#594856 (C++17) No.1321 塗るめた

• 問題: No.1321 塗るめた
• ユーザー: emthrm
• 提出日時: 2020-12-18 04:02:32
• 言語: C++17 (gcc 13.3.0 + boost 1.87.0)
• 結果: AC
• 実行時間: 461 ms / 2,000 ms
• コード長: 16,540 bytes
• コンパイル時間: 2,955 ms
• コンパイル使用メモリ: 223,236 KB
• 最終ジャッジ日時: 2025-01-17 02:43:15
• ジャッジサーバーID （参考情報）: judge5 / judge3

ファイルパターン	結果
sample	AC * 2
other	AC * 45

ソースコード

#define _USE_MATH_DEFINES
#include <bits/stdc++.h>
using namespace std;
#define FOR(i,m,n) for(int i=(m);i<(n);++i)
#define REP(i,n) FOR(i,0,n)
#define ALL(v) (v).begin(),(v).end()
using ll = long long;
constexpr int INF = 0x3f3f3f3f;
constexpr long long LINF = 0x3f3f3f3f3f3f3f3fLL;
constexpr double EPS = 1e-8;
constexpr int MOD = 998244353;
constexpr int dy[] = {1, 0, -1, 0}, dx[] = {0, -1, 0, 1};
constexpr int dy8[] = {1, 1, 0, -1, -1, -1, 0, 1}, dx8[] = {0, -1, -1, -1, 0, 1, 1, 1};
template <typename T, typename U> inline bool chmax(T &a, U b) { return a < b ? (a = b, true) : false; }
template <typename T, typename U> inline bool chmin(T &a, U b) { return a > b ? (a = b, true) : false; }
struct IOSetup {
  IOSetup() {
    std::cin.tie(nullptr);
    std::ios_base::sync_with_stdio(false);
    std::cout << fixed << setprecision(20);
  }
} iosetup;

template <int MOD>
struct MInt {
  unsigned val;
  MInt(): val(0) {}
  MInt(long long x) : val(x >= 0 ? x % MOD : x % MOD + MOD) {}
  static int get_mod() { return MOD; }
  static void set_mod(int divisor) { assert(divisor == MOD); }
  MInt pow(long long exponent) const {
    MInt tmp = *this, res = 1;
    while (exponent > 0) {
      if (exponent & 1) res *= tmp;
      tmp *= tmp;
      exponent >>= 1;
    }
    return res;
  }
  MInt &operator+=(const MInt &x) { if((val += x.val) >= MOD) val -= MOD; return *this; }
  MInt &operator-=(const MInt &x) { if((val += MOD - x.val) >= MOD) val -= MOD; return *this; }
  MInt &operator*=(const MInt &x) { val = static_cast<unsigned long long>(val) * x.val % MOD; return *this; }
  MInt &operator/=(const MInt &x) {
    // assert(std::__gcd(static_cast<int>(x.val), MOD) == 1);
    unsigned a = x.val, b = MOD; int u = 1, v = 0;
    while (b) {
      unsigned tmp = a / b;
      std::swap(a -= tmp * b, b);
      std::swap(u -= tmp * v, v);
    }
    return *this *= u;
  }
  bool operator==(const MInt &x) const { return val == x.val; }
  bool operator!=(const MInt &x) const { return val != x.val; }
  bool operator<(const MInt &x) const { return val < x.val; }
  bool operator<=(const MInt &x) const { return val <= x.val; }
  bool operator>(const MInt &x) const { return val > x.val; }
  bool operator>=(const MInt &x) const { return val >= x.val; }
  MInt &operator++() { if (++val == MOD) val = 0; return *this; }
  MInt operator++(int) { MInt res = *this; ++*this; return res; }
  MInt &operator--() { val = (val == 0 ? MOD : val) - 1; return *this; }
  MInt operator--(int) { MInt res = *this; --*this; return res; }
  MInt operator+() const { return *this; }
  MInt operator-() const { return MInt(val ? MOD - val : 0); }
  MInt operator+(const MInt &x) const { return MInt(*this) += x; }
  MInt operator-(const MInt &x) const { return MInt(*this) -= x; }
  MInt operator*(const MInt &x) const { return MInt(*this) *= x; }
  MInt operator/(const MInt &x) const { return MInt(*this) /= x; }
  friend std::ostream &operator<<(std::ostream &os, const MInt &x) { return os << x.val; }
  friend std::istream &operator>>(std::istream &is, MInt &x) { long long val; is >> val; x = MInt(val); return is; }
};
namespace std { template <int MOD> MInt<MOD> abs(const MInt<MOD> &x) { return x; } }
template <int MOD>
struct Combinatorics {
  using ModInt = MInt<MOD>;
  int val;  // "val!" and "mod" must be disjoint.
  std::vector<ModInt> fact, fact_inv, inv;
  Combinatorics(int val = 10000000) : val(val), fact(val + 1), fact_inv(val + 1), inv(val + 1) {
    fact[0] = 1;
    for (int i = 1; i <= val; ++i) fact[i] = fact[i - 1] * i;
    fact_inv[val] = ModInt(1) / fact[val];
    for (int i = val; i > 0; --i) fact_inv[i - 1] = fact_inv[i] * i;
    for (int i = 1; i <= val; ++i) inv[i] = fact[i - 1] * fact_inv[i];
  }
  ModInt nCk(int n, int k) const {
    if (n < 0 || n < k || k < 0) return 0;
    assert(n <= val && k <= val);
    return fact[n] * fact_inv[k] * fact_inv[n - k];
  }
  ModInt nPk(int n, int k) const {
    if (n < 0 || n < k || k < 0) return 0;
    assert(n <= val);
    return fact[n] * fact_inv[n - k];
  }
  ModInt nHk(int n, int k) const {
    if (n < 0 || k < 0) return 0;
    return k == 0 ? 1 : nCk(n + k - 1, k);
  }
};
using ModInt = MInt<MOD>;

struct NTT {
  NTT() {
    for (int i = 0; i < 23; ++i) {
      if (primes[i][0] == ModInt::get_mod()) {
        n_max = 1 << primes[i][2];
        root = ModInt(primes[i][1]).pow((ModInt::get_mod() - 1) >> primes[i][2]);
        return;
      }
    }
    assert(false);
  }

  void sub_dft(std::vector<ModInt> &a) {
    int n = a.size();
    assert(__builtin_popcount(n) == 1);
    calc(n);
    int shift = __builtin_ctz(butterfly.size()) - __builtin_ctz(n);
    for (int i = 0; i < n; ++i) {
      int j = butterfly[i] >> shift;
      if (i < j) std::swap(a[i], a[j]);
    }
    for (int block = 1; block < n; block <<= 1) {
      int den = __builtin_ctz(block);
      for (int i = 0; i < n; i += (block << 1)) for (int j = 0; j < block; ++j) {
        ModInt tmp = a[i + j + block] * omega[den][j];
        a[i + j + block] = a[i + j] - tmp;
        a[i + j] += tmp;
      }
    }
  }

  template <typename T>
  std::vector<ModInt> dft(const std::vector<T> &a) {
    int n = a.size(), lg = 1;
    while ((1 << lg) < n) ++lg;
    std::vector<ModInt> A(1 << lg, 0);
    for (int i = 0; i < n; ++i) A[i] = a[i];
    sub_dft(A);
    return A;
  }

  void idft(std::vector<ModInt> &a) {
    int n = a.size();
    assert(__builtin_popcount(n) == 1);
    sub_dft(a);
    std::reverse(a.begin() + 1, a.end());
    ModInt inv_n = ModInt(1) / n;
    for (int i = 0; i < n; ++i) a[i] *= inv_n;
  }

  template <typename T>
  std::vector<ModInt> convolution(const std::vector<T> &a, const std::vector<T> &b) {
    int a_sz = a.size(), b_sz = b.size(), sz = a_sz + b_sz - 1, lg = 1;
    while ((1 << lg) < sz) ++lg;
    int n = 1 << lg;
    std::vector<ModInt> A(n, 0), B(n, 0);
    for (int i = 0; i < a_sz; ++i) A[i] = a[i];
    for (int i = 0; i < b_sz; ++i) B[i] = b[i];
    sub_dft(A);
    sub_dft(B);
    for (int i = 0; i < n; ++i) A[i] *= B[i];
    idft(A);
    A.resize(sz);
    return A;
  }

private:
  int primes[23][3]{
    {16957441, 329, 14},
    {17006593, 26, 15},
    {19529729, 770, 17},
    {167772161, 3, 25},
    {469762049, 3, 26},
    {645922817, 3, 23},
    {897581057, 3, 23},
    {924844033, 5, 21},
    {935329793, 3, 22},
    {943718401, 7, 22},
    {950009857, 7, 21},
    {962592769, 7, 21},
    {975175681, 17, 21},
    {976224257, 3, 20},
    {985661441, 3, 22},
    {998244353, 3, 23},
    {1004535809, 3, 21},
    {1007681537, 3, 20},
    {1012924417, 5, 21},
    {1045430273, 3, 20},
    {1051721729, 6, 20},
    {1053818881, 7, 20},
    {1224736769, 3, 24}
  };

  int n_max;
  ModInt root;
  std::vector<int> butterfly{0};
  std::vector<std::vector<ModInt>> omega{{1}};

  void calc(int n) {
    int prev_n = butterfly.size();
    if (n <= prev_n) return;
    assert(n <= n_max);
    butterfly.resize(n);
    int prev_lg = omega.size(), lg = __builtin_ctz(n);
    for (int i = 1; i < prev_n; ++i) butterfly[i] <<= lg - prev_lg;
    for (int i = prev_n; i < n; ++i) butterfly[i] = (butterfly[i >> 1] >> 1) | ((i & 1) << (lg - 1));
    omega.resize(lg);
    for (int i = prev_lg; i < lg; ++i) {
      omega[i].resize(1 << i);
      ModInt tmp = root.pow((ModInt::get_mod() - 1) / (1 << (i + 1)));
      for (int j = 0; j < (1 << (i - 1)); ++j) {
        omega[i][j << 1] = omega[i - 1][j];
        omega[i][(j << 1) + 1] = omega[i - 1][j] * tmp;
      }
    }
  }
};

template <typename T>
struct FPS {
  using MUL = std::function<std::vector<T>(const std::vector<T>&, const std::vector<T>&)>;
  using SQR = std::function<bool(const T&, T&)>;
  std::vector<T> co;
  FPS(int deg = 0) : co(deg + 1, 0) {}
  FPS(const std::vector<T> &co) : co(co) {}
  FPS(std::initializer_list<T> init) : co(init.begin(), init.end()) {}
  template <typename InputIter> FPS(InputIter first, InputIter last) : co(first, last) {}
  inline const T &operator[](int term) const { return co[term]; }
  inline T &operator[](int term) { return co[term]; }
  static void set_mul(MUL mul) { get_mul() = mul; }
  static void set_sqr(SQR sqr) { get_sqr() = sqr; }
  void resize(int deg) { co.resize(deg + 1, 0); }
  void shrink() { while (co.size() > 1 && co.back() == 0) co.pop_back(); }
  int degree() const { return static_cast<int>(co.size()) - 1; }
  FPS &operator=(const std::vector<T> &new_co) {
    co.resize(new_co.size());
    std::copy(new_co.begin(), new_co.end(), co.begin());
    return *this;
  }
  FPS &operator=(const FPS &x) {
    co.resize(x.co.size());
    std::copy(x.co.begin(), x.co.end(), co.begin());
    return *this;
  }
  FPS &operator+=(const FPS &x) {
    int n = x.co.size();
    if (n > co.size()) resize(n - 1);
    for (int i = 0; i < n; ++i) co[i] += x.co[i];
    return *this;
  }
  FPS &operator-=(const FPS &x) {
    int n = x.co.size();
    if (n > co.size()) resize(n - 1);
    for (int i = 0; i < n; ++i) co[i] -= x.co[i];
    return *this;
  }
  FPS &operator*=(T x) {
    for (T &e : co) e *= x;
    return *this;
  }
  FPS &operator*=(const FPS &x) { return *this = get_mul()(co, x.co); }
  FPS &operator/=(T x) {
    assert(x != 0);
    T inv_x = static_cast<T>(1) / x;
    for (T &e : co) e *= inv_x;
    return *this;
  }
  FPS &operator/=(const FPS &x) {
    int sz = x.co.size();
    if (sz > co.size()) return *this = FPS();
    int n = co.size() - sz + 1;
    FPS a(co.rbegin(), co.rbegin() + n), b(x.co.rbegin(), x.co.rbegin() + std::min(sz, n));
    b = b.inv(n - 1);
    a *= b;
    return *this = FPS(a.co.rend() - n, a.co.rend());
  }
  FPS &operator%=(const FPS &x) {
    *this -= *this / x * x;
    co.resize(static_cast<int>(x.co.size()) - 1);
    if (co.empty()) co = {0};
    return *this;
  }
  FPS &operator<<=(int n) {
    co.insert(co.begin(), n, 0);
    return *this;
  }
  FPS &operator>>=(int n) {
    if (co.size() < n) return *this = FPS();
    co.erase(co.begin(), co.begin() + n);
    return *this;
  }
  bool operator==(const FPS &x) const {
    FPS a(*this), b(x);
    a.shrink(); b.shrink();
    int n = a.co.size();
    if (n != b.co.size()) return false;
    for (int i = 0; i < n; ++i) if (a.co[i] != b.co[i]) return false;
    return true;
  }
  bool operator!=(const FPS &x) const { return !(*this == x); }
  FPS operator+() const { return *this; }
  FPS operator-() const {
    FPS res(*this);
    for (T &e : res.co) e = -e;
    return res;
  }
  FPS operator+(const FPS &x) const { return FPS(*this) += x; }
  FPS operator-(const FPS &x) const { return FPS(*this) -= x; }
  FPS operator*(T x) const { return FPS(*this) *= x; }
  FPS operator*(const FPS &x) const { return FPS(*this) *= x; }
  FPS operator/(T x) const { return FPS(*this) /= x; }
  FPS operator/(const FPS &x) const { return FPS(*this) /= x; }
  FPS operator%(const FPS &x) const { return FPS(*this) %= x; }
  FPS operator<<(int n) const { return FPS(*this) <<= n; }
  FPS operator>>(int n) const { return FPS(*this) >>= n; }
  T horner(T val) const {
    T res = 0;
    for (int i = static_cast<int>(co.size()) - 1; i >= 0; --i) (res *= val) += co[i];
    return res;
  }
  FPS differential() const {
    int n = co.size();
    assert(n >= 1);
    FPS res(n - 1);
    for (int i = 1; i < n; ++i) res.co[i - 1] = co[i] * i;
    return res;
  }
  FPS integral() const {
    int n = co.size();
    FPS res(n + 1);
    for (int i = 0; i < n; ++i) res[i + 1] = co[i] / (i + 1);
    return res;
  }
  FPS exp(int deg = -1) const {
    assert(co[0] == 0);
    int n = co.size();
    if (deg == -1) deg = n - 1;
    FPS one{1}, res = one;
    for (int i = 1; i <= deg; i <<= 1) {
      res *= FPS(co.begin(), co.begin() + std::min(n, i << 1)) - res.log((i << 1) - 1) + one;
      res.co.resize(i << 1);
    }
    res.co.resize(deg + 1);
    return res;
  }
  FPS inv(int deg = -1) const {
    assert(co[0] != 0);
    int n = co.size();
    if (deg == -1) deg = n - 1;
    FPS res{static_cast<T>(1) / co[0]};
    for (int i = 1; i <= deg; i <<= 1) {
      res = res + res - res * res * FPS(co.begin(), co.begin() + std::min(n, i << 1));
      res.co.resize(i << 1);
    }
    res.co.resize(deg + 1);
    return res;
  }
  FPS log(int deg = -1) const {
    assert(co[0] == 1);
    if (deg == -1) deg = static_cast<int>(co.size()) - 1;
    FPS integrand = differential() * inv(deg - 1);
    integrand.co.resize(deg);
    return integrand.integral();
  }
  FPS pow(long long exponent, int deg = -1) const {
    int n = co.size();
    if (deg == -1) deg = n - 1;
    for (int i = 0; i < n; ++i) {
      if (co[i] != 0) {
        long long shift = exponent * i;
        if (shift > deg) break;
        T tmp = 1, base = co[i];
        long long e = exponent;
        while (e > 0) {
          if (e & 1) tmp *= base;
          base *= base;
          e >>= 1;
        }
        return ((((*this >> i) * (static_cast<T>(1) / co[i])).log(deg - shift) * static_cast<T>(exponent)).exp(deg - shift) * tmp) << shift;
      }
    }
    return FPS(deg);
  }
  FPS mod_pow(long long exponent, const FPS &md) const {
    FPS inv_rev_md = FPS(md.co.rbegin(), md.co.rend()).inv();
    int deg_of_md = md.co.size();
    auto mod_mul = [&](FPS &multiplicand, const FPS &multiplier) -> void {
      multiplicand *= multiplier;
      if (deg_of_md <= multiplicand.co.size()) {
        int n = multiplicand.co.size() - deg_of_md + 1;
        FPS quotient = FPS(multiplicand.co.rbegin(), multiplicand.co.rbegin() + n) * FPS(inv_rev_md.co.begin(), inv_rev_md.co.begin() + std::min(static_cast<int>(inv_rev_md.co.size()), n));
        multiplicand -= FPS(quotient.co.rend() - n, quotient.co.rend()) * md;
      }
      multiplicand.co.resize(deg_of_md - 1);
      if (multiplicand.co.empty()) multiplicand.co = {0};
    };
    FPS res{1}, base = *this;
    mod_mul(base, res);
    while (exponent > 0) {
      if (exponent & 1) mod_mul(res, base);
      mod_mul(base, base);
      exponent >>= 1;
    }
    return res;
  }
  FPS sqrt(int deg = -1) const {
    int n = co.size();
    if (deg == -1) deg = n - 1;
    if (co[0] == 0) {
      for (int i = 1; i < n; ++i) {
        if (co[i] == 0) continue;
        if (i & 1) return FPS(-1);
        int shift = i >> 1;
        if (deg < shift) break;
        FPS res = (*this >> i).sqrt(deg - shift);
        if (res.co.empty()) return FPS(-1);
        res <<= shift;
        res.resize(deg);
        return res;
      }
      return FPS(deg);
    }
    T s;
    if (!get_sqr()(co[0], s)) return FPS(-1);
    FPS res{s};
    T half = static_cast<T>(1) / 2;
    for (int i = 1; i <= deg; i <<= 1) {
      (res += FPS(co.begin(), co.begin() + std::min(n, i << 1)) * res.inv((i << 1) - 1)) *= half;
    }
    res.resize(deg);
    return res;
  }
  FPS translate(T c) const {
    int n = co.size();
    std::vector<T> fact(n, 1), inv_fact(n, 1);
    for (int i = 1; i < n; ++i) fact[i] = fact[i - 1] * i;
    inv_fact[n - 1] = static_cast<T>(1) / fact[n - 1];
    for (int i = n - 1; i > 0; --i) inv_fact[i - 1] = inv_fact[i] * i;
    std::vector<T> g(n), ex(n);
    for (int i = 0; i < n; ++i) g[n - 1 - i] = co[i] * fact[i];
    T pow_c = 1;
    for (int i = 0; i < n; ++i) {
      ex[i] = pow_c * inv_fact[i];
      pow_c *= c;
    }
    std::vector<T> conv = get_mul()(g, ex);
    FPS res(n - 1);
    for (int i = 0; i < n; ++i) res[i] = conv[n - 1 - i] * inv_fact[i];
    return res;
  }
private:
  static MUL &get_mul() {
    static MUL mul = [](const std::vector<T> &a, const std::vector<T> &b) -> std::vector<T> {
      int n = a.size(), m = b.size();
      std::vector<T> res(n + m - 1, 0);
      for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) res[i + j] += a[i] * b[j];
      return res;
    };
    return mul;
  }
  static SQR &get_sqr() {
    static SQR sqr = [](const T &a, T &res) -> bool { return false; };
    return sqr;
  }
};

int main() {
  NTT ntt;
  FPS<ModInt>::set_mul([&](const vector<ModInt> &a, const vector<ModInt> &b) {
    return ntt.convolution(a, b);
  });
  int n, m, k; cin >> n >> m >> k;
  Combinatorics<MOD> com(n);
  FPS<ModInt> f(n);
  REP(i, n + 1) f[i] = (ModInt(2).pow(i + 1) - 1) * com.fact_inv[i + 1];
  f = f.pow(k);
  ModInt ans = 0;
  FOR(i, k, n + 1) {
    ans += com.nCk(m, k)  // 選択したボールに含まれる色を選ぶ．
           * com.nCk(n, i)  // ボールを i 個選ぶ（色で見たとき極大な集合）．
           * com.fact[i] * f[i - k]  // 選び方と分布の通り数
           * ModInt(m - k).pow(n - i);  // 残りのボールは m - k 種類の中から選ぶ．
  }
  cout << ans << '\n';
  return 0;
}