#include #include #include #include #define rep(i,a,b) for(int i=(a);i<(b);i++) #define rrep(i,a,b) for(int i=(b)-1;i>=(a);i--) #define all(x) begin(x),end(x) #define sz(x) (int)(x).size() using namespace std; using namespace atcoder; using ll=long long; using ld=long double; using vll=vector; using vvll=vector; using vi=vector; using vvi=vector; using pll=pair; using pi=pair; // using mint=modint998244353; // using BoostPoint=boost::polygon::point_data; // using Voronoi=boost::polygon::voronoi_diagram; #line 2 "modint/montgomery-modint.hpp" template struct LazyMontgomeryModInt { using mint = LazyMontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 ret = mod; for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret; return ret; } static constexpr u32 r = get_r(); static constexpr u32 n2 = -u64(mod) % mod; static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30"); static_assert((mod & 1) == 1, "invalid, mod % 2 == 0"); static_assert(r * mod == 1, "this code has bugs."); u32 a; constexpr LazyMontgomeryModInt() : a(0) {} constexpr LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)){}; static constexpr u32 reduce(const u64 &b) { return (b + u64(u32(b) * u32(-r)) * mod) >> 32; } constexpr mint &operator+=(const mint &b) { if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } constexpr mint &operator-=(const mint &b) { if (i32(a -= b.a) < 0) a += 2 * mod; return *this; } constexpr mint &operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } constexpr mint &operator/=(const mint &b) { *this *= b.inverse(); return *this; } constexpr mint operator+(const mint &b) const { return mint(*this) += b; } constexpr mint operator-(const mint &b) const { return mint(*this) -= b; } constexpr mint operator*(const mint &b) const { return mint(*this) *= b; } constexpr mint operator/(const mint &b) const { return mint(*this) /= b; } constexpr bool operator==(const mint &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } constexpr bool operator!=(const mint &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } constexpr mint operator-() const { return mint() - mint(*this); } constexpr mint operator+() const { return mint(*this); } constexpr mint pow(u64 n) const { mint ret(1), mul(*this); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } constexpr mint inverse() const { int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0; while (y > 0) { t = x / y; x -= t * y, u -= t * v; tmp = x, x = y, y = tmp; tmp = u, u = v, v = tmp; } return mint{u}; } friend ostream &operator<<(ostream &os, const mint &b) { return os << b.get(); } friend istream &operator>>(istream &is, mint &b) { int64_t t; is >> t; b = LazyMontgomeryModInt(t); return (is); } constexpr u32 get() const { u32 ret = reduce(a); return ret >= mod ? ret - mod : ret; } static constexpr u32 get_mod() { return mod; } }; #line 2 "dp/inversion-counting.hpp" #line 2 "data-structure/binary-indexed-tree.hpp" template struct BinaryIndexedTree { int N; vector data; BinaryIndexedTree() = default; BinaryIndexedTree(int size) { init(size); } void init(int size) { N = size + 2; data.assign(N + 1, {}); } // get sum of [0,k] T sum(int k) const { if (k < 0) return T{}; // return 0 if k < 0 T ret{}; for (++k; k > 0; k -= k & -k) ret += data[k]; return ret; } // getsum of [l,r] inline T sum(int l, int r) const { return sum(r) - sum(l - 1); } // get value of k inline T operator[](int k) const { return sum(k) - sum(k - 1); } // data[k] += x void add(int k, T x) { for (++k; k < N; k += k & -k) data[k] += x; } // range add x to [l,r] void imos(int l, int r, T x) { add(l, x); add(r + 1, -x); } // minimize i s.t. sum(i) >= w int lower_bound(T w) { if (w <= 0) return 0; int x = 0; for (int k = 1 << __lg(N); k; k >>= 1) { if (x + k <= N - 1 && data[x + k] < w) { w -= data[x + k]; x += k; } } return x; } // minimize i s.t. sum(i) > w int upper_bound(T w) { if (w < 0) return 0; int x = 0; for (int k = 1 << __lg(N); k; k >>= 1) { if (x + k <= N - 1 && data[x + k] <= w) { w -= data[x + k]; x += k; } } return x; } }; /** * @brief Binary Indexed Tree(Fenwick Tree) * @docs docs/data-structure/binary-indexed-tree.md */ #line 4 "dp/inversion-counting.hpp" // 転倒数 template long long inversion_counting(const vector& v) { vector xs{v}; sort(begin(xs), end(xs)); xs.erase(unique(begin(xs), end(xs)), end(xs)); int s = xs.size(); BinaryIndexedTree bit(s + 1); long long ans = 0; for (auto& x : v) { int i = lower_bound(begin(xs), end(xs), x) - begin(xs); if (i + 1 != s) ans += bit.sum(i + 1, s - 1); bit.add(i, 1); } return ans; } // 隣接 swap によって v を w に変えるのにかかる手数 (不可能 : -1) template long long swap_distance(const vector& v, const vector& w) { if (v.size() != w.size()) return -1; int N = v.size(); vector> vv(N), ww(N); for (int i = 0; i < N; i++) { vv[i] = make_pair(v[i], i); ww[i] = make_pair(w[i], i); } sort(begin(vv), end(vv)); sort(begin(ww), end(ww)); for (int i = 0; i < N; i++) { if (vv[i].first != ww[i].first) return -1; } vector order(N); for (int i = 0; i < N; i++) { order[vv[i].second] = ww[i].second; } return inversion_counting(order); } #line 2 "modulo/binomial.hpp" #include #include #include using namespace std; // コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」 // を入れると倍速くらいになる // mod を超えて前計算して 0 割りを踏むバグは対策済み template struct Binomial { vector f, g, h; Binomial(int MAX = 0) { assert(T::get_mod() != 0 && "Binomial()"); f.resize(1, T{1}); g.resize(1, T{1}); h.resize(1, T{1}); if (MAX > 0) extend(MAX + 1); } void extend(int m = -1) { int n = f.size(); if (m == -1) m = n * 2; m = min(m, T::get_mod()); if (n >= m) return; f.resize(m); g.resize(m); h.resize(m); for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i); g[m - 1] = f[m - 1].inverse(); h[m - 1] = g[m - 1] * f[m - 2]; for (int i = m - 2; i >= n; i--) { g[i] = g[i + 1] * T(i + 1); h[i] = g[i] * f[i - 1]; } } T fac(int i) { if (i < 0) return T(0); while (i >= (int)f.size()) extend(); return f[i]; } T finv(int i) { if (i < 0) return T(0); while (i >= (int)g.size()) extend(); return g[i]; } T inv(int i) { if (i < 0) return -inv(-i); while (i >= (int)h.size()) extend(); return h[i]; } T C(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); return fac(n) * finv(n - r) * finv(r); } inline T operator()(int n, int r) { return C(n, r); } template T multinomial(const vector& r) { static_assert(is_integral::value == true); int n = 0; for (auto& x : r) { if (x < 0) return T(0); n += x; } T res = fac(n); for (auto& x : r) res *= finv(x); return res; } template T operator()(const vector& r) { return multinomial(r); } T C_naive(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); T ret = T(1); r = min(r, n - r); for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--); return ret; } T P(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); return fac(n) * finv(n - r); } // [x^r] 1 / (1-x)^n T H(int n, int r) { if (n < 0 || r < 0) return T(0); return r == 0 ? 1 : C(n + r - 1, r); } }; using mint=LazyMontgomeryModInt<998244353>; int main() { ios::sync_with_stdio(false); cin.tie(nullptr); int N,M; cin>>N>>M; vectorconstraints(M); vi fixed(N+1,0),cp(N+1,0),ck(N+1,0); rep(i,0,M){ int p,k; cin>>p>>k; fixed[k]=p; constraints[i]={p,k}; cp[p]=1; //いもす ck[k]=1; } rep(i,1,N+1){ cp[i]+=cp[i-1]; ck[i]+=ck[i-1]; } vi v; rep(i,1,N+1){ if(fixed[i])v.push_back(fixed[i]); } mint fixed_inv=inversion_counting(v); int r=N-M; Binomialcomb(r); mint ans=comb.fac(r)*fixed_inv; if(r>0){ mint R=r,sum=0; for(auto [p,k]:constraints){ mint a=p-1-cp[p-1],b=k-1-ck[k-1]; sum+=a*(R-b)+b*(R-a); } ans+=comb.fac(r-1)*sum; } cout<