#848509 (C++17) No.2245 Second Smallest

• 問題: No.2245 Second Smallest
• ユーザー: tokusakurai
• 提出日時: 2023-03-11 20:51:33
• 言語: C++17 (gcc 12.3.0 + boost 1.83.0)
• 結果: AC
• 実行時間: 30 ms / 2,000 ms
• コード長: 7,746 bytes
• コンパイル時間: 2,189 ms
• コンパイル使用メモリ: 208,596 KB
• 実行使用メモリ: 6,948 KB
• 最終ジャッジ日時: 2024-09-18 06:46:08
• 合計ジャッジ時間: 3,616 ms
• ジャッジサーバーID （参考情報）: judge1 / judge3

入力	結果	実行時間; 実行使用メモリ
testcase_00	AC	19 ms; 6,812 KB
testcase_01	AC	19 ms; 6,944 KB
testcase_02	AC	19 ms; 6,940 KB
testcase_03	AC	19 ms; 6,944 KB
testcase_04	AC	19 ms; 6,940 KB
testcase_05	AC	29 ms; 6,944 KB
testcase_06	AC	28 ms; 6,940 KB
testcase_07	AC	18 ms; 6,944 KB
testcase_08	AC	19 ms; 6,944 KB
testcase_09	AC	29 ms; 6,944 KB
testcase_10	AC	25 ms; 6,940 KB
testcase_11	AC	25 ms; 6,944 KB
testcase_12	AC	24 ms; 6,940 KB
testcase_13	AC	21 ms; 6,940 KB
testcase_14	AC	30 ms; 6,944 KB
testcase_15	AC	30 ms; 6,944 KB
testcase_16	AC	21 ms; 6,944 KB
testcase_17	AC	27 ms; 6,944 KB
testcase_18	AC	23 ms; 6,940 KB
testcase_19	AC	26 ms; 6,944 KB
testcase_20	AC	25 ms; 6,940 KB
testcase_21	AC	25 ms; 6,944 KB
testcase_22	AC	26 ms; 6,940 KB
testcase_23	AC	21 ms; 6,948 KB
testcase_24	AC	25 ms; 6,944 KB
testcase_25	AC	24 ms; 6,944 KB
testcase_26	AC	27 ms; 6,944 KB
testcase_27	AC	29 ms; 6,940 KB
testcase_28	AC	22 ms; 6,940 KB
testcase_29	AC	27 ms; 6,940 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;
}

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 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);
    }
} io_setup;

const int inf = (1 << 30) - 1;
const ll INF = (1LL << 60) - 1;
// const int MOD = 1000000007;
const 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>;

int main() {
    int H, W, K;
    cin >> H >> W >> K;

    int MAX = 100000;
    comb::init(100000);

    vector<mint> c(MAX + 1, 1);
    rep(i, MAX) {
        c[i + 1] = c[i] * mint(H * W + K - 1 - i) / mint(i + 1); //
    }

    auto beta = [&](int n, int m) {
        if (n < 0 || m < 0) return mint(0);
        assert(n + m == H * W + K - 1);
        return (c[n] * mint(n + m + 1)).inverse(); //
    };

    vector<mint> c2(W + 1, 0);
    c2[0] = 1;
    rep2(i, 1, W + 1) c2[i] = comb::C(W, i) * comb::C(H, i);

    mint ans = 0;
    rep(t, W + 1) {
        ans -= c2[t] * t * beta(K + t - 1, H * W - t);
        ans += c2[t] * (H * W - t) * beta(K + t, H * W - t - 1);
    }

    cout << ans << '\n';
}