#define _USE_MATH_DEFINES #include 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 inline bool chmax(T &a, U b) { return a < b ? (a = b, true) : false; } template 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 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(val) * x.val % MOD; return *this; } MInt &operator/=(const MInt &x) { // assert(std::__gcd(static_cast(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 MInt abs(const MInt &x) { return x; } } template struct Combinatorics { using ModInt = MInt; int val; // "val!" and "mod" must be disjoint. std::vector 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; 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 &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 std::vector dft(const std::vector &a) { int n = a.size(), lg = 1; while ((1 << lg) < n) ++lg; std::vector A(1 << lg, 0); for (int i = 0; i < n; ++i) A[i] = a[i]; sub_dft(A); return A; } void idft(std::vector &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 std::vector convolution(const std::vector &a, const std::vector &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 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 butterfly{0}; std::vector> 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 struct FPS { using MUL = std::function(const std::vector&, const std::vector&)>; using SQR = std::function; std::vector co; FPS(int deg = 0) : co(deg + 1, 0) {} FPS(const std::vector &co) : co(co) {} FPS(std::initializer_list init) : co(init.begin(), init.end()) {} template 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(co.size()) - 1; } FPS &operator=(const std::vector &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(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(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(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(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(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(1) / co[i])).log(deg - shift) * static_cast(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(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(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 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(1) / fact[n - 1]; for (int i = n - 1; i > 0; --i) inv_fact[i - 1] = inv_fact[i] * i; std::vector 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 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 &a, const std::vector &b) -> std::vector { int n = a.size(), m = b.size(); std::vector 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::set_mul([&](const vector &a, const vector &b) { return ntt.convolution(a, b); }); int n, m, k; cin >> n >> m >> k; Combinatorics com(n); FPS 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; }