#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using Int = long long; template ostream &operator<<(ostream &os, const pair &a) { return os << "(" << a.first << ", " << a.second << ")"; }; template ostream &operator<<(ostream &os, const vector &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; } template void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; } template bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; } template bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; } //////////////////////////////////////////////////////////////////////////////// template struct ModInt { static constexpr unsigned M = M_; unsigned x; constexpr ModInt() : x(0U) {} constexpr ModInt(unsigned x_) : x(x_ % M) {} constexpr ModInt(unsigned long long x_) : x(x_ % M) {} constexpr ModInt(int x_) : x(((x_ %= static_cast(M)) < 0) ? (x_ + static_cast(M)) : x_) {} constexpr ModInt(long long x_) : x(((x_ %= static_cast(M)) < 0) ? (x_ + static_cast(M)) : x_) {} ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; } ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; } ModInt &operator*=(const ModInt &a) { x = (static_cast(x) * a.x) % M; return *this; } ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); } ModInt pow(long long e) const { if (e < 0) return inv().pow(-e); ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b; } ModInt inv() const { unsigned a = M, b = x; int y = 0, z = 1; for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast(q) * z; y = z; z = w; } assert(a == 1U); return ModInt(y); } ModInt operator+() const { return *this; } ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; } ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); } ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); } ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); } ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); } template friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); } template friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); } template friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); } template friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); } explicit operator bool() const { return x; } bool operator==(const ModInt &a) const { return (x == a.x); } bool operator!=(const ModInt &a) const { return (x != a.x); } friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; } }; //////////////////////////////////////////////////////////////////////////////// constexpr unsigned MO = 998244353; using Mint = ModInt; constexpr int LIM_INV = 400'010; Mint inv[LIM_INV], fac[LIM_INV], invFac[LIM_INV]; void prepare() { inv[1] = 1; for (int i = 2; i < LIM_INV; ++i) { inv[i] = -((Mint::M / i) * inv[Mint::M % i]); } fac[0] = invFac[0] = 1; for (int i = 1; i < LIM_INV; ++i) { fac[i] = fac[i - 1] * i; invFac[i] = invFac[i - 1] * inv[i]; } } Mint binom(Int n, Int k) { if (n < 0) { if (k >= 0) { return ((k & 1) ? -1 : +1) * binom(-n + k - 1, k); } else if (n - k >= 0) { return (((n - k) & 1) ? -1 : +1) * binom(-k - 1, n - k); } else { return 0; } } else { if (0 <= k && k <= n) { assert(n < LIM_INV); return fac[n] * invFac[k] * invFac[n - k]; } else { return 0; } } } int T; vector N, M; int main() { prepare(); for (; ~scanf("%d", &T); ) { N.resize(T); M.resize(T); for (int t = 0; t < T; ++t) { scanf("%d%d", &N[t], &M[t]); } vector ans(T); for (int t = 0; t < T; ++t) { ans[t] = Mint(2).pow(N[t]) - 1; --N[t]; --M[t]; } const int maxN = *max_element(N.begin(), N.end()); const int B = max((int)(maxN / sqrt(T)), 1); vector ts(T); for (int t = 0; t < T; ++t) { ts[t] = t; } sort(ts.begin(), ts.end(), [&](int t0, int t1) -> bool { const int xa = M[t0] / B; const int xb = M[t1] / B; return ((xa != xb) ? (xa < xb) : (xa & 1) ? (N[t0] > N[t1]) : (N[t0] < N[t1])); }); Mint x = 1, y = 1; int m = 0, n = 0; for (int q = 0; q < T; ++q) { const int t = ts[q]; for (; m > M[t]; ) { --m; x -= y; y *= inv[n - m] * (m + 1); // cerr<<__LINE__<<"; "< N[t]; ) { --n; y *= inv[n + 1] * (n - m + 1); x = inv[2] * (x + y); // cerr<<__LINE__<<"; "<