) {
        x -= mod;
    }
    return x;
}
template<class T>
T power(T a, int b) {
    T res = 1;
    for (; b; b /= 2, a *= a) {
        if (b % 2) {
            res *= a;
        }
    }
    return res;
}
struct Z {
    int x;
    Z(int x = 0) : x(norm(x)) {}
    int val() const {
        return x;
    }
    Z operator-() const {
        return Z(norm(mod - x));
    }
    Z inv() const {
        assert(x != 0);
        return power(*this, mod - 2);
    }
    Z &operator*=(const Z &rhs) {
        x = i64(x) * rhs.x % mod;
        return *this;
    }
    Z &operator+=(const Z &rhs) {
        x = norm(x + rhs.x);
        return *this;
    }
    Z &operator-=(const Z &rhs) {
        x = norm(x - rhs.x);
        return *this;
    }
    Z &operator/=(const Z &rhs) {
        return *this *= rhs.inv();
    }
    friend Z operator*(const Z &lhs, const Z &rhs) {
        Z res = lhs;
        res *= rhs;
        return res;
    }
    friend Z operator+(const Z &lhs, const Z &rhs) {
        Z res = lhs;
        res += rhs;
        return res;
    }
    friend Z operator-(const Z &lhs, const Z &rhs) {
        Z res = lhs;
        res -= rhs;
        return res;
    }
    friend Z operator/(const Z &lhs, const Z &rhs) {
        Z res = lhs;
        res /= rhs;
        return res;
    }
    friend istream &operator>>(istream &is, Z &a) {
        i64 v;
        is >> v;
        a = Z(v);
        return is;
    }
    friend ostream &operator<<(ostream &os, const Z &a) {
        return os << a.val();
    }
};
vector<int> rev;
vector<Z> roots{0, 1};
void dft(vector<Z> &a) {
    int n = a.size();
    if (int(rev.size()) != n) {
        int k = __builtin_ctz(n) - 1;
        rev.resize(n);
        for (int i = 0; i < n; i ++) {
            rev[i] = rev[i >> 1] >> 1 | (i & 1) << k;
        }
    }
    for (int i = 0; i < n; i ++) {
        if (rev[i] < i) {
            swap(a[i], a[rev[i]]);
        }
    }
    if (int(roots.size()) < n) {
        int k = __builtin_ctz(roots.size());
        roots.resize(n);
        while ((1 << k) < n) {
            Z e = power(Z(3), (mod - 1) >> (k + 1));
            for (int i = 1 << (k - 1); i < (1 << k); i ++) {
                roots[i << 1] = roots[i];
                roots[i << 1 | 1] = roots[i] * e;
            }
            k ++;
        }
    }
    for (int k = 1; k < n; k *= 2) {
        for (int i = 0; i < n; i += 2 * k) {
            for (int j = 0; j < k; j ++) {
                Z u = a[i + j], v = a[i + j + k] * roots[k + j];
                a[i + j] = u + v, a[i + j + k] = u - v;
            }
        }
    }
}
void idft(vector<Z> &a) {
    int n = a.size();
    reverse(a.begin() + 1, a.end());
    dft(a);
    Z inv = (1 - mod) / n;
    for (int i = 0; i < n; i ++) {
        a[i] *= inv;
    }
}
struct Poly {
    vector<Z> a;
    Poly() {}
    Poly(const vector<Z> &a) : a(a) {}
    Poly(const initializer_list<Z> &a) : a(a) {}
    int size() const {
        return a.size();
    }
    void resize(int n) {
        a.resize(n);
    }
    Z operator[](int idx) const {
        if (idx < size()) {
            return a[idx];
        } else {
            return 0;
        }
    }
    Z &operator[](int idx) {
        return a[idx];
    }
    Poly mulxk(int k) const {
        auto b = a;
        b.insert(b.begin(), k, 0);
        return Poly(b);
    }
    Poly modxk(int k) const {
        k = min(k, size());
        return Poly(vector<Z>(a.begin(), a.begin() + k));
    }
    Poly divxk(int k) const {
        if (size() <= k) {
            return Poly();
        }
        return Poly(vector<Z>(a.begin() + k, a.end()));
    }
    friend Poly operator+(const Poly &a, const Poly &b) {
        vector<Z> res(max(a.size(), b.size()));
        for (int i = 0; i < int(res.size()); i ++) {
            res[i] = a[i] + b[i];
        }
        return Poly(res);
    }
    friend Poly operator-(const Poly &a, const Poly &b) {
        vector<Z> res(max(a.size(), b.size()));
        for (int i = 0; i < int(res.size()); i ++) {
            res[i] = a[i] - b[i];
        }
        return Poly(res);
    }
    friend Poly operator*(Poly a, Poly b) {
        if (a.size() == 0 || b.size() == 0) {
            return Poly();
        }
        int sz = 1, tot = a.size() + b.size() - 1;
        while (sz < tot) {
            sz *= 2;
        }
        a.a.resize(sz);
        b.a.resize(sz);
        dft(a.a);
        dft(b.a);
        for (int i = 0; i < sz; i ++) {
            a.a[i] = a[i] * b[i];
        }
        idft(a.a);
        a.resize(tot);
        return a;
    }
    friend Poly operator*(Z a, Poly b) {
        for (int i = 0; i < int(b.size()); i ++) {
            b[i] *= a;
        }
        return b;
    }
    friend Poly operator*(Poly a, Z b) {
        for (int i = 0; i < int(a.size()); i ++) {
            a[i] *= b;
        }
        return a;
    }
    Poly &operator+=(Poly b) {
        return (*this) = (*this) + b;
    }
    Poly &operator-=(Poly b) {
        return (*this) = (*this) - b;
    }
    Poly &operator*=(Poly b) {
        return (*this) = (*this) * b;
    }
    Poly deriv() const {
        if (a.empty()) {
            return Poly();
        }
        vector<Z> res(size() - 1);
        for (int i = 0; i < size() - 1; i ++) {
            res[i] = (i + 1) * a[i + 1];
        }
        return Poly(res);
    }
    Poly integr() const {
        vector<Z> res(size() + 1);
        for (int i = 0; i < size(); i ++) {
            res[i + 1] = a[i] / (i + 1);
        }
        return Poly(res);
    }
    Poly inv(int m) const {
        Poly x{a[0].inv()};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x * (Poly{2} - modxk(k) * x)).modxk(k);
        }
        return x.modxk(m);
    }
    Poly log(int m) const {
        return (deriv() * inv(m)).integr().modxk(m);
    }
    Poly exp(int m) const {
        Poly x{1};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x * (Poly{1} - x.log(k) + modxk(k))).modxk(k);
        }
        return x.modxk(m);
    }
    Poly pow(int k, int m) const {
        int i = 0;
        while (i < size() && a[i].val() == 0) {
            i ++;
        }
        if (i == size() || 1LL * i * k >= m) {
            return Poly(vector<Z>(m));
        }
        Z v = a[i];
        auto f = divxk(i) * v.inv();
        return (f.log(m - i * k) * k).exp(m - i * k).mulxk(i * k) * power(v, k);
    }
    Poly sqrt(int m) const {
        Poly x{1};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x + (modxk(k) * x.inv(k)).modxk(k)) * ((mod + 1) / 2);
        }
        return x.modxk(m);
    }
    Poly mulT(Poly b) const {
        if (b.size() == 0) {
            return Poly();
        }
        int n = b.size();
        reverse(b.a.begin(), b.a.end());
        return ((*this) * b).divxk(n - 1);
    }
};
signed main() {
    cin.tie(0) -> sync_with_stdio(0);
    int n;
    cin >> n;
    vector<Z> fact(n + 1), infact(n + 1);
    fact[0] = infact[0] = 1;
    for (int i = 1; i <= n; i ++) {
        fact[i] = fact[i - 1] * i;
    }
    infact[n] = fact[n].inv();
    for (int i = n; i; i --) {
        infact[i - 1] = infact[i] * i;
    }
    auto C = [&](int n, int m) {
	    if (n < 0 || m < 0 || n < m) return Z(0);
	    return fact[n] * infact[n - m] * infact[m];
	};
	vector<Z> f(n / 3 + 1), g(n / 3 + 1);
	for (int i = 0; i <= n / 3; i ++) {
		f[i] = fact[i] * C(n - 2 * i, i) * power(Z(26), n - 3 * i);
		g[i] = infact[i] * (i % 2 == 0 ? 1 : -1);
	}
	reverse(f.begin(), f.end());
	auto ans = Poly(f) * Poly(g);
	for (int i = 0; i <= n; i ++) {
		cout << (i <= n / 3 ? infact[i] * ans[n / 3 - i] : 0) << " \n"[i == n];
	}
}
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