結果

問題 No.2459 Stampaholic (Hard)
ユーザー hitonanodehitonanode
提出日時 2023-09-01 23:35:24
言語 C++23
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 574 ms / 4,000 ms
コード長 25,944 bytes
コンパイル時間 4,612 ms
コンパイル使用メモリ 232,624 KB
実行使用メモリ 54,668 KB
最終ジャッジ日時 2024-06-11 05:47:30
合計ジャッジ時間 10,978 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge5
このコードへのチャレンジ
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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
6,812 KB
testcase_01 AC 557 ms
54,536 KB
testcase_02 AC 127 ms
15,160 KB
testcase_03 AC 2 ms
6,944 KB
testcase_04 AC 2 ms
6,940 KB
testcase_05 AC 1 ms
6,944 KB
testcase_06 AC 2 ms
6,944 KB
testcase_07 AC 1 ms
6,944 KB
testcase_08 AC 275 ms
27,756 KB
testcase_09 AC 127 ms
15,460 KB
testcase_10 AC 565 ms
51,640 KB
testcase_11 AC 276 ms
27,944 KB
testcase_12 AC 567 ms
54,308 KB
testcase_13 AC 546 ms
51,604 KB
testcase_14 AC 129 ms
15,504 KB
testcase_15 AC 570 ms
54,440 KB
testcase_16 AC 559 ms
54,408 KB
testcase_17 AC 566 ms
54,408 KB
testcase_18 AC 562 ms
54,668 KB
testcase_19 AC 574 ms
52,232 KB
testcase_20 AC 29 ms
17,772 KB
testcase_21 AC 558 ms
52,912 KB
権限があれば一括ダウンロードができます

ソースコード

diff # 

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <forward_list>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iostream>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <numeric>
#include <optional>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using lint = long long;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template <typename T> bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; }
template <typename T> bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; }
const std::vector<std::pair<int, int>> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
template <class T1, class T2> T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); }
template <class T1, class T2> std::pair<T1, T2> operator+(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first + r.first, l.second + r.second); }
template <class T1, class T2> std::pair<T1, T2> operator-(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first - r.first, l.second - r.second); }
template <class T> std::vector<T> sort_unique(std::vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <class T> int arglb(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); }
template <class T> int argub(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); }
template <class IStream, class T> IStream &operator>>(IStream &is, std::vector<T> &vec) { for (auto &v : vec) is >> v; return is; }

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec);
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr);
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const pair<T, U> &pa);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa);
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp);
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp);
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl);

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; }
template <class... T> std::istream &operator>>(std::istream &is, std::tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; }
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; }
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
#ifdef HITONANODE_LOCAL
const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m";
#define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl
#define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr)
#else
#define dbg(x) ((void)0)
#define dbgif(cond, x) ((void)0)
#endif

#include <cassert>
#include <iostream>
#include <set>
#include <vector>

template <int md> struct ModInt {
    using lint = long long;
    constexpr static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val_;
    int val() const noexcept { return val_; }
    constexpr ModInt() : val_(0) {}
    constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
    constexpr ModInt(lint v) { _setval(v % md + md); }
    constexpr explicit operator bool() const { return val_ != 0; }
    constexpr ModInt operator+(const ModInt &x) const {
        return ModInt()._setval((lint)val_ + x.val_);
    }
    constexpr ModInt operator-(const ModInt &x) const {
        return ModInt()._setval((lint)val_ - x.val_ + md);
    }
    constexpr ModInt operator*(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.val_ % md);
    }
    constexpr ModInt operator/(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.inv().val() % md);
    }
    constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
    constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend constexpr ModInt operator+(lint a, const ModInt &x) {
        return ModInt()._setval(a % md + x.val_);
    }
    friend constexpr ModInt operator-(lint a, const ModInt &x) {
        return ModInt()._setval(a % md - x.val_ + md);
    }
    friend constexpr ModInt operator*(lint a, const ModInt &x) {
        return ModInt()._setval(a % md * x.val_ % md);
    }
    friend constexpr ModInt operator/(lint a, const ModInt &x) {
        return ModInt()._setval(a % md * x.inv().val() % md);
    }
    constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
    constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
    constexpr bool operator<(const ModInt &x) const {
        return val_ < x.val_;
    } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
        return os << x.val_;
    }

    constexpr ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }

    static constexpr int cache_limit = std::min(md, 1 << 21);
    static std::vector<ModInt> facs, facinvs, invs;

    constexpr static void _precalculation(int N) {
        const int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }

    constexpr ModInt inv() const {
        if (this->val_ < cache_limit) {
            if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
            while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val_];
        } else {
            return this->pow(md - 2);
        }
    }
    constexpr ModInt fac() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val_];
    }
    constexpr ModInt facinv() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val_];
    }
    constexpr ModInt doublefac() const {
        lint k = (this->val_ + 1) / 2;
        return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                                : ModInt(k).fac() * ModInt(2).pow(k);
    }

    constexpr ModInt nCr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
    }

    constexpr ModInt nPr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv();
    }

    static ModInt binom(int n, int r) {
        static long long bruteforce_times = 0;

        if (r < 0 or n < r) return ModInt(0);
        if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);

        r = std::min(r, n - r);

        ModInt ret = ModInt(r).facinv();
        for (int i = 0; i < r; ++i) ret *= n - i;
        bruteforce_times += r;

        return ret;
    }

    // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
    // Complexity: O(sum(ks))
    template <class Vec> static ModInt multinomial(const Vec &ks) {
        ModInt ret{1};
        int sum = 0;
        for (int k : ks) {
            assert(k >= 0);
            ret *= ModInt(k).facinv(), sum += k;
        }
        return ret * ModInt(sum).fac();
    }

    // Catalan number, C_n = binom(2n, n) / (n + 1)
    // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
    // https://oeis.org/A000108
    // Complexity: O(n)
    static ModInt catalan(int n) {
        if (n < 0) return ModInt(0);
        return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
    }

    ModInt sqrt() const {
        if (val_ == 0) return 0;
        if (md == 2) return val_;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val_, md - x.val_));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};

using mint = ModInt<998244353>;

// Integer convolution for arbitrary mod
// with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class.
// We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`.
// input: a (size: n), b (size: m)
// return: vector (size: n + m - 1)
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner);

constexpr int nttprimes[3] = {998244353, 167772161, 469762049};

// Integer FFT (Fast Fourier Transform) for ModInt class
// (Also known as Number Theoretic Transform, NTT)
// is_inverse: inverse transform
// ** Input size must be 2^n **
template <typename MODINT> void ntt(std::vector<MODINT> &a, bool is_inverse = false) {
    int n = a.size();
    if (n == 1) return;
    static const int mod = MODINT::mod();
    static const MODINT root = MODINT::get_primitive_root();
    assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0);

    static std::vector<MODINT> w{1}, iw{1};
    for (int m = w.size(); m < n / 2; m *= 2) {
        MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw;
        w.resize(m * 2), iw.resize(m * 2);
        for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv;
    }

    if (!is_inverse) {
        for (int m = n; m >>= 1;) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m] * w[k];
                    a[i] = x + y, a[i + m] = x - y;
                }
            }
        }
    } else {
        for (int m = 1; m < n; m *= 2) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m];
                    a[i] = x + y, a[i + m] = (x - y) * iw[k];
                }
            }
        }
        int n_inv = MODINT(n).inv().val();
        for (auto &v : a) v *= n_inv;
    }
}
template <int MOD>
std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) {
    int sz = a.size();
    assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
    std::vector<ModInt<MOD>> ap(sz), bp(sz);
    for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
    ntt(ap, false);
    if (a == b)
        bp = ap;
    else
        ntt(bp, false);
    for (int i = 0; i < sz; i++) ap[i] *= bp[i];
    ntt(ap, true);
    return ap;
}
long long garner_ntt_(int r0, int r1, int r2, int mod) {
    using mint2 = ModInt<nttprimes[2]>;
    static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
    static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv().val();
    static const long long m01_inv_m2 = mint2(m01).inv().val();

    int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
    auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2;
    return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val()) % mod;
}
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner) {
    if (a.empty() or b.empty()) return {};
    int sz = 1, n = a.size(), m = b.size();
    while (sz < n + m) sz <<= 1;
    if (sz <= 16) {
        std::vector<MODINT> ret(n + m - 1);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j];
        }
        return ret;
    }
    int mod = MODINT::mod();
    if (skip_garner or
        std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes)) {
        a.resize(sz), b.resize(sz);
        if (a == b) {
            ntt(a, false);
            b = a;
        } else {
            ntt(a, false), ntt(b, false);
        }
        for (int i = 0; i < sz; i++) a[i] *= b[i];
        ntt(a, true);
        a.resize(n + m - 1);
    } else {
        std::vector<int> ai(sz), bi(sz);
        for (int i = 0; i < n; i++) ai[i] = a[i].val();
        for (int i = 0; i < m; i++) bi[i] = b[i].val();
        auto ntt0 = nttconv_<nttprimes[0]>(ai, bi);
        auto ntt1 = nttconv_<nttprimes[1]>(ai, bi);
        auto ntt2 = nttconv_<nttprimes[2]>(ai, bi);
        a.resize(n + m - 1);
        for (int i = 0; i < n + m - 1; i++)
            a[i] = garner_ntt_(ntt0[i].val(), ntt1[i].val(), ntt2[i].val(), mod);
    }
    return a;
}

template <typename MODINT>
std::vector<MODINT> nttconv(const std::vector<MODINT> &a, const std::vector<MODINT> &b) {
    return nttconv<MODINT>(a, b, false);
}

namespace fps_nttmod {

// Calculate the inverse of f(x) mod x^d
// f(x) * g(x) = 1 mod x^d
// If d = -1, d is set to f.size()
// Complexity: O(d log d)
template <class NTTModInt> std::vector<NTTModInt> inv(const std::vector<NTTModInt> &f, int d = -1) {
    assert(d >= -1);

    const int n = f.size();
    if (d == -1) d = n;

    if (d == 0) return {};

    assert(f.front() != NTTModInt(0));

    using F = std::vector<NTTModInt>;

    F res{f.front().inv()}; // f(x) g_m(x) = 1 mod x^m

    for (int m = 1; m < d; m *= 2) { // g_2m = (2g_m - f g_m^2) mod x^2m
        F g_m{res.cbegin(), res.cbegin() + m};
        g_m.resize(2 * m);
        ntt(g_m, false);

        F f_{f.cbegin(), f.cbegin() + std::min(n, 2 * m)};

        f_.resize(2 * m);
        ntt(f_, false);
        for (int i = 0; i < 2 * m; ++i) f_.at(i) *= g_m.at(i);
        ntt(f_, true);

        std::rotate(f_.begin(), f_.begin() + m, f_.end());
        for (int i = m; i < 2 * m; ++i) f_.at(i) = 0;

        ntt(f_, false);
        for (int i = 0; i < 2 * m; ++i) f_.at(i) *= g_m.at(i);
        ntt(f_, true);

        for (int i = 0; i < m; ++i) f_.at(i) = -f_.at(i);

        res.insert(res.end(), f_.begin(), f_.begin() + m);
    }
    res.resize(d);
    return res;
}

// Calculate the integral of f(x)
// Complexity: O(len(f))
template <class NTTModInt> void integ_inplace(std::vector<NTTModInt> &f) {
    if (f.empty()) return;

    for (int i = (int)f.size() - 1; i > 0; --i) f.at(i) = f.at(i - 1) * NTTModInt(i).inv();
    f.front() = NTTModInt(0);
}

// Calculate the derivative of f(x)
// Complexity: O(len(f))
template <class NTTModInt> void deriv_inplace(std::vector<NTTModInt> &f) {
    if (f.empty()) return;

    for (int i = 1; i < (int)f.size(); ++i) f.at(i - 1) = f.at(i) * i;
    f.back() = NTTModInt(0);
}

// Calculate log f(x) mod x^d
// Require f(0) = 1 mod x^d
// Complexity: O(d log d)
template <class NTTModInt> std::vector<NTTModInt> log(const std::vector<NTTModInt> &f, int d = -1) {
    assert(d >= -1);

    const int n = f.size();
    if (d < 0) d = n;

    if (d == 0) return {};

    assert(f.front() == NTTModInt(1));

    std::vector<NTTModInt> inv_f = inv(f, d), df{f.cbegin(), f.cbegin() + std::min(d, n)};
    deriv_inplace(df);

    auto ret = nttconv(inv_f, df);
    ret.resize(d);
    integ_inplace(ret);
    return ret;
}

template <class NTTModInt> std::vector<NTTModInt> exp(const std::vector<NTTModInt> &h, int d = -1) {
    assert(d >= -1);

    const int n = h.size();
    if (d < 0) d = n;

    if (d == 0) return {};

    assert(h.empty() or h.front() == NTTModInt(0));

    using F = std::vector<NTTModInt>;

    F g{1}, g_fft;

    std::vector<NTTModInt> ret(d);
    ret.front() = 1;

    auto h_deriv = h;
    h_deriv.resize(d);
    deriv_inplace(h_deriv);

    for (int m = 1; m < d; m *= 2) {
        F f_fft = ret;
        f_fft.resize(m * 2);
        ntt(f_fft, false);

        // 2a
        if (m > 1) {
            F tmp{f_fft.cbegin(), f_fft.cbegin() + m};
            for (int i = 0; i < m; ++i) tmp.at(i) *= g_fft.at(i);
            ntt(tmp, true);
            tmp.erase(tmp.begin(), tmp.begin() + m / 2);
            tmp.resize(m);
            ntt(tmp, false);
            for (int i = 0; i < m; ++i) tmp.at(i) *= -g_fft.at(i);
            ntt(tmp, true);
            tmp.resize(m / 2);
            g.insert(g.end(), tmp.cbegin(), tmp.cbegin() + m / 2);
        }

        //
        F t{ret.cbegin(), ret.cbegin() + m};
        deriv_inplace(t);

        {
            F r{h_deriv.cbegin(), h_deriv.cbegin() + m - 1};
            r.resize(m);
            ntt(r, false);
            for (int i = 0; i < m; ++i) r.at(i) *= f_fft.at(i);
            ntt(r, true);
            for (int i = 0; i < m; ++i) t.at(i) -= r.at(i);

            std::rotate(t.begin(), t.end() - 1, t.end());
        }

        //
        t.resize(2 * m);
        ntt(t, false);

        g_fft = g;
        g_fft.resize(2 * m);
        ntt(g_fft, false);

        for (int i = 0; i < 2 * m; ++i) t.at(i) *= g_fft.at(i);
        ntt(t, true);
        t.resize(m);

        //
        F v{h.begin() + std::min(m, n), h.begin() + std::min({d, 2 * m, n})};
        v.resize(m);
        t.insert(t.begin(), m - 1, 0);
        t.push_back(0);
        integ_inplace(t);
        for (int i = 0; i < m; ++i) v.at(i) -= t.at(m + i);

        //
        v.resize(2 * m);
        ntt(v, false);
        for (int i = 0; i < 2 * m; ++i) v.at(i) *= f_fft.at(i);
        ntt(v, true);
        v.resize(m);

        for (int i = 0; i < std::min(d - m, m); ++i) ret.at(m + i) = v.at(i);
    }
    return ret;
}

// Calculate f(x)^k mod x^d
// assume 0^0 = 1
template <class NTTModInt> std::vector<NTTModInt> pow(const std::vector<NTTModInt> &A, long long k, int d = -1) {
    assert(d >= -1);

    const int n = A.size();
    if (d < 0) d = n;

    if (k == 0) {
        std::vector<NTTModInt> ret{NTTModInt(1)}; // assume 0^0 = 1
        ret.resize(d);
        return ret;
    }

    int l = 0;
    long long shift = 0;
    while (l < (int)A.size() and A.at(l) == NTTModInt(0) and shift < d) {
        ++l;
        shift += k;
    }
    if (l == (int)A.size() or shift >= d) return std::vector<NTTModInt>(d, 0);

    const NTTModInt cpow = A.at(l).pow(k), cinv = A.at(l).inv();
    std::vector<NTTModInt> tmp{A.cbegin() + l, A.cbegin() + std::min<int>(n, d - l * k + l)};

    for (auto &x : tmp) x *= cinv;

    tmp = log(tmp, d - l * k);

    for (auto &x : tmp) x *= k;

    tmp = exp(tmp, d - l * k);

    for (auto &x : tmp) x *= cpow;

    tmp.insert(tmp.begin(), l * k, NTTModInt(0));

    tmp.resize(d);

    return tmp;
}

} // namespace fps_nttmod

// 1 から n までの p 乗和を p <= pub まで
vector<mint> sum_deg(int n, int pub) {
    vector<mint> bernoulli(pub + 10);
    REP(i, bernoulli.size()) bernoulli.at(i) = mint(i).facinv();
    bernoulli.erase(bernoulli.begin());
    bernoulli = fps_nttmod::inv(bernoulli);
    REP(i, bernoulli.size()) bernoulli.at(i) *= mint(i).fac();
    bernoulli.at(1) = -bernoulli.at(1);
    assert(bernoulli.at(1) * 2 == 1);

    vector<mint> f(pub + 1);
    const mint invn = mint(n).inv();

    mint pp = 1;

    REP(k, f.size()) {
        f.at(k) = bernoulli.at(k) * mint(k).facinv() * pp;
        pp *= invn;
    }
    vector<mint> g(pub + 3);
    FOR(i, 1, g.size()) g.at(i) = mint(i).facinv();
    const auto h = nttconv(f, g);

    vector<mint> sums(pub + 1);
    pp = 1;
    REP(p, pub + 1) {
        pp *= n;
        sums.at(p) = h.at(p + 1) * pp * mint(p).fac();
    }
    return sums;
}

vector<mint> gen(int H, int N, int K) {
    const int h = min({K, H / 2, H - K});

    auto h_sums = sum_deg(h, N);

    {
        const mint ainv = mint(H - K + 1).inv();
        dbg(ainv);
        mint den = 1;
        REP(i, h_sums.size()) {
            h_sums.at(i) *= den;
            den *= ainv;
        }
    }
    for (auto &x : h_sums) x *= 2;

    if (h * 2 < H) {
        int ilo = max(0, h - K + 1);
        int ihi = min(h, H - 1 - (K - 1));
        const mint p = mint(ihi - ilo + 1) / mint(H - K + 1);
        const mint weight = H - h * 2;
        mint ppow = 1;
        REP(i, h_sums.size()) {
            h_sums.at(i) += ppow * weight;
            ppow *= p;
        }
    }
    return h_sums;
}

mint solve(int H, int W, int N, int K) {

    auto h_sums = gen(H, N, K);
    auto w_sums = gen(W, N, K);

    mint ret = 0;
    FOR(d, 1, N + 1) ret += mint(N).nCr(d) * h_sums.at(d) * w_sums.at(d) * (d % 2 ? 1 : -1);
    return ret;
}


int main() {
    int H, W, N, K;
    cin >> H >> W >> N >> K;
    cout << solve(H, W, N, K) << endl;
}
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