Stirling Number of the First Kind (Small Prime Mod)
(library/sequence/stirling_number1_small_prime_mod.hpp)

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• Last update: 2024-01-30 20:57:42+09:00
• Include: #include "library/sequence/stirling_number1_small_prime_mod.hpp"

Stirling Number of the First Kind (Small Prime Mod)

(符号無し) 第一種スターリング数 $\begin{bmatrix} n \newline k \end{bmatrix}$ を以下で定義する:

\[\begin{bmatrix} n \newline k \end{bmatrix} := [x^k] \prod _ {i = 0} ^ {n - 1} (x + i).\]

$\displaystyle \prod _ {i = 0} ^ {p - 1}(x - i)$ の根は $0,\ldots,p-1$ である。一方、Fermat の小定理より $x ^ p \equiv x \pmod{p}$ より $x ^ p - x$ の根も $0,\ldots,p-1$ である。従って、次が成り立つ。

\[\prod _ {i = 0} ^ {p - 1}(x - i) = x ^ p - x.\]

従って、$n=ap+b$ とすれば、次が成り立つ:

\[\prod _ {i = 0} ^ {n - 1} (x + i) = x ^ a (x ^ {p - 1} - 1) ^ a \prod _ {i = 0} ^ {b - 1} (x + i).\]

即ち、$k=a+c(p-1)+d$ とすれば、次が成り立つ:

\[\begin{bmatrix} n \newline k \end{bmatrix} = \sum _ {c,d} (-1) ^ {a - c} \cdot \binom{a}{c} \cdot \begin{bmatrix} b \newline d \end{bmatrix}.\]

$\begin{bmatrix} b \newline d \end{bmatrix}$ が非零になりうるのは $0\leq d\leq b\leq p-1$ の範囲なので、右辺は高々 $2$ 項の和として計算できる。

$\displaystyle \binom{a}{c}$ は Lucas の定理を用いることで $\langle O(p ^ 2),O(\log _p n)\rangle$ で計算できる [1]。また、全ての $0\leq n\lt p,0\leq k\lt p$ に対する $\begin{bmatrix} n \newline k \end{bmatrix}$ も $O(p ^ 2)$ 時間で計算できる [2]。

以上より、任意の整数 $n,k$ に対する $\begin{bmatrix} n \newline k \end{bmatrix}$ を $\langle O(p ^ 2),O(\log _p n)\rangle$ で計算できた。

Depends on

• 階乗テーブル (library/math/factorial.hpp)
• 逆元テーブル (library/math/inv_mods.hpp)
• Binomial Coefficient Small Prime Mod (library/sequence/binomial_coefficient_small_prime_mod.hpp)
• Stirling Number1 (library/sequence/stirling_number1.hpp)

Verified with

• test/src/sequence/stirling_number1_small_prime_mod/stirling_number_of_the_first_kind_small_p_large_n.test.cpp

Code

#ifndef SUISEN_STIRLING_NUMBER1_SMALL_P
#define SUISEN_STIRLING_NUMBER1_SMALL_P

#include "library/sequence/stirling_number1.hpp"
#include "library/sequence/binomial_coefficient_small_prime_mod.hpp"

namespace suisen {
    template <typename mint>
    struct StirlingNumber1SmallPrimeMod {
        mint operator()(long long n, long long k) const {
            return s1(n, k);
        }
        static mint s1(long long n, long long k) {
            static const std::vector<std::vector<mint>> table = stirling_number1_table<mint>(mint::mod() - 1);
            static const BinomialCoefficientSmallPrimeMod<mint> binom{};
            static const int p = mint::mod();
            if (k < 0 or n < k) return 0;
            long long a = n / p, b = n % p;
            if (k < a) return 0;
            long long c = (k - a) / (p - 1), d = (k - a) % (p - 1);
            return ((a - c) & 1 ? -1 : 1) * (b == p - 1 and d == 0 ? -binom(a, c - 1) : d <= b ? table[b][d] * binom(a, c): 0);
        }
    };
} // namespace suisen


#endif // SUISEN_STIRLING_NUMBER1_SMALL_P

#line 1 "library/sequence/stirling_number1_small_prime_mod.hpp"



#line 1 "library/sequence/stirling_number1.hpp"



#include <algorithm>
#line 1 "library/math/inv_mods.hpp"



#include <vector>

namespace suisen {
    template <typename mint>
    class inv_mods {
    public:
        inv_mods() = default;
        inv_mods(int n) { ensure(n); }
        const mint& operator[](int i) const {
            ensure(i);
            return invs[i];
        }
        static void ensure(int n) {
            int sz = invs.size();
            if (sz < 2) invs = { 0, 1 }, sz = 2;
            if (sz < n + 1) {
                invs.resize(n + 1);
                for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i];
            }
        }
    private:
        static std::vector<mint> invs;
        static constexpr int mod = mint::mod();
    };
    template <typename mint>
    std::vector<mint> inv_mods<mint>::invs{};

    template <typename mint>
    std::vector<mint> get_invs(const std::vector<mint>& vs) {
        const int n = vs.size();

        mint p = 1;
        for (auto& e : vs) {
            p *= e;
            assert(e != 0);
        }
        mint ip = p.inv();

        std::vector<mint> rp(n + 1);
        rp[n] = 1;
        for (int i = n - 1; i >= 0; --i) {
            rp[i] = rp[i + 1] * vs[i];
        }
        std::vector<mint> res(n);
        for (int i = 0; i < n; ++i) {
            res[i] = ip * rp[i + 1];
            ip *= vs[i];
        }
        return res;
    }
}


#line 1 "library/math/factorial.hpp"



#include <cassert>
#line 6 "library/math/factorial.hpp"

namespace suisen {
    template <typename T, typename U = T>
    struct factorial {
        factorial() = default;
        factorial(int n) { ensure(n); }

        static void ensure(const int n) {
            int sz = _fac.size();
            if (n + 1 <= sz) return;
            int new_size = std::max(n + 1, sz * 2);
            _fac.resize(new_size), _fac_inv.resize(new_size);
            for (int i = sz; i < new_size; ++i) _fac[i] = _fac[i - 1] * i;
            _fac_inv[new_size - 1] = U(1) / _fac[new_size - 1];
            for (int i = new_size - 1; i > sz; --i) _fac_inv[i - 1] = _fac_inv[i] * i;
        }

        T fac(const int i) {
            ensure(i);
            return _fac[i];
        }
        T operator()(int i) {
            return fac(i);
        }
        U fac_inv(const int i) {
            ensure(i);
            return _fac_inv[i];
        }
        U binom(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[r] * _fac_inv[n - r];
        }
        template <typename ...Ds, std::enable_if_t<std::conjunction_v<std::is_integral<Ds>...>, std::nullptr_t> = nullptr>
        U polynom(const int n, const Ds& ...ds) {
            if (n < 0) return 0;
            ensure(n);
            int sumd = 0;
            U res = _fac[n];
            for (int d : { ds... }) {
                if (d < 0 or d > n) return 0;
                sumd += d;
                res *= _fac_inv[d];
            }
            if (sumd > n) return 0;
            res *= _fac_inv[n - sumd];
            return res;
        }
        U perm(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[n - r];
        }
    private:
        static std::vector<T> _fac;
        static std::vector<U> _fac_inv;
    };
    template <typename T, typename U>
    std::vector<T> factorial<T, U>::_fac{ 1 };
    template <typename T, typename U>
    std::vector<U> factorial<T, U>::_fac_inv{ 1 };
} // namespace suisen


#line 7 "library/sequence/stirling_number1.hpp"

namespace suisen {
    /**
     * return:
     *   vector<mint> v s.t. v[i] = S1[n,n-i] for i=0,...,k (unsigned)
     * constraints:
     *   0 <= n <= 10^6
     */
    template <typename FPSType>
    std::vector<typename FPSType::value_type> stirling_number1_reversed(int n) {
        using mint = typename FPSType::value_type;
        factorial<mint> fac(n);
        int l = 0;
        while ((n >> l) != 0) ++l;
        FPSType a{ 1 };
        int m = 0;
        while (l-- > 0) {
            FPSType f(m + 1), g(m + 1);
            mint powm = 1;
            for (int i = 0; i <= m; ++i, powm *= m) {
                f[i] = powm * fac.fac_inv(i);
                g[i] = a[i] * fac.fac(m - i);
            }
            f *= g, f.cut(m + 1);
            for (int i = 0; i <= m; ++i) f[i] *= fac.fac_inv(m - i);
            a *= f, m *= 2, a.cut(m + 1);
            if ((n >> l) & 1) {
                a.push_back(0);
                for (int i = m; i > 0; --i) a[i] += m * a[i - 1];
                ++m;
            }
        }
        return a;
    }
    template <typename FPSType>
    std::vector<typename FPSType::value_type> stirling_number1(int n) {
        std::vector<typename FPSType::value_type> a(stirling_number1_reversed<FPSType>(n));
        std::reverse(a.begin(), a.end());
        return a;
    }
    /**
     * return:
     *   vector<mint> v s.t. v[i] = S1[n,n-i] for i=0,...,k, where S1 is the stirling number of the first kind (unsigned).
     * constraints:
     * - 0 <= n <= 10^18
     * - 0 <= k <= 5000
     * - k < mod
     */
    template <typename mint>
    std::vector<mint> stirling_number1_reversed(const long long n, const int k) {
        inv_mods<mint> invs(k + 1);
        std::vector<mint> a(k + 1, 0);
        a[0] = 1;
        int l = 0;
        while (n >> l) ++l;
        mint m = 0;
        while (l-- > 0) {
            std::vector<mint> b(k + 1, 0);
            for (int j = 0; j <= k; ++j) {
                mint tmp = 1;
                for (int i = j; i <= k; ++i) {
                    b[i] += a[j] * tmp;
                    tmp *= (m - i) * invs[i - j + 1] * m;
                }
            }
            for (int i = k + 1; i-- > 0;) {
                mint sum = 0;
                for (int j = 0; j <= i; ++j) sum += a[j] * b[i - j];
                a[i] = sum;
            }
            m *= 2;
            if ((n >> l) & 1) {
                for (int i = k; i > 0; --i) a[i] += m * a[i - 1];
                ++m;
            }
        }
        return a;
    }
    template <typename mint>
    std::vector<std::vector<mint>> stirling_number1_table(int n) {
        std::vector dp(n + 1, std::vector<mint>{});
        for (int i = 0; i <= n; ++i) {
            dp[i].resize(i + 1);
            dp[i][0] = 0, dp[i][i] = 1;
            for (int j = 1; j < i; ++j) dp[i][j] = dp[i - 1][j - 1] + (i - 1) * dp[i - 1][j];
        }
        return dp;
    }
} // namespace suisen


#line 1 "library/sequence/binomial_coefficient_small_prime_mod.hpp"



#line 5 "library/sequence/binomial_coefficient_small_prime_mod.hpp"

namespace suisen {
    template <typename mint>
    struct BinomialCoefficientSmallPrimeMod {
        mint operator()(long long n, long long r) const {
            return binom(n, r);
        }
        static mint binom(long long n, long long r) {
            factorial<mint> fac(mint::mod() - 1);

            if (r < 0 or n < r) return 0;
            r = std::min(r, n - r);
            // Lucas's theorem
            mint res = 1;
            while (r) {
                int ni = n % mint::mod(), ri = r % mint::mod();
                if (ni < ri) return 0;
                res *= fac.binom(ni, ri);
                n = n / mint::mod(), r = r / mint::mod();
            }
            return res;
        }
    };
} // namespace suisen



#line 6 "library/sequence/stirling_number1_small_prime_mod.hpp"

namespace suisen {
    template <typename mint>
    struct StirlingNumber1SmallPrimeMod {
        mint operator()(long long n, long long k) const {
            return s1(n, k);
        }
        static mint s1(long long n, long long k) {
            static const std::vector<std::vector<mint>> table = stirling_number1_table<mint>(mint::mod() - 1);
            static const BinomialCoefficientSmallPrimeMod<mint> binom{};
            static const int p = mint::mod();
            if (k < 0 or n < k) return 0;
            long long a = n / p, b = n % p;
            if (k < a) return 0;
            long long c = (k - a) / (p - 1), d = (k - a) % (p - 1);
            return ((a - c) & 1 ? -1 : 1) * (b == p - 1 and d == 0 ? -binom(a, c - 1) : d <= b ? table[b][d] * binom(a, c): 0);
        }
    };
} // namespace suisen

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