結果
| 問題 | No.2459 Stampaholic (Hard) |
| ユーザー |
 Aeren
|
| 提出日時 | 2023-09-02 07:19:12 |
| 言語 | C++23 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 749 ms / 4,000 ms |
| コード長 | 32,559 bytes |
| コンパイル時間 | 5,078 ms |
| コンパイル使用メモリ | 387,140 KB |
| 実行使用メモリ | 56,184 KB |
| 最終ジャッジ日時 | 2024-06-11 14:13:54 |
| 合計ジャッジ時間 | 13,106 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge3 |
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テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 2 ms
5,248 KB |
| testcase_01 | AC | 749 ms
55,592 KB |
| testcase_02 | AC | 88 ms
12,492 KB |
| testcase_03 | AC | 2 ms
5,376 KB |
| testcase_04 | AC | 2 ms
5,376 KB |
| testcase_05 | AC | 2 ms
5,376 KB |
| testcase_06 | AC | 2 ms
5,376 KB |
| testcase_07 | AC | 2 ms
5,376 KB |
| testcase_08 | AC | 336 ms
22,392 KB |
| testcase_09 | AC | 92 ms
14,012 KB |
| testcase_10 | AC | 717 ms
47,356 KB |
| testcase_11 | AC | 192 ms
26,344 KB |
| testcase_12 | AC | 741 ms
56,184 KB |
| testcase_13 | AC | 718 ms
47,736 KB |
| testcase_14 | AC | 176 ms
17,004 KB |
| testcase_15 | AC | 745 ms
55,420 KB |
| testcase_16 | AC | 741 ms
55,464 KB |
| testcase_17 | AC | 391 ms
51,624 KB |
| testcase_18 | AC | 395 ms
51,592 KB |
| testcase_19 | AC | 397 ms
51,636 KB |
| testcase_20 | AC | 1 ms
5,376 KB |
| testcase_21 | AC | 693 ms
42,556 KB |
ソースコード
#include <bits/stdc++.h>
#include <x86intrin.h>
using namespace std;
using namespace numbers;
using uint = unsigned int;
template<uint _mod>
struct modular_fixed_base{
static constexpr uint mod(){
return _mod;
}
template<class T>
static vector<modular_fixed_base> precalc_power(T base, int SZ){
vector<modular_fixed_base> res(SZ + 1, 1);
for(auto i = 1; i <= SZ; ++ i) res[i] = res[i - 1] * base;
return res;
}
static vector<modular_fixed_base> _INV;
static void precalc_inverse(int SZ){
if(_INV.empty()) _INV.assign(2, 1);
for(auto x = _INV.size(); x <= SZ; ++ x) _INV.push_back(_mod / x * -_INV[_mod % x]);
}
// _mod must be a prime
static modular_fixed_base _primitive_root;
static modular_fixed_base primitive_root(){
if(_primitive_root) return _primitive_root;
if(_mod == 2) return _primitive_root = 1;
if(_mod == 998244353) return _primitive_root = 3;
uint divs[20] = {};
divs[0] = 2;
int cnt = 1;
uint x = (_mod - 1) / 2;
while(x % 2 == 0) x /= 2;
for(auto i = 3; 1LL * i * i <= x; i += 2){
if(x % i == 0){
divs[cnt ++] = i;
while(x % i == 0) x /= i;
}
}
if(x > 1) divs[cnt ++] = x;
for(auto g = 2; ; ++ g){
bool ok = true;
for(auto i = 0; i < cnt; ++ i){
if((modular_fixed_base(g).power((_mod - 1) / divs[i])) == 1){
ok = false;
break;
}
}
if(ok) return _primitive_root = g;
}
}
constexpr modular_fixed_base(): data(){ }
modular_fixed_base(const double &x){ data = normalize(llround(x)); }
modular_fixed_base(const long double &x){ data = normalize(llround(x)); }
template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base(const T &x){ data = normalize(x); }
template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> static uint normalize(const T &x){
int sign = x >= 0 ? 1 : -1;
uint v = _mod <= sign * x ? sign * x % _mod : sign * x;
if(sign == -1 && v) v = _mod - v;
return v;
}
const uint &operator()() const{ return data; }
template<class T> operator T() const{ return data; }
modular_fixed_base &operator+=(const modular_fixed_base &otr){ if((data += otr.data) >= _mod) data -= _mod; return *this; }
modular_fixed_base &operator-=(const modular_fixed_base &otr){ if((data += _mod - otr.data) >= _mod) data -= _mod; return *this; }
template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base &operator+=(const T &otr){ return *this += modular_fixed_base(otr); }
template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base &operator-=(const T &otr){ return *this -= modular_fixed_base(otr); }
modular_fixed_base &operator++(){ return *this += 1; }
modular_fixed_base &operator--(){ return *this += _mod - 1; }
modular_fixed_base operator++(int){ modular_fixed_base result(*this); *this += 1; return result; }
modular_fixed_base operator--(int){ modular_fixed_base result(*this); *this += _mod - 1; return result; }
modular_fixed_base operator-() const{ return modular_fixed_base(_mod - data); }
modular_fixed_base &operator*=(const modular_fixed_base &rhs){
data = (unsigned long long)data * rhs.data % _mod;
return *this;
}
template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr>
modular_fixed_base &inplace_power(T e){
if(!data) return *this = {};
if(data == 1) return *this;
if(data == mod() - 1) return e % 2 ? *this : *this = -*this;
if(e < 0) *this = 1 / *this, e = -e;
modular_fixed_base res = 1;
for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this;
return *this = res;
}
template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr>
modular_fixed_base power(T e) const{
return modular_fixed_base(*this).inplace_power(e);
}
modular_fixed_base &operator/=(const modular_fixed_base &otr){
int a = otr.data, m = _mod, u = 0, v = 1;
if(a < _INV.size()) return *this *= _INV[a];
while(a){
int t = m / a;
m -= t * a; swap(a, m);
u -= t * v; swap(u, v);
}
assert(m == 1);
return *this *= u;
}
uint data;
};
template<uint _mod> vector<modular_fixed_base<_mod>> modular_fixed_base<_mod>::_INV;
template<uint _mod> modular_fixed_base<_mod> modular_fixed_base<_mod>::_primitive_root;
template<uint _mod> bool operator==(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data == rhs.data; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator==(const modular_fixed_base<_mod> &lhs, T rhs){ return lhs == modular_fixed_base<_mod>(rhs); }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator==(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) == rhs; }
template<uint _mod> bool operator!=(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return !(lhs == rhs); }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator!=(const modular_fixed_base<_mod> &lhs, T rhs){ return !(lhs == rhs); }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator!=(T lhs, const modular_fixed_base<_mod> &rhs){ return !(lhs == rhs); }
template<uint _mod> bool operator<(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data < rhs.data; }
template<uint _mod> bool operator>(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data > rhs.data; }
template<uint _mod> bool operator<=(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data <= rhs.data; }
template<uint _mod> bool operator>=(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data >= rhs.data; }
template<uint _mod> modular_fixed_base<_mod> operator+(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) += rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator+(const modular_fixed_base<_mod> &lhs, T rhs){ return modular_fixed_base<_mod>(lhs) += rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator+(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) += rhs; }
template<uint _mod> modular_fixed_base<_mod> operator-(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) -= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator-(const modular_fixed_base<_mod> &lhs, T rhs){ return modular_fixed_base<_mod>(lhs) -= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator-(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) -= rhs; }
template<uint _mod> modular_fixed_base<_mod> operator*(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) *= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator*(const modular_fixed_base<_mod> &lhs, T rhs){ return modular_fixed_base<_mod>(lhs) *= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator*(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) *= rhs; }
template<uint _mod> modular_fixed_base<_mod> operator/(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs) { return modular_fixed_base<_mod>(lhs) /= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator/(const modular_fixed_base<_mod> &lhs, T rhs) { return modular_fixed_base<_mod>(lhs) /= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator/(T lhs, const modular_fixed_base<_mod> &rhs) { return modular_fixed_base<_mod>(lhs) /= rhs; }
template<uint _mod> istream &operator>>(istream &in, modular_fixed_base<_mod> &number){
long long x;
in >> x;
number.data = modular_fixed_base<_mod>::normalize(x);
return in;
}
#define _PRINT_AS_FRACTION
template<uint _mod> ostream &operator<<(ostream &out, const modular_fixed_base<_mod> &number){
#if defined(LOCAL) && defined(_PRINT_AS_FRACTION)
out << number();
cerr << "(";
for(auto d = 1; ; ++ d){
if((number * d).data <= 1000000){
cerr << (number * d).data << "/" << d;
break;
}
else if((-number * d).data <= 1000000){
cerr << "-" << (-number * d).data << "/" << d;
break;
}
}
cerr << ")";
return out;
#else
return out << number();
#endif
}
#undef _PRINT_AS_FRACTION
// const uint mod = 1e9 + 7; // 1000000007
const uint mod = (119 << 23) + 1; // 998244353
// const uint mod = 1e9 + 9; // 1000000009
using modular = modular_fixed_base<mod>;
template<class T>
struct combinatorics{
// O(n)
static vector<T> precalc_fact(int n){
vector<T> f(n + 1, 1);
for(auto i = 1; i <= n; ++ i) f[i] = f[i - 1] * i;
return f;
}
// O(n * m)
static vector<vector<T>> precalc_C(int n, int m){
vector<vector<T>> c(n + 1, vector<T>(m + 1));
for(auto i = 0; i <= n; ++ i) for(auto j = 0; j <= min(i, m); ++ j) c[i][j] = i && j ? c[i - 1][j - 1] + c[i - 1][j] : T(1);
return c;
}
int SZ = 0;
vector<T> inv, fact, invfact;
combinatorics(){ }
// O(SZ)
combinatorics(int SZ): SZ(SZ), inv(SZ + 1, 1), fact(SZ + 1, 1), invfact(SZ + 1, 1){
for(auto i = 1; i <= SZ; ++ i) fact[i] = fact[i - 1] * i;
invfact[SZ] = 1 / fact[SZ];
for(auto i = SZ - 1; i >= 0; -- i){
invfact[i] = invfact[i + 1] * (i + 1);
inv[i + 1] = invfact[i + 1] * fact[i];
}
}
// O(1)
T C(int n, int k) const{
return n < 0 ? C(-n + k - 1, k) * (k & 1 ? -1 : 1) : n < k || k < 0 ? T() : fact[n] * invfact[k] * invfact[n - k];
}
// O(1)
T P(int n, int k) const{
return n < k ? T() : fact[n] * invfact[n - k];
}
// O(1)
T H(int n, int k) const{
return C(n + k - 1, k);
}
// O(min(k, n - k))
T naive_C(long long n, long long k) const{
if(n < k) return 0;
T res = 1;
k = min(k, n - k);
for(auto i = n; i > n - k; -- i) res *= i;
return res * invfact[k];
}
// O(k)
T naive_P(long long n, int k) const{
if(n < k) return 0;
T res = 1;
for(auto i = n; i > n - k; -- i) res *= i;
return res;
}
// O(k)
T naive_H(long long n, int k) const{
return naive_C(n + k - 1, k);
}
// O(1)
bool parity_C(long long n, long long k) const{
return n < k ? false : (n & k) == k;
}
// Number of ways to place n '('s and k ')'s starting with m copies of '(' such that in each prefix, number of '(' is equal or greater than ')'
// Catalan(n, n, 0): n-th catalan number
// Catalan(s, s+k-1, k-1): sum of products of k catalan numbers where the index of product sums up to s.
// O(1)
T Catalan(int n, int k, int m = 0) const{
assert(0 <= min({n, k, m}));
return k <= m ? C(n + k, n) : k <= n + m ? C(n + k, n) - C(n + k, k - m - 1) : T();
}
};
// T must be of modular type
// mod must be a prime
// Requires modular
template<class T>
struct number_theoric_transform_wrapper{
// i \in [2^k, 2^{k+1}) holds w_{2^k+1}^{i-2^k}
static vector<T> root, buffer1, buffer2;
static void adjust_root(int n){
if(root.empty()) root = {1, 1};
for(auto k = (int)root.size(); k < n; k <<= 1){
root.resize(n, 1);
T w = T::primitive_root().power((T::mod() - 1) / (k << 1));
for(auto i = k; i < k << 1; ++ i) root[i] = i & 1 ? root[i >> 1] * w : root[i >> 1];
}
}
// n must be a power of two
// p must have next n memories allocated
// O(n * log(n))
static void transform(int n, T *p, bool invert = false){
assert(n && __builtin_popcount(n) == 1 && (T::mod() - 1) % n == 0);
for(auto i = 1, j = 0; i < n; ++ i){
int bit = n >> 1;
for(; j & bit; bit >>= 1) j ^= bit;
j ^= bit;
if(i < j) swap(p[i], p[j]);
}
adjust_root(n);
for(auto len = 1; len < n; len <<= 1) for(auto i = 0; i < n; i += len << 1) for(auto j = 0; j < len; ++ j){
T x = p[i + j], y = p[len + i + j] * root[len + j];
p[i + j] = x + y, p[len + i + j] = x - y;
}
if(invert){
reverse(p + 1, p + n);
T inv_n = T(1) / n;
for(auto i = 0; i < n; ++ i) p[i] *= inv_n;
}
}
static void transform(vector<T> &p, bool invert = false){
transform((int)p.size(), p.data(), invert);
}
// Double the length of the ntt array
// n must be a power of two
// p must have next 2n memories allocated
// O(n * log(n))
static void double_up(int n, T *p){
assert(n && __builtin_popcount(n) == 1 && (T().mod() - 1) % (n << 1) == 0);
buffer1.resize(n << 1);
for(auto i = 0; i < n; ++ i) buffer1[i << 1] = p[i];
transform(n, p, true);
adjust_root(n << 1);
for(auto i = 0; i < n; ++ i) p[i] *= root[n | i];
transform(n, p);
for(auto i = 0; i < n; ++ i) buffer1[i << 1 | 1] = p[i];
copy(buffer1.begin(), buffer1.begin() + 2 * n, p);
}
static void double_up(vector<T> &p){
int n = (int)p.size();
p.resize(n << 1);
double_up(n, p.data());
}
// O(n * m)
static vector<T> convolute_naive(const vector<T> &p, const vector<T> &q){
vector<T> res(max((int)p.size() + (int)q.size() - 1, 0));
for(auto i = 0; i < (int)p.size(); ++ i) for(auto j = 0; j < (int)q.size(); ++ j) res[i + j] += p[i] * q[j];
return res;
}
// O((n + m) * log(n + m))
static vector<T> convolute(const vector<T> &p, const vector<T> &q){
if(min(p.size(), q.size()) < 55) return convolute_naive(p, q);
int m = (int)p.size() + (int)q.size() - 1, n = 1 << __lg(m) + 1;
buffer1.assign(n, 0);
copy(p.begin(), p.end(), buffer1.begin());
transform(buffer1);
buffer2.assign(n, 0);
copy(q.begin(), q.end(), buffer2.begin());
transform(buffer2);
for(auto i = 0; i < n; ++ i) buffer1[i] *= buffer2[i];
transform(buffer1, true);
return vector<T>(buffer1.begin(), buffer1.begin() + m);
}
// O((n + m) * log(n + m))
static vector<T> arbitrarily_convolute(const vector<T> &p, const vector<T> &q){
using modular0 = modular_fixed_base<1045430273>;
using modular1 = modular_fixed_base<1051721729>;
using modular2 = modular_fixed_base<1053818881>;
using ntt0 = number_theoric_transform_wrapper<modular0>;
using ntt1 = number_theoric_transform_wrapper<modular1>;
using ntt2 = number_theoric_transform_wrapper<modular2>;
vector<modular0> p0((int)p.size()), q0((int)q.size());
for(auto i = 0; i < (int)p.size(); ++ i) p0[i] = p[i].data;
for(auto i = 0; i < (int)q.size(); ++ i) q0[i] = q[i].data;
auto xy0 = ntt0::convolute(p0, q0);
vector<modular1> p1((int)p.size()), q1((int)q.size());
for(auto i = 0; i < (int)p.size(); ++ i) p1[i] = p[i].data;
for(auto i = 0; i < (int)q.size(); ++ i) q1[i] = q[i].data;
auto xy1 = ntt1::convolute(p1, q1);
vector<modular2> p2((int)p.size()), q2((int)q.size());
for(auto i = 0; i < (int)p.size(); ++ i) p2[i] = p[i].data;
for(auto i = 0; i < (int)q.size(); ++ i) q2[i] = q[i].data;
auto xy2 = ntt2::convolute(p2, q2);
static const modular1 r01 = 1 / modular1(modular0::mod());
static const modular2 r02 = 1 / modular2(modular0::mod());
static const modular2 r12 = 1 / modular2(modular1::mod());
static const modular2 r02r12 = r02 * r12;
static const T w1 = modular0::mod();
static const T w2 = w1 * modular1::mod();
int n = (int)p.size() + (int)q.size() - 1;
vector<T> res(n);
for(auto i = 0; i < n; ++ i){
using ull = unsigned long long;
ull a = xy0[i].data;
ull b = (xy1[i].data + modular1::mod() - a) * r01.data % modular1::mod();
ull c = ((xy2[i].data + modular2::mod() - a) * r02r12.data + (modular2::mod() - b) * r12.data) % modular2::mod();
res[i] = xy0[i].data + w1 * b + w2 * c;
}
return res;
}
};
template<class T> vector<T> number_theoric_transform_wrapper<T>::root;
template<class T> vector<T> number_theoric_transform_wrapper<T>::buffer1;
template<class T> vector<T> number_theoric_transform_wrapper<T>::buffer2;
using ntt = number_theoric_transform_wrapper<modular>;
// T must support *=, *, +=, -=, and -
// T{} should return the additive identity
// Requires modular and number_theoric_transform
template<class T, class FFT>
struct power_series_base: vector<T>{
#define data (*this)
template<class ...Args>
power_series_base(Args... args): vector<T>(args...){}
power_series_base(initializer_list<T> init): vector<T>(init){}
int degree() const{
return data.empty() ? numeric_limits<int>::max() : data.size() - 1;
}
// Returns \sum_{i=0}^{n-1} a_i/i! * X^i
static power_series_base EGF(vector<T> a){
int n = (int)a.size();
T fact = 1;
for(auto x = 2; x < n; ++ x) fact *= x;
fact = 1 / fact;
for(auto i = n - 1; i >= 0; -- i) a[i] *= fact, fact *= i;
return power_series_base(a);
}
// Returns exp(coef * X).prefix(n) = \sum_{i=0}^{n-1} coef^i/i! * X^i
static power_series_base EGF(int n, T coef = 1){
vector<T> a(n, 1);
for(auto i = 1; i < n; ++ i) a[i] = a[i - 1] * coef;
return EGF(a);
}
vector<T> EGF_to_seq() const{
int n = (int)data.size();
vector<T> seq(n);
T fact = 1;
for(auto i = 0; i < n; ++ i){
seq[i] = data[i] * fact;
fact *= i + 1;
}
return seq;
}
power_series_base &reduce(){
while(!data.empty() && !data.back()) data.pop_back();
return *this;
}
power_series_base reduced() const{
return power_series_base(*this).reduce();
}
friend ostream &operator<<(ostream &out, const power_series_base &p){
if(p.empty()) return out << "{}";
else{
out << "{";
for(auto i = 0; i < (int)p.size(); ++ i){
out << p[i];
i + 1 < (int)p.size() ? out << ", " : out << "}";
}
return out;
}
}
power_series_base &inplace_take(int n){
data.erase(data.begin() + min((int)data.size(), n), data.end());
data.resize(n, T{});
return *this;
}
power_series_base take(int n) const{
return power_series_base(*this).inplace_take(n);
}
power_series_base &inplace_drop(int n){
data.erase(data.begin(), data.begin() + min((int)data.size(), n));
return *this;
}
power_series_base drop(int n) const{
return power_series_base(*this).inplace_drop(n);
}
power_series_base &inplace_reverse(int n){
data.resize(max(n, (int)data.size()));
std::reverse(data.begin(), data.begin() + n);
return *this;
}
power_series_base reverse(int n) const{
return power_series_base(*this).inplace_reverse(n);
}
T evaluate(T x) const{
T res = {};
for(auto i = (int)data.size() - 1; i >= 0; -- i) res = res * x + data[i];
return res;
}
power_series_base operator*(const power_series_base &p) const{
return FFT::convolute(data, p);
}
power_series_base &operator*=(const power_series_base &p){
return *this = *this * p;
}
template<class U>
power_series_base &operator*=(U x){
for(auto &c: data) c *= x;
return *this;
}
template<class U>
power_series_base operator*(U x) const{
return power_series_base(*this) *= x;
}
template<class U>
friend power_series_base operator*(U x, power_series_base p){
for(auto &c: p) c = x * c;
return p;
}
power_series_base &operator+=(const power_series_base &p){
data.resize(max(data.size(), p.size()));
for(auto i = 0; i < (int)p.size(); ++ i) data[i] += p[i];
return *this;
}
power_series_base operator+(const power_series_base &p) const{
return power_series_base(*this) += p;
}
template<class U>
power_series_base &operator+=(const U &x){
if(data.empty()) data.emplace_back();
data[0] += x;
return *this;
}
template<class U>
power_series_base operator+(const U &x) const{
return power_series_base(*this) += x;
}
template<class U>
friend power_series_base operator+(const U &x, const power_series_base &p){
return p + x;
}
power_series_base &operator-=(const power_series_base &p){
data.resize(max(data.size(), p.size()));
for(auto i = 0; i < (int)p.size(); ++ i) data[i] -= p[i];
return *this;
}
power_series_base operator-(const power_series_base &p) const{
return power_series_base(*this) -= p;
}
template<class U>
power_series_base &operator-=(const U &x){
if(data.empty()) data.emplace_back();
data[0] -= x;
return *this;
}
template<class U>
power_series_base operator-(const U &x) const{
return power_series_base(*this) -= x;
}
template<class U>
friend power_series_base operator-(const U &x, const power_series_base &p){
return -p + x;
}
power_series_base operator-() const{
power_series_base res = *this;
for(auto i = 0; i < data.size(); ++ i) res[i] = T{} - res[i];
return res;
}
power_series_base &operator++(){
if(data.empty()) data.push_back(1);
else ++ data[0];
return *this;
}
power_series_base &operator--(){
if(data.empty()) data.push_back(-1);
else -- data[0];
return *this;
}
power_series_base operator++(int){
power_series_base result(*this);
if(data.empty()) data.push_back(1);
else ++ data[0];
return result;
}
power_series_base operator--(int){
power_series_base result(*this);
if(data.empty()) data.push_back(-1);
else -- data[0];
return result;
}
power_series_base &clear_range(int l, int r){
assert(0 <= l && l <= r && r <= data.size());
for(auto i = l; i < r; ++ i) data[i] = 0;
return *this;
}
power_series_base &transform(bool invert = false){
FFT::transform(data, invert);
return *this;
}
power_series_base transformed(bool invert = false) const{
auto res = *this;
res.transform(invert);
return res;
}
power_series_base &inplace_dot_product(const power_series_base &p){
for(auto i = 0; i < min(data.size(), p.size()); ++ i) data[i] *= p[i];
return *this;
}
power_series_base dot_product(const power_series_base &p) const{
return power_series_base(*this).inplace_power_series_product(p);
}
power_series_base &inverse_doubled_up(power_series_base &f, const power_series_base &freq) const{
assert((f.size() & -f.size()) == f.size());
int s = f.size();
power_series_base buffer = take(s << 1);
buffer.transform();
buffer.inplace_dot_product(freq);
buffer.transform(true);
buffer.clear_range(0, s);
buffer.transform();
buffer.inplace_dot_product(freq);
buffer.transform(true);
f.resize(s << 1);
return f -= buffer.clear_range(0, s);
}
power_series_base &inverse_doubled_up(power_series_base &f) const{
assert((f.size() & -f.size()) == f.size());
return inverse_doubled_up(f, f.take(f.size() << 1).transformed());
}
// Returns the first n terms of the inverse series
// O(n * log(n))
power_series_base inverse(int n) const{
assert(!data.empty() && data[0]);
auto inv = 1 / data[0];
power_series_base res{inv};
for(auto s = 1; s < n; s <<= 1) inverse_doubled_up(res);
res.resize(n);
return res;
}
// Returns the first n terms of the inverse series
// O(n * log(n))
power_series_base &inplace_inverse(int n){
return *this = this->inverse(n);
}
// O(n * log(n))
power_series_base &inplace_power_series_division(power_series_base p, int n){
int i = 0;
while(i < min(data.size(), p.size()) && !data[i] && !p[i]) ++ i;
data.erase(data.begin(), data.begin() + i);
p.erase(p.begin(), p.begin() + i);
(*this *= p.inverse(n)).resize(n);
return *this;
}
// O(n * log(n))
power_series_base power_series_division(const power_series_base &p, int n){
return power_series_base(*this).inplace_power_series_division(p, n);
}
// Euclidean division
// O(n * log(n))
power_series_base &operator/=(const power_series_base &p){
assert(!p.empty() && p.back() != T{0});
reduce();
if(data.size() < p.size()){
data.clear();
return *this;
}
if((int)p.size() - count(p.begin(), p.end(), 0) <= 100){
T inv = 1 / p.back();
static vector<int> indices;
for(auto i = 0; i < (int)p.size() - 1; ++ i) if(p[i]) indices.push_back(i);
power_series_base res((int)data.size() - (int)p.size() + 1);
for(auto i = (int)data.size() - 1; i >= (int)p.size() - 1; -- i) if(data[i]){
T x = data[i] * inv;
res[i - (int)p.size() + 1] = x;
for(auto j: indices) data[i - ((int)p.size() - 1 - j)] -= x * p[j];
}
indices.clear();
return *this = res;
}
power_series_base b;
int n = data.size() - p.size() + 1;
b.assign(n, {});
copy(p.rbegin(), p.rbegin() + min(p.size(), b.size()), b.begin());
std::reverse(data.begin(), data.end());
data = FFT::convolute(data, b.inverse(n));
data.erase(data.begin() + n, data.end());
std::reverse(data.begin(), data.end());
return *this;
}
power_series_base operator/(const power_series_base &p) const{
return power_series_base(*this) /= p;
}
template<class U>
power_series_base &operator/=(U x){
assert(x);
T inv_x = T(1) / x;
for(auto &c: data) c *= inv_x;
return *this;
}
template<class U>
power_series_base operator/(U x) const{
return power_series_base(*this) /= x;
}
pair<power_series_base, power_series_base> divrem(const power_series_base &p) const{
auto q = *this / p, r = *this - q * p;
while(!r.empty() && r.back() == 0) r.pop_back();
return {q, r};
}
power_series_base &operator%=(const power_series_base &p){
assert(!p.empty() && p.back() != T{0});
reduce();
if(data.size() < p.size()) return *this;
if((int)p.size() - count(p.begin(), p.end(), 0) <= 100){
T inv = 1 / p.back();
static vector<int> indices;
for(auto i = 0; i < (int)p.size() - 1; ++ i) if(p[i]) indices.push_back(i);
for(auto i = (int)data.size() - 1; i >= (int)p.size() - 1; -- i) if(data[i]){
T x = data[i] * inv;
data[i] = 0;
for(auto j: indices) data[i - ((int)p.size() - 1 - j)] -= x * p[j];
}
indices.clear();
return reduce();
}
return *this = this->divrem(p).second;
}
power_series_base operator%(const power_series_base &p) const{
return power_series_base(*this) %= p;
}
power_series_base &inplace_derivative(){
if(!data.empty()){
for(auto i = 0; i < data.size(); ++ i) data[i] *= i;
data.erase(data.begin());
}
return *this;
}
// p'
power_series_base derivative() const{
return power_series_base(*this).inplace_derivative();
}
power_series_base &inplace_derivative_shift(){
for(auto i = 0; i < data.size(); ++ i) data[i] *= i;
return *this;
}
// xP'
power_series_base derivative_shift() const{
return power_series_base(*this).inplace_derivative_shift();
}
power_series_base &inplace_antiderivative(){
T::precalc_inverse(data.size());
data.push_back(0);
for(auto i = (int)data.size() - 1; i >= 1; -- i) data[i] = data[i - 1] / i;
data[0] = 0;
return *this;
}
// Integral(P)
power_series_base antiderivative() const{
return power_series_base(*this).inplace_antiderivative();
}
power_series_base &inplace_shifted_antiderivative(){
T::precalc_inverse(data.size());
if(!data.empty()) data[0] = 0;
for(auto i = 1; i < data.size(); ++ i) data[i] /= i;
return *this;
}
// Integral(P/x)
power_series_base shifted_antiderivative() const{
return power_series_base(*this).inplace_shifted_antiderivative();
}
// O(n * log(n))
power_series_base &inplace_log(int n){
assert(!data.empty() && data[0] == 1);
if(!n){
data.clear();
return *this;
}
(*this = derivative() * inverse(n)).resize(n - 1);
inplace_antiderivative();
return *this;
}
// O(n * log(n))
power_series_base log(int n) const{
return power_series_base(*this).inplace_log(n);
}
// O(n * log(n))
power_series_base exp(int n) const{
assert(data.empty() || data[0] == 0);
power_series_base f{1}, g{1};
for(auto s = 1; s < n; s <<= 1){
power_series_base f2 = f.take(s << 1).transform();
power_series_base g2 = g.take(s << 1).transform();
power_series_base dt = take(s).inplace_derivative_shift();
power_series_base w = dt;
w.transform();
for(auto i = 0; i < s; ++ i) w[i] *= f2[i << 1];
w.transform(true);
w -= f.derivative_shift();
w.resize(s << 1);
w.transform();
w.inplace_dot_product(g2);
w.transform(true);
w.resize(s);
w.insert(w.begin(), s, 0);
w -= dt;
power_series_base z = take(s << 1);
z += w.inplace_shifted_antiderivative();
z.transform();
z.inplace_dot_product(f2);
z.transform(true);
f.resize(s << 1);
f += z.clear_range(0, s);
if(s << 1 < n) f.inverse_doubled_up(g, g2);
}
f.resize(n);
return f;
}
// O(n * log(n))
power_series_base &inplace_exp(int n){
return *this = this->exp(n);
}
// O(n * log(n))
template<class U>
power_series_base &inplace_power(U e, int n){
data.resize(n);
if(e == 0 || n == 0){
if(n) data[0] = 1;
return *this;
}
if(e < 0) return inplace_inverse(n).inplace_power(-e, n);
if(all_of(data.begin(), data.end(), [&](auto x){ return x == 0; })) return *this;
int pivot = find_if(data.begin(), data.end(), [&](auto x){ return x; }) - data.begin();
if(pivot && e >= (n + pivot - 1) / pivot){
fill(data.begin(), data.end(), 0);
return *this;
}
data.erase(data.begin(), data.begin() + pivot);
n -= pivot * e;
T pivot_c = data[0].power(e);
((*this /= data[0]).inplace_log(n) *= e).inplace_exp(n);
data.insert(data.begin(), pivot * e, 0);
return *this *= pivot_c;
}
// O(n * log(n))
template<class U>
power_series_base power(U e, int n) const{
return power_series_base(*this).inplace_power(e, n);
}
static power_series_base multiply_all(const vector<power_series_base> &a){
if(a.empty()) return {1};
auto solve = [&](auto self, int l, int r)->power_series_base{
if(r - l == 1) return a[l];
int m = l + (r - l >> 1);
return self(self, l, m) * self(self, m, r);
};
return solve(solve, 0, (int)a.size());
}
#undef data
};
using power_series = power_series_base<modular, ntt>;
int main(){
cin.tie(0)->sync_with_stdio(0);
cin.exceptions(ios::badbit | ios::failbit);
int nr, nc, n, k;
cin >> nr >> nc >> n >> k;
combinatorics<modular> C(n);
if(nr > nc){
swap(nr, nc);
}
auto inv = modular(1) / (nr - (k - 1)) / (nc - (k - 1));
auto invpower = modular::precalc_power(inv, n);
modular res = 0;
auto get_sum = [&](int k)->vector<modular>{
return ((1 - power_series::EGF(n + 2, k + 1)).drop(1) * (1 - power_series::EGF(n + 2)).drop(1).inverse(n + 1)).EGF_to_seq();
};
if(2 * k <= nr){
auto power_k = modular::precalc_power(k, n);
auto sum_k = get_sum(k);
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_k[i] * sum_k[i] * 4;
}
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_k[i] * power_k[i] * 2 * (nr + nc - 4 * k);
}
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * power_k[i] * power_k[i] * (nr - 2 * k) * (nc - 2 * k);
}
}
else if(2 * k <= nc){
auto power_k = modular::precalc_power(k, n);
auto sum_k = get_sum(k);
auto power_r = modular::precalc_power(nr - k + 1, n);
auto sum_r = get_sum(nr - k);
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_k[i] * sum_r[i] * 4;
}
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_k[i] * power_r[i] * (2 * k - nr) * 2;
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_r[i] * power_k[i] * (nc - 2 * k) * 2;
}
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * power_r[i] * power_k[i] * (2 * k - nr) * (nc - 2 * k);
}
}
else{
auto power_r = modular::precalc_power(nr - k + 1, n);
auto sum_r = get_sum(nr - k);
auto power_c = modular::precalc_power(nc - k + 1, n);
auto sum_c = get_sum(nc - k);
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_r[i] * sum_c[i] * 4;
}
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_c[i] * power_r[i] * (2 * k - nr) * 2;
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * sum_r[i] * power_c[i] * (2 * k - nc) * 2;
}
for(auto i = 1; i <= n; ++ i){
res += C.C(n, i) * invpower[i] * modular(-1).power(i - 1) * power_r[i] * power_c[i] * (2 * k - nr) * (2 * k - nc);
}
}
cout << res << "\n";
return 0;
}
/*
*/
////////////////////////////////////////////////////////////////////////////////////////
// //
// Coded by Aeren //
// //
////////////////////////////////////////////////////////////////////////////////////////