the Stirling number of the second kind (前処理 $O(NK)$ + $O(1)$)
(modulus/stirling_number_of_the_second_kind_table.hpp)

• View this file on GitHub
• Last update: 2021-08-30 04:35:37+09:00
• Include: #include "modulus/stirling_number_of_the_second_kind_table.hpp"
• Link: http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html
• Link: http://oeis.org/A008277
• Link: https://ja.wikipedia.org/wiki/%E3%82%B9%E3%82%BF%E3%83%BC%E3%83%AA%E3%83%B3%E3%82%B0%E6%95%B0#.E7.AC.AC2.E7.A8.AE.E3.82.B9.E3.82.BF.E3.83.BC.E3.83.AA.E3.83.B3.E3.82.B0.E6.95.B0

Depends on

• quotient ring / 剰余環 $\mathbb{Z}/n\mathbb{Z}$ (modulus/mint.hpp)
• modulus/modinv.hpp
• modulus/modpow.hpp
• utils/macros.hpp

Required by

• the Bell number (前処理 $O(NK)$ + $O(1)$) (modulus/bell_number.hpp)
• the partition number (前処理 $O(NK)$ + $O(1)$) (modulus/partition_number.hpp)
• twelvefold way / 写像12相 (modulus/twelvefold_way.hpp)

Verified with

• modulus/twelvefold_way.balls_and_boxes_1.test.cpp
• modulus/twelvefold_way.balls_and_boxes_10.test.cpp
• modulus/twelvefold_way.balls_and_boxes_11.test.cpp
• modulus/twelvefold_way.balls_and_boxes_2.test.cpp
• modulus/twelvefold_way.balls_and_boxes_3.test.cpp
• modulus/twelvefold_way.balls_and_boxes_4.test.cpp
• modulus/twelvefold_way.balls_and_boxes_5.test.cpp
• modulus/twelvefold_way.balls_and_boxes_6.test.cpp
• modulus/twelvefold_way.balls_and_boxes_7.test.cpp
• modulus/twelvefold_way.balls_and_boxes_8.test.cpp
• modulus/twelvefold_way.balls_and_boxes_9.test.cpp

Code

#pragma once
#include <cassert>
#include <vector>
#include "../utils/macros.hpp"
#include "../modulus/mint.hpp"

/**
 * @brief the Stirling number of the second kind (前処理 $O(NK)$ + $O(1)$)
 * @description the number of ways of partitioning a set of n elements into k nonempty sets
 * @see http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html
 * @see http://oeis.org/A008277
 * @see https://ja.wikipedia.org/wiki/%E3%82%B9%E3%82%BF%E3%83%BC%E3%83%AA%E3%83%B3%E3%82%B0%E6%95%B0#.E7.AC.AC2.E7.A8.AE.E3.82.B9.E3.82.BF.E3.83.BC.E3.83.AA.E3.83.B3.E3.82.B0.E6.95.B0
 */
template <int MOD>
mint<MOD> stirling_number_of_the_second_kind_table(int n, int k) {
    assert (0 <= n and 0 <= k);
    if (n  < k) return 0;
    if (n == k) return 1;
    if (k == 0) return 0;
    static std::vector<std::vector<mint<MOD> > > memo;
    if (memo.size() <= n) {
        int l = memo.size();
        memo.resize(n + 1);
        REP3 (i, l, n + 1) {
            memo[i].resize(i);
        }
    }
    if (memo[n][k]) return memo[n][k];
    return memo[n][k] =
        stirling_number_of_the_second_kind_table<MOD>(n - 1, k - 1) +
        stirling_number_of_the_second_kind_table<MOD>(n - 1, k) * k;
}

#line 2 "modulus/stirling_number_of_the_second_kind_table.hpp"
#include <cassert>
#include <vector>
#line 2 "utils/macros.hpp"
#define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i))
#define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i))
#define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i))
#define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i))
#define ALL(x) std::begin(x), std::end(x)
#line 2 "modulus/mint.hpp"
#include <cstdint>
#include <iostream>
#line 4 "modulus/modpow.hpp"

inline int32_t modpow(uint_fast64_t x, uint64_t k, int32_t MOD) {
    assert (/* 0 <= x and */ x < (uint_fast64_t)MOD);
    uint_fast64_t y = 1;
    for (; k; k >>= 1) {
        if (k & 1) (y *= x) %= MOD;
        (x *= x) %= MOD;
    }
    assert (/* 0 <= y and */ y < (uint_fast64_t)MOD);
    return y;
}
#line 2 "modulus/modinv.hpp"
#include <algorithm>
#line 5 "modulus/modinv.hpp"

inline int32_t modinv_nocheck(int32_t value, int32_t MOD) {
    assert (0 <= value and value < MOD);
    if (value == 0) return -1;
    int64_t a = value, b = MOD;
    int64_t x = 0, y = 1;
    for (int64_t u = 1, v = 0; a; ) {
        int64_t q = b / a;
        x -= q * u; std::swap(x, u);
        y -= q * v; std::swap(y, v);
        b -= q * a; std::swap(b, a);
    }
    if (not (value * x + MOD * y == b and b == 1)) return -1;
    if (x < 0) x += MOD;
    assert (0 <= x and x < MOD);
    return x;
}

inline int32_t modinv(int32_t x, int32_t MOD) {
    int32_t y = modinv_nocheck(x, MOD);
    assert (y != -1);
    return y;
}
#line 6 "modulus/mint.hpp"

/**
 * @brief quotient ring / 剰余環 $\mathbb{Z}/n\mathbb{Z}$
 */
template <int32_t MOD>
struct mint {
    int32_t value;
    mint() : value() {}
    mint(int64_t value_) : value(value_ < 0 ? value_ % MOD + MOD : value_ >= MOD ? value_ % MOD : value_) {}
    mint(int32_t value_, std::nullptr_t) : value(value_) {}
    explicit operator bool() const { return value; }
    inline mint<MOD> operator + (mint<MOD> other) const { return mint<MOD>(*this) += other; }
    inline mint<MOD> operator - (mint<MOD> other) const { return mint<MOD>(*this) -= other; }
    inline mint<MOD> operator * (mint<MOD> other) const { return mint<MOD>(*this) *= other; }
    inline mint<MOD> & operator += (mint<MOD> other) { this->value += other.value; if (this->value >= MOD) this->value -= MOD; return *this; }
    inline mint<MOD> & operator -= (mint<MOD> other) { this->value -= other.value; if (this->value <    0) this->value += MOD; return *this; }
    inline mint<MOD> & operator *= (mint<MOD> other) { this->value = (uint_fast64_t)this->value * other.value % MOD; return *this; }
    inline mint<MOD> operator - () const { return mint<MOD>(this->value ? MOD - this->value : 0, nullptr); }
    inline bool operator == (mint<MOD> other) const { return value == other.value; }
    inline bool operator != (mint<MOD> other) const { return value != other.value; }
    inline mint<MOD> pow(uint64_t k) const { return mint<MOD>(modpow(value, k, MOD), nullptr); }
    inline mint<MOD> inv() const { return mint<MOD>(modinv(value, MOD), nullptr); }
    inline mint<MOD> operator / (mint<MOD> other) const { return *this * other.inv(); }
    inline mint<MOD> & operator /= (mint<MOD> other) { return *this *= other.inv(); }
};
template <int32_t MOD> mint<MOD> operator + (int64_t value, mint<MOD> n) { return mint<MOD>(value) + n; }
template <int32_t MOD> mint<MOD> operator - (int64_t value, mint<MOD> n) { return mint<MOD>(value) - n; }
template <int32_t MOD> mint<MOD> operator * (int64_t value, mint<MOD> n) { return mint<MOD>(value) * n; }
template <int32_t MOD> mint<MOD> operator / (int64_t value, mint<MOD> n) { return mint<MOD>(value) / n; }
template <int32_t MOD> std::istream & operator >> (std::istream & in, mint<MOD> & n) { int64_t value; in >> value; n = value; return in; }
template <int32_t MOD> std::ostream & operator << (std::ostream & out, mint<MOD> n) { return out << n.value; }
#line 6 "modulus/stirling_number_of_the_second_kind_table.hpp"

/**
 * @brief the Stirling number of the second kind (前処理 $O(NK)$ + $O(1)$)
 * @description the number of ways of partitioning a set of n elements into k nonempty sets
 * @see http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html
 * @see http://oeis.org/A008277
 * @see https://ja.wikipedia.org/wiki/%E3%82%B9%E3%82%BF%E3%83%BC%E3%83%AA%E3%83%B3%E3%82%B0%E6%95%B0#.E7.AC.AC2.E7.A8.AE.E3.82.B9.E3.82.BF.E3.83.BC.E3.83.AA.E3.83.B3.E3.82.B0.E6.95.B0
 */
template <int MOD>
mint<MOD> stirling_number_of_the_second_kind_table(int n, int k) {
    assert (0 <= n and 0 <= k);
    if (n  < k) return 0;
    if (n == k) return 1;
    if (k == 0) return 0;
    static std::vector<std::vector<mint<MOD> > > memo;
    if (memo.size() <= n) {
        int l = memo.size();
        memo.resize(n + 1);
        REP3 (i, l, n + 1) {
            memo[i].resize(i);
        }
    }
    if (memo[n][k]) return memo[n][k];
    return memo[n][k] =
        stirling_number_of_the_second_kind_table<MOD>(n - 1, k - 1) +
        stirling_number_of_the_second_kind_table<MOD>(n - 1, k) * k;
}

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