import std.conv, std.functional, std.range, std.stdio, std.string; import std.algorithm, std.array, std.bigint, std.bitmanip, std.complex, std.container, std.math, std.mathspecial, std.numeric, std.regex, std.typecons; import core.bitop; class EOFException : Throwable { this() { super("EOF"); } } string[] tokens; string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; } int readInt() { return readToken.to!int; } long readLong() { return readToken.to!long; } real readReal() { return readToken.to!real; } bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } } bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } } int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; } int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); } int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); } struct ModInt(int M_) { import std.conv : to; alias M = M_; int x; this(ModInt a) { x = a.x; } this(long a) { x = cast(int)(a % M); if (x < 0) x += M; } ref ModInt opAssign(long a) { return (this = ModInt(a)); } ref ModInt opOpAssign(string op)(ModInt a) { static if (op == "+") { x += a.x; if (x >= M) x -= M; } else static if (op == "-") { x -= a.x; if (x < 0) x += M; } else static if (op == "*") { x = cast(int)((cast(long)(x) * a.x) % M); } else static if (op == "/") { this *= a.inv(); } else static assert(false); return this; } ref ModInt opOpAssign(string op)(long a) { static if (op == "^^") { if (a < 0) return (this = inv()^^(-a)); ModInt t2 = this, te = ModInt(1); for (long e = a; e > 0; e >>= 1) { if (e & 1) te *= t2; t2 *= t2; } x = cast(int)(te.x); return this; } else return mixin("this " ~ op ~ "= ModInt(a)"); } ModInt inv() const { int a = x, b = M, y = 1, z = 0, t; for (; ; ) { t = a / b; a -= t * b; if (a == 0) { assert(b == 1 || b == -1); return ModInt(b * z); } y -= t * z; t = b / a; b -= t * a; if (b == 0) { assert(a == 1 || a == -1); return ModInt(a * y); } z -= t * y; } } ModInt opUnary(string op: "-")() const { return ModInt(-x); } ModInt opBinary(string op, T)(T a) const { return mixin("ModInt(this) " ~ op ~ "= a"); } ModInt opBinaryRight(string op)(long a) const { return mixin("ModInt(a) " ~ op ~ "= this"); } bool opCast(T: bool)() const { return (x != 0); } string toString() const { return x.to!string; } } enum MO = 998244353; alias Mint = ModInt!MO; enum LIM = 2 * 10^^5 + 100; Mint[] inv, fac, invFac; void prepare() { inv = new Mint[LIM]; fac = new Mint[LIM]; invFac = new Mint[LIM]; inv[1] = 1; foreach (i; 2 .. LIM) { inv[i] = -(Mint.M / i) * inv[cast(size_t)(Mint.M % i)]; } fac[0] = invFac[0] = 1; foreach (i; 1 .. LIM) { fac[i] = fac[i - 1] * i; invFac[i] = invFac[i - 1] * inv[i]; } } Mint binom(long n, long k) { if (0 <= k && k <= n) { assert(n < LIM); return fac[cast(size_t)(n)] * invFac[cast(size_t)(k)] * invFac[cast(size_t)(n - k)]; } else { return Mint(0); } } int root(int[] uf, int u) { return (uf[u] < 0) ? u : (uf[u] = uf.root(uf[u])); } bool connect(int[] uf, int u, int v) { u = uf.root(u); v = uf.root(v); if (u == v) return false; if (uf[u] > uf[v]) swap(u, v); uf[u] += uf[v]; uf[v] = u; return true; } void main() { prepare; auto two = new Mint[LIM]; two[0] = 1; foreach (i; 1 .. LIM) { two[i] = two[i - 1] * 2; } try { for (; ; ) { const N = readInt(); const M = readInt(); auto A = new int[M]; auto B = new int[M]; foreach (i; 0 .. M) { A[i] = readInt() - 1; B[i] = readInt() - 1; } int[] us = A.dup ~ B.dup; us = us.sort.uniq.array; const usLen = cast(int)(us.length); foreach (i; 0 .. M) { A[i] = us.lowerBound(A[i]); B[i] = us.lowerBound(B[i]); } Mint ans; // 0 foreach (k; 3 .. N + 1) { ans += binom(N, k) * fac[k - 1] * inv[2]; } // 1 foreach (j; 1 .. (N - 2) + 1) { ans -= M * (binom(N - 2, j) * fac[j]); debug { writeln(j, ": ", binom(N - 2, j) * fac[j]); } } auto mem = new bool[][](2 * M + 1, 2 * M + 1); auto cache = new Mint[][](2 * M + 1, 2 * M + 1); Mint calc(int x, int y) { Mint ret; if (!mem[x][y]) { foreach (j; 0 .. (N - y) + 1) { ret += binom(N - y, j) * two[x - 1] * fac[x + j - 1]; } debug { writefln("calc %s %s = %s", x, y, ret); } mem[x][y] = true; cache[x][y] = ret; } else { ret = cache[x][y]; } return ret; } foreach (p; 0 .. 1 << M) { if (popcnt(p) >= 2) { auto uf = new int[usLen]; uf[] = -1; auto deg = new int[usLen]; foreach (i; 0 .. M) { if (p & 1 << i) { uf.connect(A[i], B[i]); ++deg[A[i]]; ++deg[B[i]]; } } auto uss = new int[][usLen]; foreach (u; 0 .. usLen) { uss[uf.root(u)] ~= u; } int numComps, numVertices; int numCycles; foreach (r; 0 .. usLen) { if (uss[r].length >= 2) { numComps += 1; numVertices += uss[r].length; if (uss[r].any!(u => (deg[u] >= 3))) { goto failed; } if (uss[r].all!(u => (deg[u] == 2))) { ++numCycles; } } } { Mint num; switch (numCycles) { case 0: { num = calc(numComps, numVertices); } break; case 1: { num = 1; } break; default: {} } ans += (-1)^^(popcnt(p) & 1) * num; } failed: } } writeln(ans); } } catch (EOFException e) { } }