test/yosupo/stirling_number_of_the_second_kind.test.cpp

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• Last update: 2024-01-08 00:27:17+09:00
• Problem: https://judge.yosupo.jp/problem/stirling_number_of_the_second_kind

Depends on

• Number Theoretic Transform (convolution/ntt.hpp)
• Combination (math/combination.cpp)
• Mod int (math/modint.hpp)
• Stirling Number of the Second Kind (math/stirling_second.hpp)

Code

#define PROBLEM "https://judge.yosupo.jp/problem/stirling_number_of_the_second_kind"


#include "../../math/modint.hpp"
#include "../../math/stirling_second.hpp"

#include <bits/stdc++.h>
using namespace std;
using ll = long long;

using mint = Modint<998244353>;

int main() {
    int N;
    cin >> N;
    auto ans = stirling_second_table<mint>(N);
    for (int i = 0; i <= N; ++i) {
        cout << ans[i] << (i < N ? " " : "\n");
    }
}

#line 1 "test/yosupo/stirling_number_of_the_second_kind.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/stirling_number_of_the_second_kind"


#line 2 "math/modint.hpp"
#include <algorithm>
#include <iostream>

/**
 * @brief Mod int
 */
template <int m>
class Modint {
    using mint = Modint;
    static_assert(m > 0, "Modulus must be positive");

   public:
    static constexpr int mod() { return m; }

    constexpr Modint(long long y = 0) : x(y >= 0 ? y % m : (y % m + m) % m) {}

    constexpr int val() const { return x; }

    constexpr mint& operator+=(const mint& r) {
        if ((x += r.x) >= m) x -= m;
        return *this;
    }
    constexpr mint& operator-=(const mint& r) {
        if ((x += m - r.x) >= m) x -= m;
        return *this;
    }
    constexpr mint& operator*=(const mint& r) {
        x = static_cast<int>(1LL * x * r.x % m);
        return *this;
    }
    constexpr mint& operator/=(const mint& r) { return *this *= r.inv(); }

    constexpr bool operator==(const mint& r) const { return x == r.x; }

    constexpr mint operator+() const { return *this; }
    constexpr mint operator-() const { return mint(-x); }

    constexpr friend mint operator+(const mint& l, const mint& r) {
        return mint(l) += r;
    }
    constexpr friend mint operator-(const mint& l, const mint& r) {
        return mint(l) -= r;
    }
    constexpr friend mint operator*(const mint& l, const mint& r) {
        return mint(l) *= r;
    }
    constexpr friend mint operator/(const mint& l, const mint& r) {
        return mint(l) /= r;
    }

    constexpr mint inv() const {
        int a = x, b = m, u = 1, v = 0;
        while (b > 0) {
            int t = a / b;
            std::swap(a -= t * b, b);
            std::swap(u -= t * v, v);
        }
        return mint(u);
    }

    constexpr mint pow(long long n) const {
        mint ret(1), mul(x);
        while (n > 0) {
            if (n & 1) ret *= mul;
            mul *= mul;
            n >>= 1;
        }
        return ret;
    }

    friend std::ostream& operator<<(std::ostream& os, const mint& r) {
        return os << r.x;
    }

    friend std::istream& operator>>(std::istream& is, mint& r) {
        long long t;
        is >> t;
        r = mint(t);
        return is;
    }

   private:
    int x;
};
#line 2 "math/stirling_second.hpp"
#include <vector>
#line 2 "convolution/ntt.hpp"
#include <bit>
#line 4 "convolution/ntt.hpp"

constexpr int get_primitive_root(int mod) {
    if (mod == 167772161) return 3;
    if (mod == 469762049) return 3;
    if (mod == 754974721) return 11;
    if (mod == 998244353) return 3;
    if (mod == 1224736769) return 3;
}

template <typename mint>
void ntt(std::vector<mint>& a) {
    constexpr int mod = mint::mod();
    constexpr mint primitive_root = get_primitive_root(mod);

    const int n = a.size();
    for (int m = n; m > 1; m >>= 1) {
        mint omega = primitive_root.pow((mod - 1) / m);
        for (int s = 0; s < n / m; ++s) {
            mint w = 1;
            for (int i = 0; i < m / 2; ++i) {
                mint l = a[s * m + i];
                mint r = a[s * m + i + m / 2];
                a[s * m + i] = l + r;
                a[s * m + i + m / 2] = (l - r) * w;
                w *= omega;
            }
        }
    }
}

template <typename mint>
void intt(std::vector<mint>& a) {
    constexpr int mod = mint::mod();
    constexpr mint primitive_root = get_primitive_root(mod);

    const int n = a.size();
    for (int m = 2; m <= n; m <<= 1) {
        mint omega = primitive_root.pow((mod - 1) / m).inv();
        for (int s = 0; s < n / m; ++s) {
            mint w = 1;
            for (int i = 0; i < m / 2; ++i) {
                mint l = a[s * m + i];
                mint r = a[s * m + i + m / 2] * w;
                a[s * m + i] = l + r;
                a[s * m + i + m / 2] = l - r;
                w *= omega;
            }
        }
    }
}

template <typename mint>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
    const int size = a.size() + b.size() - 1;
    const int n = std::bit_ceil((unsigned int)size);
    a.resize(n);
    b.resize(n);
    ntt(a);
    ntt(b);
    for (int i = 0; i < n; ++i) a[i] *= b[i];
    intt(a);
    a.resize(size);
    mint n_inv = mint(n).inv();
    for (int i = 0; i < size; ++i) a[i] *= n_inv;
    return a;
}
#line 3 "math/combination.cpp"

template <typename mint>
class Combination {
   public:
    Combination() = default;
    Combination(int n) : fact_(n + 1), fact_inv_(n + 1) {
        fact_[0] = 1;
        for (int i = 1; i <= n; ++i) fact_[i] = fact_[i - 1] * i;
        fact_inv_[n]=fact_[n].inv();
        for (int i = n; i > 0; --i) fact_inv_[i - 1] = fact_inv_[i] * i;
    }

    mint perm(int n, int k) const {
        if (k < 0 || n < k) return 0;
        return fact_[n] * fact_inv_[n - k];
    }

    mint comb(int n, int k) const {
        if (k < 0 || n < k) return 0;
        return fact_[n] * fact_inv_[k] * fact_inv_[n - k];
    }

    mint fact(int n) const { return fact_[n]; }
    mint fact_inv(int n) const { return fact_inv_[n]; }

   private:
    std::vector<mint> fact_, fact_inv_;
};

template <typename T>
T comb(long long n, int k) {
    if (k < 0 || n < k) return 0;
    T num = 1, den = 1;
    for (int i = 1; i <= k; ++i) {
        num = num * (n - i + 1);
        den = den * i;
    }
    return num / den;
}
#line 5 "math/stirling_second.hpp"

template <typename T>
std::vector<T> stirling_second_table(int n) {
    T f = 1;
    for (int i = 1; i <= n; ++i) f *= i;
    f = T(1) / f;
    std::vector<T> a(n + 1), b(n + 1);
    for (int i = n; i >= 0; --i) {
        a[i] = f * (i % 2 ? -1 : 1);
        b[i] = f * T(i).pow(n);
        f *= i;
    }
    auto c = convolution(a, b);
    return std::vector(c.begin(), c.begin() + n + 1);
}

template <typename T>
T stirling_second(int n, int k) {
    Combination<T> comb(n);
    T res = 0;
    for (int i = 0; i <= k; ++i) {
        T tmp = comb.comb(k, i) * T(i).pow(n);
        if ((k - i) & 1) res -= tmp;
        else res += tmp;
    }
    res /= comb.fact(k);
    return res;
}
#line 6 "test/yosupo/stirling_number_of_the_second_kind.test.cpp"

#include <bits/stdc++.h>
using namespace std;
using ll = long long;

using mint = Modint<998244353>;

int main() {
    int N;
    cin >> N;
    auto ans = stirling_second_table<mint>(N);
    for (int i = 0; i <= N; ++i) {
        cout << ans[i] << (i < N ? " " : "\n");
    }
}

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