結果
| 問題 | No.2459 Stampaholic (Hard) |
| ユーザー |
 tokusakurai
|
| 提出日時 | 2023-09-02 00:18:19 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 2,949 ms / 4,000 ms |
| コード長 | 24,949 bytes |
| コンパイル時間 | 2,618 ms |
| コンパイル使用メモリ | 221,064 KB |
| 実行使用メモリ | 38,844 KB |
| 最終ジャッジ日時 | 2024-06-11 06:42:45 |
| 合計ジャッジ時間 | 34,262 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge4 |
（要ログイン）
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 | 16 ms
11,008 KB |
| testcase_01 | AC | 2,684 ms
38,728 KB |
| testcase_02 | AC | 555 ms
16,436 KB |
| testcase_03 | AC | 17 ms
11,136 KB |
| testcase_04 | AC | 16 ms
11,008 KB |
| testcase_05 | AC | 17 ms
11,088 KB |
| testcase_06 | AC | 16 ms
11,136 KB |
| testcase_07 | AC | 16 ms
11,008 KB |
| testcase_08 | AC | 1,147 ms
21,372 KB |
| testcase_09 | AC | 592 ms
16,952 KB |
| testcase_10 | AC | 2,491 ms
35,256 KB |
| testcase_11 | AC | 1,232 ms
24,344 KB |
| testcase_12 | AC | 2,697 ms
38,128 KB |
| testcase_13 | AC | 2,486 ms
35,004 KB |
| testcase_14 | AC | 621 ms
18,104 KB |
| testcase_15 | AC | 2,674 ms
38,824 KB |
| testcase_16 | AC | 2,949 ms
38,844 KB |
| testcase_17 | AC | 2,805 ms
38,836 KB |
| testcase_18 | AC | 2,636 ms
38,688 KB |
| testcase_19 | AC | 2,680 ms
38,824 KB |
| testcase_20 | AC | 16 ms
11,060 KB |
| testcase_21 | AC | 2,344 ms
32,184 KB |
ソースコード
#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for (int i = 0; i < (n); i++)
#define per(i, n) for (int i = (n)-1; i >= 0; i--)
#define rep2(i, l, r) for (int i = (l); i < (r); i++)
#define per2(i, l, r) for (int i = (r)-1; i >= (l); i--)
#define each(e, v) for (auto &e : v)
#define MM << " " <<
#define pb push_back
#define eb emplace_back
#define all(x) begin(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define sz(x) (int)x.size()
using ll = long long;
using pii = pair<int, int>;
using pil = pair<int, ll>;
using pli = pair<ll, int>;
using pll = pair<ll, ll>;
template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using maxheap = priority_queue<T>;
template <typename T>
bool chmax(T &x, const T &y) {
return (x < y) ? (x = y, true) : false;
}
template <typename T>
bool chmin(T &x, const T &y) {
return (x > y) ? (x = y, true) : false;
}
template <typename T>
int flg(T x, int i) {
return (x >> i) & 1;
}
int pct(int x) { return __builtin_popcount(x); }
int pct(ll x) { return __builtin_popcountll(x); }
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int botbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int botbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
template <typename T>
void print(const vector<T> &v, T x = 0) {
int n = v.size();
for (int i = 0; i < n; i++) cout << v[i] + x << (i == n - 1 ? '\n' : ' ');
if (v.empty()) cout << '\n';
}
template <typename T>
void printn(const vector<T> &v, T x = 0) {
int n = v.size();
for (int i = 0; i < n; i++) cout << v[i] + x << '\n';
}
template <typename T>
int lb(const vector<T> &v, T x) {
return lower_bound(begin(v), end(v), x) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, T x) {
return upper_bound(begin(v), end(v), x) - begin(v);
}
template <typename T>
void rearrange(vector<T> &v) {
sort(begin(v), end(v));
v.erase(unique(begin(v), end(v)), end(v));
}
template <typename T>
vector<int> id_sort(const vector<T> &v, bool greater = false) {
int n = v.size();
vector<int> ret(n);
iota(begin(ret), end(ret), 0);
sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; });
return ret;
}
template <typename T>
void reorder(vector<T> &a, const vector<int> &ord) {
int n = a.size();
vector<T> b(n);
for (int i = 0; i < n; i++) b[i] = a[ord[i]];
swap(a, b);
}
template <typename T>
T floor(T x, T y) {
assert(y != 0);
if (y < 0) x = -x, y = -y;
return (x >= 0 ? x / y : (x - y + 1) / y);
}
template <typename T>
T ceil(T x, T y) {
assert(y != 0);
if (y < 0) x = -x, y = -y;
return (x >= 0 ? (x + y - 1) / y : x / y);
}
template <typename S, typename T>
pair<S, T> operator+(const pair<S, T> &p, const pair<S, T> &q) {
return make_pair(p.first + q.first, p.second + q.second);
}
template <typename S, typename T>
pair<S, T> operator-(const pair<S, T> &p, const pair<S, T> &q) {
return make_pair(p.first - q.first, p.second - q.second);
}
template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &p) {
S a;
T b;
is >> a >> b;
p = make_pair(a, b);
return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, const pair<S, T> &p) {
return os << p.first << ' ' << p.second;
}
struct io_setup {
io_setup() {
ios_base::sync_with_stdio(false);
cin.tie(NULL);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(15);
}
} io_setup;
constexpr int inf = (1 << 30) - 1;
constexpr ll INF = (1LL << 60) - 1;
// constexpr int MOD = 1000000007;
constexpr int MOD = 998244353;
template <int mod>
struct Mod_Int {
int x;
Mod_Int() : x(0) {}
Mod_Int(long long y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}
static int get_mod() { return mod; }
Mod_Int &operator+=(const Mod_Int &p) {
if ((x += p.x) >= mod) x -= mod;
return *this;
}
Mod_Int &operator-=(const Mod_Int &p) {
if ((x += mod - p.x) >= mod) x -= mod;
return *this;
}
Mod_Int &operator*=(const Mod_Int &p) {
x = (int)(1LL * x * p.x % mod);
return *this;
}
Mod_Int &operator/=(const Mod_Int &p) {
*this *= p.inverse();
return *this;
}
Mod_Int &operator++() { return *this += Mod_Int(1); }
Mod_Int operator++(int) {
Mod_Int tmp = *this;
++*this;
return tmp;
}
Mod_Int &operator--() { return *this -= Mod_Int(1); }
Mod_Int operator--(int) {
Mod_Int tmp = *this;
--*this;
return tmp;
}
Mod_Int operator-() const { return Mod_Int(-x); }
Mod_Int operator+(const Mod_Int &p) const { return Mod_Int(*this) += p; }
Mod_Int operator-(const Mod_Int &p) const { return Mod_Int(*this) -= p; }
Mod_Int operator*(const Mod_Int &p) const { return Mod_Int(*this) *= p; }
Mod_Int operator/(const Mod_Int &p) const { return Mod_Int(*this) /= p; }
bool operator==(const Mod_Int &p) const { return x == p.x; }
bool operator!=(const Mod_Int &p) const { return x != p.x; }
Mod_Int inverse() const {
assert(*this != Mod_Int(0));
return pow(mod - 2);
}
Mod_Int pow(long long k) const {
Mod_Int now = *this, ret = 1;
for (; k > 0; k >>= 1, now *= now) {
if (k & 1) ret *= now;
}
return ret;
}
friend ostream &operator<<(ostream &os, const Mod_Int &p) { return os << p.x; }
friend istream &operator>>(istream &is, Mod_Int &p) {
long long a;
is >> a;
p = Mod_Int<mod>(a);
return is;
}
};
using mint = Mod_Int<MOD>;
template <typename T>
struct Combination {
static vector<T> _fac, _ifac;
Combination() {}
static void init(int n) {
_fac.resize(n + 1), _ifac.resize(n + 1);
_fac[0] = 1;
for (int i = 1; i <= n; i++) _fac[i] = _fac[i - 1] * i;
_ifac[n] = _fac[n].inverse();
for (int i = n; i >= 1; i--) _ifac[i - 1] = _ifac[i] * i;
}
static T fac(int k) { return _fac[k]; }
static T ifac(int k) { return _ifac[k]; }
static T inv(int k) { return fac(k - 1) * ifac(k); }
static T P(int n, int k) {
if (k < 0 || n < k) return 0;
return fac(n) * ifac(n - k);
}
static T C(int n, int k) {
if (k < 0 || n < k) return 0;
return fac(n) * ifac(n - k) * ifac(k);
}
// n 個の区別できる箱に、k 個の区別できない玉を入れる場合の数
static T H(int n, int k) {
if (n < 0 || k < 0) return 0;
return k == 0 ? 1 : C(n + k - 1, k);
}
// n 個の区別できる玉を、k 個の区別しない箱に、各箱に 1 個以上玉が入るように入れる場合の数
static T second_stirling_number(int n, int k) {
T ret = 0;
for (int i = 0; i <= k; i++) {
T tmp = C(k, i) * T(i).pow(n);
ret += ((k - i) & 1) ? -tmp : tmp;
}
return ret * ifac(k);
}
// n 個の区別できる玉を、k 個の区別しない箱に入れる場合の数
static T bell_number(int n, int k) {
if (n == 0) return 1;
k = min(k, n);
vector<T> pref(k + 1);
pref[0] = 1;
for (int i = 1; i <= k; i++) {
if (i & 1) {
pref[i] = pref[i - 1] - ifac(i);
} else {
pref[i] = pref[i - 1] + ifac(i);
}
}
T ret = 0;
for (int i = 1; i <= k; i++) ret += T(i).pow(n) * ifac(i) * pref[k - i];
return ret;
}
};
template <typename T>
vector<T> Combination<T>::_fac = vector<T>();
template <typename T>
vector<T> Combination<T>::_ifac = vector<T>();
using comb = Combination<mint>;
// n 次多項式 f の f(0),...,f(n) を与えて f(c) を計算
// comb を n まで初期化する
template <typename T>
T single_point_interpolation(vector<T> ys, T c) {
using comb_ = Combination<T>;
int n = ys.size();
T coef = 1;
for (int i = 0; i < n; i++) {
ys[i] *= coef * comb_::ifac(i);
coef *= c - i;
}
coef = 1;
T ret = 0;
for (int i = n - 1; i >= 0; i--) {
ret += ys[i] * coef * comb_::ifac(n - 1 - i) * ((n - 1 - i) & 1 ? -1 : 1);
coef *= c - i;
}
return ret;
}
template <typename T>
struct Number_Theoretic_Transform {
static int max_base;
static T root;
static vector<T> r, ir;
Number_Theoretic_Transform() {}
static void init() {
if (!r.empty()) return;
int mod = T::get_mod();
int tmp = mod - 1;
root = 2;
while (root.pow(tmp >> 1) == 1) root++;
max_base = 0;
while (tmp % 2 == 0) tmp >>= 1, max_base++;
r.resize(max_base), ir.resize(max_base);
for (int i = 0; i < max_base; i++) {
r[i] = -root.pow((mod - 1) >> (i + 2)); // r[i] := 1 の 2^(i+2) 乗根
ir[i] = r[i].inverse(); // ir[i] := 1/r[i]
}
}
static void ntt(vector<T> &a) {
init();
int n = a.size();
assert((n & (n - 1)) == 0);
assert(n <= (1 << max_base));
for (int k = n; k >>= 1;) {
T w = 1;
for (int s = 0, t = 0; s < n; s += 2 * k) {
for (int i = s, j = s + k; i < s + k; i++, j++) {
T x = a[i], y = w * a[j];
a[i] = x + y, a[j] = x - y;
}
w *= r[__builtin_ctz(++t)];
}
}
}
static void intt(vector<T> &a) {
init();
int n = a.size();
assert((n & (n - 1)) == 0);
assert(n <= (1 << max_base));
for (int k = 1; k < n; k <<= 1) {
T w = 1;
for (int s = 0, t = 0; s < n; s += 2 * k) {
for (int i = s, j = s + k; i < s + k; i++, j++) {
T x = a[i], y = a[j];
a[i] = x + y, a[j] = w * (x - y);
}
w *= ir[__builtin_ctz(++t)];
}
}
T inv = T(n).inverse();
for (auto &e : a) e *= inv;
}
static vector<T> convolve(vector<T> a, vector<T> b) {
if (a.empty() || b.empty()) return {};
if (min(a.size(), b.size()) < 40) {
int n = a.size(), m = b.size();
vector<T> c(n + m - 1, 0);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) c[i + j] += a[i] * b[j];
}
return c;
}
int k = (int)a.size() + (int)b.size() - 1, n = 1;
while (n < k) n <<= 1;
a.resize(n, 0), b.resize(n, 0);
ntt(a), ntt(b);
for (int i = 0; i < n; i++) a[i] *= b[i];
intt(a), a.resize(k);
return a;
}
};
template <typename T>
int Number_Theoretic_Transform<T>::max_base = 0;
template <typename T>
T Number_Theoretic_Transform<T>::root = T();
template <typename T>
vector<T> Number_Theoretic_Transform<T>::r = vector<T>();
template <typename T>
vector<T> Number_Theoretic_Transform<T>::ir = vector<T>();
using NTT = Number_Theoretic_Transform<mint>;
template <typename T>
struct Formal_Power_Series : vector<T> {
using NTT_ = Number_Theoretic_Transform<T>;
using vector<T>::vector;
Formal_Power_Series(const vector<T> &f) : vector<T>(f) {}
// f(x) mod x^n
Formal_Power_Series pre(int n) const {
Formal_Power_Series ret(begin(*this), begin(*this) + min((int)this->size(), n));
ret.resize(n, 0);
return ret;
}
// f(1/x)x^{n-1}
Formal_Power_Series rev(int n = -1) const {
Formal_Power_Series ret = *this;
if (n != -1) ret.resize(n, 0);
reverse(begin(ret), end(ret));
return ret;
}
void normalize() {
while (!this->empty() && this->back() == 0) this->pop_back();
}
Formal_Power_Series operator-() const {
Formal_Power_Series ret = *this;
for (int i = 0; i < (int)ret.size(); i++) ret[i] = -ret[i];
return ret;
}
Formal_Power_Series &operator+=(const T &t) {
if (this->empty()) this->resize(1, 0);
(*this)[0] += t;
return *this;
}
Formal_Power_Series &operator+=(const Formal_Power_Series &g) {
if (g.size() > this->size()) this->resize(g.size());
for (int i = 0; i < (int)g.size(); i++) (*this)[i] += g[i];
this->normalize();
return *this;
}
Formal_Power_Series &operator-=(const T &t) {
if (this->empty()) this->resize(1, 0);
(*this)[0] -= t;
return *this;
}
Formal_Power_Series &operator-=(const Formal_Power_Series &g) {
if (g.size() > this->size()) this->resize(g.size());
for (int i = 0; i < (int)g.size(); i++) (*this)[i] -= g[i];
this->normalize();
return *this;
}
Formal_Power_Series &operator*=(const T &t) {
for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= t;
return *this;
}
Formal_Power_Series &operator*=(const Formal_Power_Series &g) {
if (empty(*this) || empty(g)) {
this->clear();
return *this;
}
return *this = NTT_::convolve(*this, g);
}
Formal_Power_Series &operator/=(const T &t) {
assert(t != 0);
T inv = t.inverse();
return *this *= inv;
}
// f(x) を g(x) で割った商
Formal_Power_Series &operator/=(const Formal_Power_Series &g) {
if (g.size() > this->size()) {
this->clear();
return *this;
}
int n = this->size(), m = g.size();
return *this = (rev() * g.rev().inv(n - m + 1)).pre(n - m + 1).rev();
}
// f(x) を g(x) で割った余り
Formal_Power_Series &operator%=(const Formal_Power_Series &g) { return *this -= (*this / g) * g; }
// f(x)/x^k
Formal_Power_Series &operator<<=(int k) {
Formal_Power_Series ret(k, 0);
ret.insert(end(ret), begin(*this), end(*this));
return *this = ret;
}
// f(x)x^k
Formal_Power_Series &operator>>=(int k) {
Formal_Power_Series ret;
ret.insert(end(ret), begin(*this) + k, end(*this));
return *this = ret;
}
Formal_Power_Series operator+(const T &t) const { return Formal_Power_Series(*this) += t; }
Formal_Power_Series operator+(const Formal_Power_Series &g) const { return Formal_Power_Series(*this) += g; }
Formal_Power_Series operator-(const T &t) const { return Formal_Power_Series(*this) -= t; }
Formal_Power_Series operator-(const Formal_Power_Series &g) const { return Formal_Power_Series(*this) -= g; }
Formal_Power_Series operator*(const T &t) const { return Formal_Power_Series(*this) *= t; }
Formal_Power_Series operator*(const Formal_Power_Series &g) const { return Formal_Power_Series(*this) *= g; }
Formal_Power_Series operator/(const T &t) const { return Formal_Power_Series(*this) /= t; }
Formal_Power_Series operator/(const Formal_Power_Series &g) const { return Formal_Power_Series(*this) /= g; }
Formal_Power_Series operator%(const Formal_Power_Series &g) const { return Formal_Power_Series(*this) %= g; }
Formal_Power_Series operator<<(int k) const { return Formal_Power_Series(*this) <<= k; }
Formal_Power_Series operator>>(int k) const { return Formal_Power_Series(*this) >>= k; }
// f(c)
T val(const T &c) const {
T ret = 0;
for (int i = (int)this->size() - 1; i >= 0; i--) ret *= c, ret += (*this)[i];
return ret;
}
// df/dx
Formal_Power_Series derivative() const {
if (empty(*this)) return *this;
int n = this->size();
Formal_Power_Series ret(n - 1);
for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * i;
return ret;
}
// ∫f(x)dx
Formal_Power_Series integral() const {
if (empty(*this)) return *this;
int n = this->size();
vector<T> inv(n + 1, 0);
inv[1] = 1;
int mod = T::get_mod();
for (int i = 2; i <= n; i++) inv[i] = -inv[mod % i] * T(mod / i);
Formal_Power_Series ret(n + 1, 0);
for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] * inv[i + 1];
return ret;
}
// 1/f(x) mod x^n (f[0] != 0)
Formal_Power_Series inv(int n = -1) const {
assert((*this)[0] != 0);
if (n == -1) n = this->size();
Formal_Power_Series ret(1, (*this)[0].inverse());
for (int m = 1; m < n; m <<= 1) {
Formal_Power_Series f = pre(2 * m), g = ret;
f.resize(2 * m), g.resize(2 * m);
NTT_::ntt(f), NTT_::ntt(g);
Formal_Power_Series h(2 * m);
for (int i = 0; i < 2 * m; i++) h[i] = f[i] * g[i];
NTT_::intt(h);
for (int i = 0; i < m; i++) h[i] = 0;
NTT_::ntt(h);
for (int i = 0; i < 2 * m; i++) h[i] *= g[i];
NTT_::intt(h);
for (int i = 0; i < m; i++) h[i] = 0;
ret -= h;
}
ret.resize(n);
return ret;
}
// log(f(x)) mod x^n (f[0] = 1)
Formal_Power_Series log(int n = -1) const {
assert((*this)[0] == 1);
if (n == -1) n = this->size();
Formal_Power_Series ret = (derivative() * inv(n)).pre(n - 1).integral();
ret.resize(n);
return ret;
}
// exp(f(x)) mod x^n (f[0] = 0)
Formal_Power_Series exp(int n = -1) const {
assert((*this)[0] == 0);
if (n == -1) n = this->size();
vector<T> inv(2 * n + 1, 0);
inv[1] = 1;
int mod = T::get_mod();
for (int i = 2; i <= 2 * n; i++) inv[i] = -inv[mod % i] * T(mod / i);
auto inplace_integral = [inv](Formal_Power_Series &f) {
if (empty(f)) return;
int m = f.size();
f.insert(begin(f), 0);
for (int i = 1; i <= m; i++) f[i] *= inv[i];
};
auto inplace_derivative = [](Formal_Power_Series &f) {
if (empty(f)) return;
int m = f.size();
f.erase(begin(f));
for (int i = 0; i < m - 1; i++) f[i] *= T(i + 1);
};
Formal_Power_Series ret{1, this->size() > 1 ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
for (int m = 2; m < n; m *= 2) {
auto y = ret;
y.resize(2 * m);
NTT_::ntt(y);
z1 = z2;
Formal_Power_Series z(m);
for (int i = 0; i < m; i++) z[i] = y[i] * z1[i];
NTT_::intt(z);
fill(begin(z), begin(z) + m / 2, 0);
NTT_::ntt(z);
for (int i = 0; i < m; i++) z[i] *= -z1[i];
NTT_::intt(z);
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c, z2.resize(2 * m);
NTT_::ntt(z2);
Formal_Power_Series x(begin(*this), begin(*this) + min((int)this->size(), m));
inplace_derivative(x);
x.resize(m, 0);
NTT_::ntt(x);
for (int i = 0; i < m; i++) x[i] *= y[i];
NTT_::intt(x);
x -= ret.derivative(), x.resize(2 * m);
for (int i = 0; i < m - 1; i++) x[m + i] = x[i], x[i] = 0;
NTT_::ntt(x);
for (int i = 0; i < 2 * m; i++) x[i] *= z2[i];
NTT_::intt(x);
x.pop_back();
inplace_integral(x);
for (int i = m; i < min((int)this->size(), 2 * m); i++) x[i] += (*this)[i];
fill(begin(x), begin(x) + m, 0);
NTT_::ntt(x);
for (int i = 0; i < 2 * m; i++) x[i] *= y[i];
NTT_::intt(x);
ret.insert(end(ret), begin(x) + m, end(x));
}
ret.resize(n);
return ret;
}
// f(x)^k mod x^n
Formal_Power_Series pow(long long k, int n = -1) const {
if (n == -1) n = this->size();
int m = this->size();
for (int i = 0; i < m; i++) {
if ((*this)[i] == 0) continue;
T inv = (*this)[i].inverse();
Formal_Power_Series g(m - i, 0);
for (int j = i; j < m; j++) g[j - i] = (*this)[j] * inv;
g = (g.log(n) * k).exp(n) * ((*this)[i].pow(k));
Formal_Power_Series ret(n, 0);
if (i > 0 && k > n / i) return ret;
long long d = i * k;
for (int j = 0; j + d < n && j < (int)g.size(); j++) ret[j + d] = g[j];
return ret;
}
Formal_Power_Series ret(n, 0);
if (k == 0) ret[0] = 1;
return ret;
}
// √f(x) mod x^n (存在しなければ空)
Formal_Power_Series sqrt(int n = -1) const {
if (n == -1) n = this->size();
int mod = T::get_mod();
auto sqrt_mod = [mod](const T &a) {
if (mod == 2) return a;
int s = mod - 1, t = 0;
while (s % 2 == 0) s /= 2, t++;
T root = 2;
while (root.pow((mod - 1) / 2) == 1) root++;
T x = a.pow((s + 1) / 2);
T u = root.pow(s);
T y = x * x * a.inverse();
while (y != 1) {
int k = 0;
T z = y;
while (z != 1) k++, z *= z;
for (int i = 0; i < t - k - 1; i++) u *= u;
x *= u, u *= u, y *= u;
t = k;
}
return x;
};
if ((*this)[0] == 0) {
for (int i = 1; i < (int)this->size(); i++) {
if ((*this)[i] != 0) {
if (i & 1) return {};
if ((*this)[i].pow((mod - 1) / 2) != 1) return {};
if (n <= i / 2) break;
return ((*this) >> i).sqrt(n - i / 2) << (i / 2);
}
}
return Formal_Power_Series(n, 0);
}
if ((*this)[0].pow((mod - 1) / 2) != 1) return {};
T tw = T(2).inverse();
Formal_Power_Series ret{sqrt_mod((*this)[0])};
for (int m = 1; m < n; m *= 2) {
Formal_Power_Series g = (*this).pre(m * 2) * ret.inv(m * 2);
g.resize(2 * m);
ret = (ret + g) * tw;
}
ret.resize(n);
return ret;
}
// f(x+c)
Formal_Power_Series Taylor_shift(T c) const {
int n = this->size();
vector<T> ifac(n, 1);
Formal_Power_Series f(n), g(n);
for (int i = 0; i < n; i++) {
f[n - 1 - i] = (*this)[i] * ifac[n - 1];
if (i < n - 1) ifac[n - 1] *= i + 1;
}
ifac[n - 1] = ifac[n - 1].inverse();
for (int i = n - 1; i > 0; i--) ifac[i - 1] = ifac[i] * i;
T pw = 1;
for (int i = 0; i < n; i++) {
g[i] = pw * ifac[i];
pw *= c;
}
f *= g;
Formal_Power_Series b(n);
for (int i = 0; i < n; i++) b[i] = f[n - 1 - i] * ifac[i];
return b;
}
};
using fps = Formal_Power_Series<mint>;
void solve() {
ll H, W, N, K;
cin >> H >> W >> N >> K;
ll s = (H - K + 1) * (W - K + 1);
// cout << s << '\n';
comb::init(1000000);
fps g(N + 1, 0);
rep(i, N + 1) g[i] = comb::ifac(i + 1);
fps ig = g.inv();
// ∑[1<=i<=n] i^k を k=0,...,N について計算する
auto get = [&](ll n) {
fps f(N + 1, 0), g(N + 1, 0);
rep(i, N + 1) f[i] = mint(n + 1).pow(i + 1) * comb::ifac(i + 1);
auto h = f * ig;
h.resize(N + 1);
h[0]--;
rep(i, N + 1) h[i] *= comb::fac(i);
return h;
};
auto calc = [&](ll a, ll x, ll y) {
mint ret = 0;
auto f = get(x), g = get(y);
vector<mint> pa(N + 1, 1);
rep(i, N) pa[i + 1] = pa[i] * a;
rep(k, N + 1) {
ret += f[k] * g[k] * (k & 1 ? -1 : 1) * pa[N - k] * comb::C(N, k); //
}
return ret;
};
ll lx = min(K - 1, H - K), rx = min(K - 1, H - K);
ll ly = min(K - 1, W - K), ry = min(K - 1, W - K);
// cout << lx MM rx MM ly MM ry << '\n';
// print(get(1)), print(get(2)), print(get(3));
mint ans = 0;
for (auto x : {lx, rx}) {
for (auto y : {ly, ry}) {
ans += calc(s, x, y); //
}
}
// cout << ans << '\n';
if (ly + ry < W) {
mint tmp = calc(s, lx, ly + 1) - calc(s, lx, ly);
ans += tmp * mint(W - ly - ry) * 2;
}
if (lx + rx < H) {
mint tmp = calc(s, lx + 1, ly) - calc(s, lx, ly);
ans += tmp * mint(H - lx - rx) * 2;
}
// cout << ans << '\n';
if (ly + ry < W && lx + rx < H) {
mint tmp = mint(s - (lx + 1) * (ly + 1)).pow(N);
ans += tmp * mint(H - lx - rx) * mint(W - ly - ry); //
}
// cout << ans << '\n';
ans = mint(s).pow(N) * mint(H * W) - ans;
// cout << ans << ' ';
cout << ans / mint(s).pow(N) << '\n';
}
int main() {
int T = 1;
// cin >> T;
while (T--) solve();
}