2021CCPC威海 M.810975

2021CCPC威海 M.810975

题意：问构造出长度为 \(n\) 的01串，有 \(m\) 个1，其中最长连续 \(1\) 的段长度恰好为 \(k\) 的方案数。

知识点：容斥，多项式快速幂

先推荐一个类似的题目 HDU6397 Character Encoding

这题有两种方法，先说简单的那种

可以先解决将 \(m\) 个 \(1\) 插入到 \(n - m + 1\) 个空中，其中最长连续 \(1\) 的段长度不超过 \(k\) 的方案数。

这个问题等价与问

\[\sum_{i=1}^{n-m+1}x_i = m(0 \le x_i \le k) \]

方程组非负整数解的个数

于是对于每个 \(x_i\) 有生成函数 \(\sum_{i=0}^k x^i\)

得到答案多项式为

\[G(x) = (\sum_{i=0}^k x^i)^{n-m+1} \]

这个可以用多项式快速幂求出

\([x^m]G(x)\) 即是答案

当然这个是最长段不超过 \(k\) 的答案，只需要减去最长段不超过 \(k-1\) 的答案即可

注意特判即可

#include <bits/stdc++.h>
#define endl '\n'
#define ls u << 1
#define rs u << 1 | 1
using namespace std;
typedef long long LL;
typedef pair<int, int> PII;
typedef pair<LL,LL> PLL;
const int INF = 0x3f3f3f3f, N = 1e5 + 10;
const double eps = 1e-6;
const double PI = acos(-1);
inline int lowbit(int x) {return x & (-x);}

namespace polybase {//范围为1e9需要先取模,别忘记改模数,原根和数组长度
    constexpr LL MOD = 998244353, G = 3, L = 1 << 22;

    //分别表示模数，原根以及默认数组长度
    LL fpow(LL x, LL r) {
        LL result = 1;
        while (r) {
            if (r & 1) result = result * x % MOD;
            r >>= 1;
            x = x * x % MOD;
        }
        return result;
    }

    int w[L], _ = [] {
        LL x = fpow(G, (MOD - 1) / L);
        w[L / 2] = 1;
        for (int i = L / 2 + 1; i < L; i++)
            w[i] = 1ll * w[i - 1] * x % MOD;
        for (int i = L / 2 - 1; i >= 0; i--)
            w[i] = w[i << 1];
        return 0;
    }();

    inline int norm(int n) { return 1 << __lg(n * 2 - 1); }

    void dft(LL *a, int n) {
        for (int k = n >> 1; k; k >>= 1)
            for (int i = 0; i < n; i += k << 1)
                for (int j = 0; j < k; j++) {
                    LL &x = a[i + j], y = a[i + j + k];
                    a[i + j + k] = (x - y + MOD) * w[k + j] % MOD;
                    x += y;
                    if (x >= MOD)x -= MOD;
                }
    }

    void idft(LL *a, int n) {
        for (int k = 1; k < n; k <<= 1)
            for (int i = 0; i < n; i += k << 1)
                for (int j = 0; j < k; j++) {
                    LL x = a[i + j], y = a[i + j + k] * w[k + j] % MOD;
                    a[i + j + k] = x - y < 0 ? x - y + MOD : x - y;
                    a[i + j] += y;
                    if (a[i + j] >= MOD)a[i + j] -= MOD;
                }
        for (int i = 0, inv = MOD - (MOD - 1) / n; i < n; i++)
            a[i] = a[i] * inv % MOD;
        reverse(a + 1, a + n);
    }

    struct poly : public vector<LL> {
        using vector<LL>::vector;
#define T (*this)

        poly MODxk(int k) const {
            k = min(k, (int) size());
            return poly(begin(), begin() + k);
        }

        poly rev() const { return poly(rbegin(), rend()); }

        friend void dft(poly &a) { dft(a.data(), a.size()); }

        friend void idft(poly &a) { idft(a.data(), a.size()); }

        friend poly operator*(const poly &x, const poly &y) {
            if (x.empty() || y.empty())return poly();
            poly a(x), b(y);
            int len = a.size() + b.size() - 1;
            int n = norm(len);
            a.resize(n), b.resize(n);
            dft(a), dft(b);
            for (int i = 0; i < n; i++)
                a[i] = a[i] * b[i] % MOD;
            idft(a);
            a.resize(len);
            return a;
        }

        poly operator+(const poly &b) {
            poly a(T);
            if (a.size() < b.size())
                a.resize(b.size());
            for (int i = 0; i < b.size(); i++)//用b.size()防止越界
            {
                a[i] += b[i];
                if (a[i] >= MOD)a[i] -= MOD;
            }
            return a;
        }

        poly operator-(const poly &b) {
            poly a(T);
            if (a.size() < b.size())
                a.resize(b.size());
            for (int i = 0; i < b.size(); i++) {
                a[i] -= b[i];
                if (a[i] < 0)a[i] += MOD;
            }
            return a;
        }

        poly operator*(const LL p) {
            poly a(T);
            for (auto &x: a)
                x = x * p % MOD;
            return a;
        }

        poly &operator<<=(int r) { return insert(begin(), r, 0), T; }//注意逗号,F(x)*(x^r)

        poly operator<<(int r) const { return poly(T) <<= r; }

        poly operator>>(int r) const { return r >= size() ? poly() : poly(begin() + r, end()); }

        poly &operator>>=(int r) { return T = T >> r; }//F[x]/(x^r)

        poly deriv() {
            //求导
            if (empty())return T;
            poly a(size() - 1);
            for (int i = 1; i < size(); i++)//注意是size()
                a[i - 1] = T[i] * i % MOD;
            return a;
        }

        poly integ() {
            //积分
            poly a(size() + 1);
            for (int i = 1; i < a.size(); i++)//注意是a.size()
                a[i] = T[i - 1] * fpow(i, MOD - 2) % MOD;
            return a;
        }

        poly inv(int n) {
            poly a{fpow(T[0], MOD - 2)};
            int k = 1;
            while (k < n) {
                k <<= 1;
                a = (a * 2 - MODxk(k) * a * a).MODxk(k);
            }
            return a.MODxk(n);
        }

        poly sqrt(int n) {
            //f[0]必须等于1
            poly a{1};
            int k = 1;
            const LL inv2 = fpow(2, MOD - 2);
            while (k < n) {
                k <<= 1;
                a = ((MODxk(k) * a.inv(k)).MODxk(k) + a) * inv2;
            }
            return a.MODxk(n);
        }

        poly ln(int n) {
            //需要保证f[0]=1
            return (deriv() * inv(n)).integ().MODxk(n);
        }

        poly exp(int n) {
            //需要保证f[0]=0
            poly a{1};
            int k = 1;
            while (k < n) {
                k <<= 1;
                a = (a * (poly{1} - a.ln(k) + MODxk(k))).MODxk(k);
            }
            return a.MODxk(n);
        }
#undef T
    };
}
using namespace polybase;

inline void solve() {
    int n, m, k; cin >> n >> m >> k;
    if (m > n || k > m) {cout << 0 << endl; return ;}
    if (m == 0) {cout << 1 << endl; return ;}
    if (n == 0) {cout << 1 << endl; return ;}
    if (k == 0) {cout << 0 << endl; return ;}
    poly a(k + 1);
    for (int i = 0; i <= k; i ++ ) a[i] = 1;
    a = (a.ln(m + 1) * (n - m + 1)).exp(m + 1);
    if (k == 1) {cout << a[m] << endl; return ;}
    poly b(k);
    for (int i = 0; i < k; i ++ ) b[i] = 1;
    b = (b.ln(m + 1) * (n - m + 1)).exp(m + 1);
    cout << (a[m] - b[m] + MOD) % MOD << endl;
}
signed main() {
#ifdef DEBUG
    freopen("in.txt", "r", stdin);
    freopen("out.txt", "w", stdout);
    auto now = clock();
#endif
    ios::sync_with_stdio(false), cin.tie(nullptr);
    cout << fixed << setprecision(2);
//    int T; cin >> T;
//    while (T -- )
        solve();
#ifdef DEBUG
    cout << "============================" << endl;
    cout << "Program run for " << (clock() - now) / (double)CLOCKS_PER_SEC * 1000 << " ms." << endl;
#endif
    return 0;
}

接下来说另外一种方法

看回这个式子

\[\sum_{i=1}^{n-m+1}x_i = m(0 \le x_i \le k) \]

先考虑没有 \(k\) 的限制，问题等价于 \(\sum_{i=1}^{n-m+1} x_i = m + k\) 的正整数解个数

可以隔板法得到答案 \(m + k - 1 \choose n-m+1\)

考虑容斥原理，分割成这样一个问题

设 $ans_k $ 为 \(\sum_{i=1}^{n-m+1} x_i = m (0 \le x_i \le k)\) 解的个数

这个解的方法可以参考 HDU6397 Character Encoding

则答案为 \(ans_k - ans_{k-1}\)

注意特判

#include <bits/stdc++.h>
#define endl '\n'
#define ls u << 1
#define rs u << 1 | 1
using namespace std;
typedef long long LL;
typedef pair<int, int> PII;
typedef pair<LL,LL> PLL;
const int INF = 0x3f3f3f3f, N = 3e5 + 10;
const int MOD = 998244353;
const double eps = 1e-6;
const double PI = acos(-1);
inline int lowbit(int x) {return x & (-x);}

LL f1[N], f2[N];
LL q_pow(LL a, LL b, LL p) {
    LL res = 1;
    for (; b; b >>= 1) {
        if (b & 1) res = res * a % p;
        a = a * a % p;
    }
    return res;
}
LL C(LL n, LL m) {
    if (n < m) return 0;
    if (n < 0 || m < 0) return 0;
    if (n == m) return 1;
    return f1[n] * f2[m] % MOD * f2[n - m] % MOD;
}
LL calc(LL n, LL m, LL k) {
    LL res = 0;
    int ops = 1;
    for (LL i = 0; i <= m; i ++ ) {
        res = (res + 1ll * ops * C(m, i) * C(m + k - 1 - i * n, m - 1) % MOD + MOD) % MOD;
        ops *= -1;
    }
    return res;
}
inline void solve() {
    LL n, m, k; cin >> n >> m >> k;
    if (m > n || k > m) {cout << 0 << endl; return ;}
    if (m == 0) {cout << 1 << endl; return ;}
    if (n == 0) {cout << 1 << endl; return ;}
    if (k == 0) {cout << 0 << endl; return ;}
    cout << (calc(k + 1, n - m + 1, m) - calc(k, n - m + 1, m) + MOD) % MOD << endl;
}
signed main() {
#ifdef DEBUG
    freopen("in.txt", "r", stdin);
    freopen("out.txt", "w", stdout);
    auto now = clock();
#endif
    ios::sync_with_stdio(false), cin.tie(nullptr);
    cout << fixed << setprecision(2);
    int n = 300000;
    f1[0] = 1;
    for (int i = 1; i <= n; i ++ ) f1[i] = f1[i - 1] * i % MOD;
    f2[n] = q_pow(f1[n], MOD - 2, MOD);
    for (int i = n; i; i -- ) f2[i - 1] = f2[i] * i % MOD;
//    int T; cin >> T;
//    while (T -- )
        solve();
#ifdef DEBUG
    cout << "============================" << endl;
    cout << "Program run for " << (clock() - now) / (double)CLOCKS_PER_SEC * 1000 << " ms." << endl;
#endif
    return 0;
}