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#define PROBLEM "http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=DPL_5_F" #include <iostream> #include "../modulus/twelvefold_way.hpp" using namespace std; constexpr int MOD = 1e9 + 7; int main() { int n, k; cin >> n >> k; cout << twelvefold_uls<MOD>(n, k) << endl; return 0; }
#line 1 "modulus/twelvefold_way.balls_and_boxes_6.test.cpp" #define PROBLEM "http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=DPL_5_F" #include <iostream> #line 2 "modulus/mint.hpp" #include <cstdint> #line 2 "modulus/modpow.hpp" #include <cassert> #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 2 "modulus/factorial.hpp" #include <vector> #line 4 "modulus/factorial.hpp" template <int32_t MOD> mint<MOD> fact(int n) { static std::vector<mint<MOD> > memo(1, 1); while (n >= memo.size()) { memo.push_back(memo.back() * mint<MOD>(memo.size())); } return memo[n]; } template <int32_t MOD> mint<MOD> inv_fact(int n) { static std::vector<mint<MOD> > memo; if (memo.size() <= n) { int l = memo.size(); int r = n * 1.3 + 100; memo.resize(r); memo[r - 1] = fact<MOD>(r - 1).inv(); for (int i = r - 2; i >= l; -- i) { memo[i] = memo[i + 1] * (i + 1); } } return memo[n]; } #line 5 "modulus/choose.hpp" /** * @brief combination / 組合せ ${} _ n C _ r$ (前処理 $O(n)$ + $O(1)$) */ template <int32_t MOD> mint<MOD> choose(int n, int r) { assert (0 <= r and r <= n); return fact<MOD>(n) * inv_fact<MOD>(n - r) * inv_fact<MOD>(r); } #line 5 "modulus/permute.hpp" /** * @brief permutation / 順列 ${} _ n P _ r$ (前処理 $O(n)$ + $O(1)$) */ template <int32_t MOD> mint<MOD> permute(int n, int r) { assert (0 <= r and r <= n); return fact<MOD>(n) * inv_fact<MOD>(n - r); } #line 5 "modulus/multichoose.hpp" /** * @brief 重複組合せ ${} _ n H _ r = {} _ {n + r - 1} C _ r$ (前処理 $O(n)$ + $O(1)$) */ template <int32_t MOD> mint<MOD> multichoose(int n, int r) { assert (0 <= n and 0 <= r); if (n == 0 and r == 0) return 1; return choose<MOD>(n + r - 1, r); } #line 4 "modulus/stirling_number_of_the_second_kind_direct.hpp" #include <map> #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 9 "modulus/stirling_number_of_the_second_kind_direct.hpp" /** * @brief the Stirling number of the second kind ($O(K \log N)$) */ template <int MOD> mint<MOD> stirling_number_of_the_second_kind_direct(int n, int k) { assert (0 <= n and 0 <= k); mint<MOD> acc = 0; REP (i, k + 1) { int parity = ((k - i) % 2 == 0 ? +1 : -1); acc += choose<MOD>(k, i) * mint<MOD>(i).pow(n) * parity; } return acc * inv_fact<MOD>(k); } #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; } #line 5 "modulus/bell_number.hpp" /** * @brief the Bell number (前処理 $O(NK)$ + $O(1)$) * @description the number of ways a set of n elements can be partitioned into nonempty subsets * @see http://mathworld.wolfram.com/BellNumber.html * @see https://oeis.org/A110 * @see https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E6%95%B0 */ template <int PRIME> mint<PRIME> bell_number(int n, int k) { static std::vector<std::vector<mint<PRIME> > > memo; if (memo.size() <= n) { memo.resize(n + 1); } if (memo[n].empty()) { memo[n].push_back(0); } while (memo[n].size() <= k) { int i = memo[n].size(); memo[n].push_back(memo[n].back() + stirling_number_of_the_second_kind_table<PRIME>(n, i)); } return memo[n][k]; } template <int PRIME> mint<PRIME> unary_bell_number(int n) { return bell_number<PRIME>(n, n); } #line 5 "modulus/partition_number.hpp" /** * @brief the partition number (前処理 $O(NK)$ + $O(1)$) * @description the number of non-decreasing sequences with the length k which the sum is n */ template <int MOD> mint<MOD> partition_number(int n, int k) { static std::vector<std::vector<mint<MOD> > > memo; if (memo.size() <= n) { memo.resize(n + 1); } while (memo[n].size() <= k) { if (n == 0) { memo[0].resize(k + 1, 1); } else if (memo[n].empty()) { memo[n].push_back(0); } else { int j = memo[n].size(); auto a = (n - j >= 0 ? partition_number<MOD>(n - j, j) : 0); auto b = (j - 1 >= 0 ? partition_number<MOD>(n, j - 1) : 0); memo[n].push_back(a + b); } } return memo[n][k]; } template <int MOD> mint<MOD> unary_partition_number(int n) { return partition_number<MOD>(n, n); } #line 9 "modulus/twelvefold_way.hpp" /** * @brief twelvefold way / 写像12相 * @sa https://en.wikipedia.org/wiki/Twelvefold_way * @sa https://mathtrain.jp/twelveway * @note the numbers of mapprings f putting N balls into K boxes */ /** * @brief labeled-N labeled-K any-f * @note the number of f for all f : N \to K * @note O(log K) */ template <int MOD> mint<MOD> twelvefold_lla(int n, int k) { return mint<MOD>(k).pow(n); } /** * @brief labeled-N labeled-K injective-f * @note the number of f for injective f : N \rightarrowtail K * @note O(1) with precomputation O(N + K) */ template <int MOD> mint<MOD> twelvefold_lli(int n, int k) { if (n > k) return 0; return permute<MOD>(k, n); } /** * @brief labeled-N labeled-K surjective-f * @note the number of f for surjective f : N \twoheadrightarrow K * @note O(NK) or O(N \log K) */ template <int MOD> mint<MOD> twelvefold_lls(int n, int k) { return stirling_number_of_the_second_kind_direct<MOD>(n, k) * fact<MOD>(k); } /** * @brief unlabeled-N labeled-K any-f * @note the number of f \circ S_N for any f : N \to K * @note O(1) with precomputation O(N + K) */ template <int MOD> mint<MOD> twelvefold_ula(int n, int k) { return choose<MOD>(n + k - 1, n); } /** * @brief unlabeled-N labeled-K injective-f * @note the number of f \circ S_N for injective f : N \rightarrowtail K * @note O(1) with precomputation O(K) */ template <int MOD> mint<MOD> twelvefold_uli(int n, int k) { if (n > k) return 0; return choose<MOD>(k, n); } /** * @brief unlabeled-N labeled-K surjective-f * @note the number of f \circ S_N for surjective f : N \twoheadrightarrow K * @note O(1) with precomputation O(N) */ template <int MOD> mint<MOD> twelvefold_uls(int n, int k) { if (n < k) return 0; return choose<MOD>(n - 1, n - k); } /** * @brief labeled-N unlabeled-K any-f * @note the number of S_K \circ f for all f : N \to K */ template <int MOD> mint<MOD> twelvefold_lua(int n, int k) { return bell_number<MOD>(n, k); } /** * @brief labeled-N unlabeled-K injective-f * @note the number of S_K \circ f for injective f : N \rightarrowtail K * @note O(1) */ template <int MOD> mint<MOD> twelvefold_lui(int n, int k) { if (n > k) return 0; return 1; } /** * @brief labeled-N unlabeled-K surjective-f * @note the number of S_K \circ f for surjective f : N \twoheadrightarrow K * @note O(NK) or O(N \log K) */ template <int MOD> mint<MOD> twelvefold_lus(int n, int k) { return stirling_number_of_the_second_kind_direct<MOD>(n, k); } /** * @brief unlabeled-N unlabeled-K any-f */ template <int MOD> mint<MOD> twelvefold_uua(int n, int k) { return partition_number<MOD>(n, k); } /** * @brief unlabeled-N unlabeled-K injective-f * @note the number of S_K \circ f \circ S_N for injective f : N \rightarrowtail K * @note O(1) */ template <int MOD> mint<MOD> twelvefold_uui(int n, int k) { if (n > k) return 0; return 1; } #line 4 "modulus/twelvefold_way.balls_and_boxes_6.test.cpp" using namespace std; constexpr int MOD = 1e9 + 7; int main() { int n, k; cin >> n >> k; cout << twelvefold_uls<MOD>(n, k) << endl; return 0; }