結果
| 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 |
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| 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;}