#ifdef __LOCAL #include // プリコンパイル済みヘッダ ~/local/include/mytemplate.hpp.gch #else #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #endif using namespace std; // #pragma GCC target "no-avx" // gcc12.2のバグ回避用。boost使用時はコメントアウトの事 #include using bint = boost::multiprecision::cpp_int; using ll = long long; using vl = vector; using vvl = vector; using vb = vector; using vvb = vector; using vd = vector; using vvd = vector; using vc = vector; using vvc = vector; using pll = pair; template using min_priority_queue = priority_queue, greater>; #define REP1(i, n) REP3(i, 0, n, 1) #define REP2(i, s, n) REP3(i, s, n, 1) #define REP3(i, s, n, d) for (ll i = (ll)(s); i < (ll)(n); i += (d)) #define REP_OVERLOAD(e1, e2, e3, e4, NAME, ...) NAME #define rep(...) REP_OVERLOAD(__VA_ARGS__, REP3, REP2, REP1)(__VA_ARGS__) #define DEP1(i, n) DEP3(i, n, -1, 1) #define DEP2(i, n, s) DEP3(i, n, s, 1) #define DEP3(i, n, s, d) for (ll i = (ll)(n); (ll)(s) < i; i -= (d)) #define DEP_OVERLOAD(e1, e2, e3, e4, NAME, ...) NAME #define dep(...) DEP_OVERLOAD(__VA_ARGS__, DEP3, DEP2, DEP1)(__VA_ARGS__) #define fore(e, a) for (auto&& e : (a)) #define len(a) (ll)(a).size() #define all(a) (a).begin(), (a).end() #define rall(a) (a).rbegin(), (a).rend() #define pb push_back #define eb emplace_back #define fi first #define se second #define th third #define fo fourth /********* C O N S T A N T **********/ constexpr ll INF = LONG_LONG_MAX / 2 - 10000LL; // 4,611,686,018,427,377,903 ~= 4.6e+18, 19桁 const double PI = acos(-1); /********* 1 2 8 b i t 整数 **********/ using int128 = __int128_t; istream& operator>>(istream& stream, int128& val); ostream& operator<<(ostream& stream, const int128& val_); /********* P A I R **********/ // 単項演算子（+ -） template pair operator+(pair a); template pair operator-(pair a); // 複合代入演算子 template pair& operator+=(pair& a, const pair b); template pair& operator-=(pair& a, const pair b); template pair& operator*=(pair& a, const pair b); template pair& operator/=(pair& a, const pair b); template pair& operator%=(pair& a, const pair b); template pair& operator+=(pair& a, const V b); template pair& operator-=(pair& a, const V b); template pair& operator*=(pair& a, const V b); template pair& operator/=(pair& a, const V b); template pair& operator%=(pair& a, const V b); // 算術演算子 template pair operator+(pair a, const V& b); template pair operator-(pair a, const V& b); template pair operator*(pair a, const V& b); template pair operator/(pair a, const V& b); template pair operator%(pair a, const V& b); // 入出力 template istream& operator>>(istream& stream, pair& a); template ostream& operator<<(ostream& stream, const pair& a); /********* V E C T O R **********/ // 単項演算子（+ -） template vector operator+(vector a); template vector operator-(vector a); // 複合代入演算子 template vector& operator+=(vector& a, const vector b); template vector& operator-=(vector& a, const vector b); template vector& operator*=(vector& a, const vector b); template vector& operator/=(vector& a, const vector b); template vector& operator%=(vector& a, const vector b); template vector& operator+=(vector& a, const U b); template vector& operator-=(vector& a, const U b); template vector& operator*=(vector& a, const U b); template vector& operator/=(vector& a, const U b); template vector& operator%=(vector& a, const U b); // 算術演算子 template vector operator+(vector a, const U& b); template vector operator-(vector a, const U& b); template vector operator*(vector a, const U& b); template vector operator/(vector a, const U& b); template vector operator%(vector a, const U& b); // 入出力 template istream& operator>>(istream& stream, vector& a); template ostream& operator<<(ostream& stream, const vector& v); template ostream& operator<<(ostream& stream, const vector>& vv); // 計算 template T Sum(const vector& v); template T Max(const vector& v); template T Min(const vector& v); template ll Argmax(const vector& v); template ll Argmin(const vector& v); template bool Contains(const vector& v, const U& a); template void Unique(vector& v); // 生成 template vector make_vector(int n, U v); template auto make_vector(int n, Args... args) { auto val = make_vector(args...); return make_vector(n, move(val)); } /********* S E T **********/ template istream& operator>>(istream& stream, set& st); template ostream& operator<<(ostream& stream, const set& st); template T Max(const set& st); template T Min(const set& st); /********* M U L T I S E T **********/ template istream& operator>>(istream& stream, multiset& st); template ostream& operator<<(ostream& stream, const multiset& st); template T Max(const multiset& st); template T Min(const multiset& st); template bool EraseOne(multiset& st, const T& x); /********* C H A R **********/ bool IsUpper(const char& c); bool IsLower(const char& c); char ToUpper(const char& c); char ToLower(const char& c); /********* S T R I N G **********/ string ToUpper(const string& s); string ToLower(const string& s); bool IsPalindrome(const string& S); /********* T R I O **********/ template struct trio { T1 first; T2 second; T3 third; // コンストラクタ・代入演算子 trio(); trio(const T1& x); trio(const T1& x, const T2& y, const T3& z); trio(const trio& t); trio& operator=(const trio& t); // 比較演算子 bool operator<(const trio& t) const; bool operator==(const trio& t) const; bool operator!=(const trio& t) const; bool operator>(const trio& t) const; bool operator<=(const trio& t) const; bool operator>=(const trio& t) const; // 単項演算子（+ -） trio operator+() const; trio operator-() const; // 入出力 template friend istream& operator>>(istream&, trio&); template friend ostream& operator<<(ostream&, const trio&); // `tie(a, b, c) = tll{3, 5, 7}`をするのに必要（コンパイラが警告？を出すのでここで定義） operator tuple() { return tie(first, second, third); } }; // 複合代入演算子 template trio& operator+=(trio& a, const trio b); template trio& operator-=(trio& a, const trio b); template trio& operator*=(trio& a, const trio b); template trio& operator/=(trio& a, const trio b); template trio& operator%=(trio& a, const trio b); template trio& operator+=(trio& a, const U b); template trio& operator-=(trio& a, const U b); template trio& operator*=(trio& a, const U b); template trio& operator/=(trio& a, const U b); template trio& operator%=(trio& a, const U b); // 算術演算子 template trio operator+(trio a, const U& b); template trio operator-(trio a, const U& b); template trio operator*(trio a, const U& b); template trio operator/(trio a, const U& b); template trio operator%(trio a, const U& b); using tll = trio; /********* Q U A R T E T **********/ template struct quartet { T1 first; T2 second; T3 third; T4 fourth; // コンストラクタ・代入演算子 quartet(); quartet(const T1& x); quartet(const T1& x, const T2& y, const T3& z, const T4& w); quartet(const quartet& t); quartet& operator=(const quartet& t); // 比較演算子 bool operator<(const quartet& t) const; bool operator==(const quartet& t) const; bool operator!=(const quartet& other) const; bool operator>(const quartet& other) const; bool operator<=(const quartet& other) const; bool operator>=(const quartet& other) const; // 単項演算子（+ -） quartet operator+() const; quartet operator-() const; // 入出力 template friend istream& operator>>(istream&, quartet&); template friend ostream& operator<<(ostream&, const quartet&); // `tie(a, b, c, d) = qll{3, 5, 7, 9}`をするのに必要（コンパイラが警告？を出すのでここで定義） operator tuple() { return tie(first, second, third, fourth); } }; // 複合代入演算子 template quartet& operator+=(quartet& a, const quartet b); template quartet& operator-=(quartet& a, const quartet b); template quartet& operator*=(quartet& a, const quartet b); template quartet& operator/=(quartet& a, const quartet b); template quartet& operator%=(quartet& a, const quartet b); template quartet& operator+=(quartet& a, const U b); template quartet& operator-=(quartet& a, const U b); template quartet& operator*=(quartet& a, const U b); template quartet& operator/=(quartet& a, const U b); template quartet& operator%=(quartet& a, const U b); // 算術演算子 template quartet operator+(quartet a, const U& b); template quartet operator-(quartet a, const U& b); template quartet operator*(quartet a, const U& b); template quartet operator/(quartet a, const U& b); template quartet operator%(quartet a, const U& b); using qll = quartet; /********* R A N D O M **********/ struct Random { mt19937_64 rnd; Random(); Random(ll seed); ll randint(ll a, ll b); double randreal(double a, double b); char randchar(char a, char b); template void shuffle(vector& vec); }; /********* T I M E R **********/ struct Timer { int time_limit; int start_time; Timer(double time_limit_sec); bool is_time_up() const; double get_elapsed_ratio() const; }; /********* P R I N T **********/ template void print(const T& e); template void print(const H& h, const T&... t); template void End(const T&... t); /********* D E B U G **********/ #ifdef __LOCAL #define debug(...) \ if (DEBUG) do { \ cout << '[' << #__VA_ARGS__ << "] "; \ debug_(__VA_ARGS__); \ } while (0) #else #define debug(...) #endif void dbg_(const long long& e); template void dbg_(const T& e); template void dbg_(const pair& p); template void dbg_(const trio& t); template void dbg_(const quartet& t); template void debug_(const T& e); template void debug_(const vector& v); template void debug_(const vector>& vv); template void debug_(const map& mp); template void debug_(const vector>& vm); template void debug_(const set& st); template void debug_(const multiset& st); template void debug_(const vector>& vs); template void debug_(const vector>& vs); template void debug_(const H& h, const T&... t); /********* O T H E R S **********/ template bool chmin(T& a, U b); template bool chmax(T& a, U b); template T Mod(const T& a, const U& m); template T Ceil(const T& x, const U& y); template T Floor(const T& x, const U& y); template ll Sign(const T& x); ll Isqrt(ll n); ll Comb(ll n, ll r); template T Pow(T x, ll n); template T Aseries(T a, T d, ll n); template T Gseries(T a, T r, ll n); bool Bit(ll b, int i); ll Popcount(ll b); ll Mask(ll n); ll Mask(ll n, ll m); pll GetOverlap(ll l1, ll r1, ll l2, ll r2); bool IsOverlap(ll l1, ll r1, ll l2, ll r2); void PrintYesNo(bool b); /********* M O D **********/ // #include // https://atcoder.github.io/ac-library/production/document_ja/ // using namespace atcoder; // using mint = modint998244353; // modint1000000007; // istream& operator >>(istream& stream, mint& e) {ll n; stream >> n; e = n; return stream; } // ostream& operator <<(ostream& stream, const mint& e) { stream << e.val(); return stream; } // using vm = vector; // using vvm = vector; /***************************************/ constexpr bool DEBUG = true; // 分数構造体 // 利用可能演算子：+ - * / == != < > <= >= // 利用可能関数：sq(), inv(), abs(), val() // 特殊な値： 0: 0/1, +inf: 1/0, -inf: -1/0 // Examples: // Rational r(-6, 20); // -6/20 = -3/10 // Rational s(4, 30); // 4/30 = 2/15 // cout << ((r + s) * 2 / 3) << endl; // (-3/10 + 2/15) * 2 / 3 = -1/9 // cout << s.val() << endl; // 2/15 = 0.1333333333 template struct Rational { private: T num, den; // 分子, 分母 public: T get_num() { return num; } T get_den() { return den; } // コンストラクタ Rational() { num = 0; den = 1; } Rational(const T& num_) { num = num_; den = 1; } Rational(const Rational& r) {num = r.num; den = r.den; } // コピーコンストラクタ Rational(const T& num_, const T& den_) { assert(num_ != 0 || den_ != 0); if (num_ == 0) { num = 0; den = 1; } else { // 分母 >= 0にする & 約分する. // 等号での比較のため約分（値を一意にする）は必要. T sign = den_ < 0 ? -1 : 1; T g = gcd(num_, den_); num = sign * num_ / g; den = sign * den_ / g; } } // 代入演算子 Rational& operator =(const Rational& other) { num = other.num; den = other.den; return *this; } // 比較演算子（== != < > <= >=） bool operator <(const Rational& other) const { if (*this == other) { // 両辺が（+/-infも含めて）同一のとき return false; } else if ((num == -1 && den == 0) || (other.num == 1 && other.den == 0)) { // 左辺が-inf か、右辺が+inf return true; } else if ((num == 1 && den == 0) || (other.num == -1 && other.den == 0)) { // 左辺が+infか、右辺が-inf return false; } else { return num * other.den < other.num * den; } } bool operator ==(const Rational& other) const { return (num == other.num) && (den == other.den); } bool operator !=(const Rational& other) const { return !(*this == other); } bool operator >(const Rational& other) const { return other < *this; } bool operator <=(const Rational& other) const { return !(*this > other); } bool operator >=(const Rational& other) const { return !(*this < other); } // 単項演算子（+ -） Rational operator +() const { return *this; } Rational operator -() const { return Rational(-num, den); } // 複合代入演算子（+ - * /） Rational& operator +=(const Rational& other) { // inf + inf = inf, -inf + (-inf) = -inf // inf + (-inf) と (-inf) + inf は未定義 if (den == 0) { // そのまま返す } else if (other.den == 0) { num = other.num; den = other.den; } else { *this = Rational(num * other.den + other.num * den, den * other.den); } return *this; } Rational& operator *=(const Rational& other) { *this = Rational(num * other.num, den * other.den); return *this; } Rational& operator -=(const Rational& other) {*this += -other; return *this;} Rational& operator /=(const Rational& other) {*this *= other.inv(); return *this;} // 算術演算子（+ - * /） Rational operator +(const Rational& other) const {return Rational(*this) += other;} Rational operator -(const Rational& other) const {return Rational(*this) -= other;} Rational operator *(const Rational& other) const {return Rational(*this) *= other;} Rational operator /(const Rational& other) const {return Rational(*this) /= other;} // 二乗、絶対値、逆数 Rational sq() const { return Rational(num * num, den * den); } Rational abs() const { if (num < 0) return Rational(-num, den); else return *this; } Rational inv() const { return Rational(den, num); } // double型に変換 double val() const { return (double)num / (double)den; } // 整数型との演算 // Rational + intのような整数型が異なる場合に呼び出される template Rational& operator +=(const U& x) { *this += Rational((T)x); return *this; } template Rational& operator -=(const U& x) { *this -= Rational((T)x); return *this; } template Rational& operator *=(const U& x) { *this *= Rational((T)x); return *this; } template Rational& operator /=(const U& x) { *this /= Rational((T)x); return *this; } template Rational operator +(const U& x) const { return Rational(*this) += x; } template Rational operator -(const U& x) const { return Rational(*this) -= x; } template Rational operator *(const U& x) const { return Rational(*this) *= x; } template Rational operator /(const U& x) const { return Rational(*this) /= x; } // 入出力。参考：https://stackoverflow.com/questions/4039817/friend-declaration-declares-a-non-template-function // 入力時はインスタンスを生成 & 変更するのでconstをつけない. 出力時は変更しないのでconstをつける. friend istream& operator >>(istream &stream, Rational& r) { return stream >> r.num; } friend ostream& operator <<(ostream &stream, const Rational& r) { stream << r.num << '/' << r.den; return stream; } }; // 整数と分数の演算 template Rational operator +(const U& x, const Rational& r) { return Rational((T)x) + r; } template Rational operator -(const U& x, const Rational& r) { return Rational((T)x) - r; } template Rational operator *(const U& x, const Rational& r) { return Rational((T)x) * r; } template Rational operator /(const U& x, const Rational& r) { return Rational((T)x) / r; } int main() { cin.tie(nullptr); cout << fixed << setprecision(10); ll K, N; cin >> K >> N; using R = Rational; vl P(K + 1, 1); rep(i, 2, K + 1) { P[i] = P[i - 1] * i; } debug(P); vector rest(K + 1); rep(k, 1, K + 1) { rest[k] = R(1) - R(k, N); } ll cnt = 0; ll ans = 0; auto f = [&](auto&& f, ll p, ll last, ll sum, R val) -> void { cnt++; if (p == K) { if (val == R(1)) ans += sum; return; } rep(n, last + 1, N + 1) { rep(k, 1, K - p + 1) { R val2 = val + R(k, n); if (val2 > R(1)) break; f(f, p + k, n, sum / P[k], val2); } } }; f(f, 0, 0, P[K], R(0)); debug(cnt); print(ans); } /********* 1 2 8 b i t 整数 *********/ // 128bit整数を入力. LONG_LONG_MAX超過/LONG_LONG_MIN未満もOK. istream& operator>>(istream& stream, int128& val) { string s; stream >> s; assert(s.size() > 0); val = 0; bool is_minus = false; for (int i = 0; i < (int)s.size(); i++) { if (i == 0 && s[i] == '-') { assert(s.size() > 1); is_minus = true; } else { assert('0' <= s[i] && s[i] <= '9'); val *= 10; val += s[i] - '0'; } } if (is_minus) val *= -1; return stream; } // 128bit整数を出力. ostream& operator<<(ostream& stream, const int128& val_) { auto val(val_); if (val == 0) return stream << 0; if (val < 0) { stream << '-'; val *= -1; } string s; while (val > 0) { s += (val % 10) + '0'; val /= 10; } reverse(all(s)); return stream << s; } /********* P A I R **********/ /* 単項演算子（+ -） */ template pair operator+(pair a) { return a; } template pair operator-(pair a) { return a * (-1); } /* 複合代入演算子 (pair) */ template pair& operator+=(pair& a, const pair b) { a.fi += b.fi; a.se += b.se; return a; } template