static_modint.cpp

template<uint32_t m> class static_modint {
    using mint = static_modint;
    uint32_t _v = 0;
    static constexpr bool prime = []() -> bool {
        if (m == 1) return 0;
        if (m == 2 || m == 7 || m == 61) return 1;
        if (m % 2 == 0) return 0;
        uint32_t d = m - 1;
        while (d % 2 == 0) d /= 2;
        for (uint32_t a : {2, 7, 61}) {
            uint32_t t = d;
            mint y = mint(a).pow(t);
            while (t != m - 1 && y != 1 && y != m - 1) {
                y *= y; t <<= 1;
            }
            if (y != m - 1 && t % 2 == 0) return 0;
        }
        return 1;
    }();
public:
    static constexpr mint raw(uint32_t v) { mint a; a._v = v; return a; }
    constexpr static_modint() {}
    template <class T>
    constexpr static_modint(T v) {
        static_assert(is_integral_v<T>, "T is not integral type.");
        if constexpr (is_signed_v<T>) {
            int64_t x = int64_t(v % int64_t(m));
            if (x < 0) x += m; _v = uint32_t(x);
        }
        else _v = uint32_t(v % m);
    }
    static constexpr uint32_t mod() { return m; }
    constexpr uint32_t val() const { return _v; }
    constexpr mint& operator++() { return *this += 1; }
    constexpr mint& operator--() { return *this -= 1; }
    constexpr mint operator++(int) { mint res = *this; ++*this; return res; }
    constexpr mint operator--(int) { mint res = *this; --*this; return res; }
    constexpr mint& operator+=(mint rhs) {
        if (_v >= m - rhs._v) _v -= m;
        _v += rhs._v; return *this;
    }
    constexpr mint& operator-=(mint rhs) {
        if (_v < rhs._v) _v += m;
        _v -= rhs._v; return *this;
    }
    constexpr mint& operator*=(mint rhs) { return *this = *this * rhs; }
    constexpr mint& operator/=(mint rhs) { return *this *= rhs.inv(); }
    constexpr mint operator+() const { return *this; }
    constexpr mint operator-() const { return mint{} - *this; }
    constexpr mint pow(long long n) const {
        assert(0 <= n);
        if(n == 0) return 1;
        mint x = *this, r = 1;
        while (1) {
            if (n & 1) r *= x; n >>= 1;
            if (n == 0) return r;
            x *= x;
        }
    }
    static constexpr pair<int32_t, int32_t> inv_gcd(int32_t a, int32_t b) {
        if (a == 0) return {b, 0};
        int32_t s = b, t = a, m0 = 0, m1 = 1;
        while (t) {
            const int32_t u = s / t;
            s -= t * u; m0 -= m1 * u;
            swap(s, t); swap(m0, m1);
        }
        if (m0 < 0) m0 += b / s;
        return {s, m0};
    }
    constexpr mint inv() const {
        if constexpr (prime) {
            assert(_v);
            return pow(m - 2);
        } else {
            auto eg = inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }
    friend constexpr mint operator+(mint lhs, mint rhs) { return lhs += rhs; }
    friend constexpr mint operator-(mint lhs, mint rhs) { return lhs -= rhs; }
    friend constexpr mint operator*(mint lhs, mint rhs) { return uint64_t(lhs._v) * rhs._v; }
    friend constexpr mint operator/(mint lhs, mint rhs) { return lhs /= rhs; }
    friend constexpr bool operator==(mint lhs, mint rhs) { return lhs._v == rhs._v; }
    friend constexpr bool operator!=(mint lhs, mint rhs) { return lhs._v != rhs._v; }
};
using mint = static_modint<998244353>;
istream& operator>>(istream& in, mint& x) { long long a; in >> a; x = a; return in; }
ostream& operator<<(ostream& out, mint x) { return out << x.val(); }
constexpr mint operator""_M(unsigned long long x) { return x; }

constexpr uint32_t fact_mx = min<uint32_t>(2e7, mint::mod() - 1);
mint fac[fact_mx + 1], inv[fact_mx + 1];
struct factorial {
    factorial() {
        fac[0] = 1;
        for(uint32_t i = 1; i <= fact_mx; i++) fac[i] = fac[i - 1] * mint::raw(i);
        inv[fact_mx] = fac[fact_mx].inv();
        for(uint32_t i = fact_mx; i; i--) inv[i - 1] = inv[i] * mint::raw(i);
    }
} factorial;
mint inverse(long long n) { return inv[n] * fac[n - 1]; }
mint perm(long long n, long long r) {
    if(n < r || r < 0) return 0;
    if(n > fact_mx) [[unlikely]] {
        mint ans = 1, x = n;
        while(r--) ans *= x--;
        return ans;
    }
    return fac[n] * inv[n - r];
}
mint comb(long long n, long long r) {
    if(n < r || r < 0) return 0;
    r = min(r, n - r);
    const mint res = perm(n, r);
    return res * inv[r];
}
mint Mcomb(long long n, long long r){ return comb(n + r - 1, r); } // r balls into n boxes