Problem:
Score : 1100 points

Problem Statement

We have an H \times W grid, where each square is painted white or black in the initial state.
Given are strings A_1, A_2, ..., A_H representing the colors of the squares in the initial state.
For each pair (i, j) (1 \leq i \leq H, 1 \leq j \leq W), if the j-th character of A_i is ., the square at the i-th row and j-th column is painted white; if that character is #, that square is painted black.

Among the 2^{HW} ways for each square in the grid to be painted white or black, how many can be obtained from the initial state by performing the operations below any number of times (possibly zero) in any order? Find this count modulo 998,244,353.

Choose one row, then paint all the squares in that row white.

Choose one row, then paint all the squares in that row black.

Choose one column, then paint all the squares in that column white.

Choose one column, then paint all the squares in that column black.

Constraints

1 \leq H, W \leq 10

|A_i| = W (1 \leq i \leq H)

All strings A_i consist of . and #.

H and W are integers.

Input

Input is given from Standard Input in the following format:

H W
A_1
A_2
\vdots
A_H

Output

Print the answer.

Sample Input 1

2 2
#.
.#

Sample Output 1

15

For example, if we paint the second row black, the grid becomes:

#.
##

Sample Input 2

3 3
...
...
...

Sample Output 2

230

Sample Input 3

2 4
#...
...#

Sample Output 3

150

Sample Input 4

6 7
.......
.......
.#.....
..#....
.#.#...
.......

Sample Output 4

203949910

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
 
typedef long long ll;
typedef long double ld; 
typedef double db; 
typedef string str; 

typedef pair<int, int> pi;
typedef pair<ll,ll> pl; 
typedef pair<ld,ld> pd; 
#define mp make_pair
#define f first
#define s second

typedef vector<int> vi; 
typedef vector<ll> vl; 
typedef vector<ld> vd; 
typedef vector<str> vs; 
typedef vector<pi> vpi; 
typedef vector<pl> vpl; 
typedef vector<pd> vpd; 

#define sz(x) (int)x.size()
#define all(x) begin(x), end(x)
#define rall(x) (x).rbegin(), (x).rend() 
#define rsz resize
#define ins insert 
#define ft front() 
#define bk back() 
#define pf push_front 
#define pb push_back
#define eb emplace_back 
#define lb lower_bound 
#define ub upper_bound 

#define FOR(i,a,b) for (int i = (a); i < (b); ++i)
#define F0R(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i)
#define R0F(i,a) ROF(i,0,a)
#define trav(a,x) for (auto& a: x)

const int MOD = 998244353; // = (119<<23)+1
const int MX = 2e5+5; 
const ll INF = 1e18; 
const ld PI = acos((ld)-1);
const int xd[4] = {1,0,-1,0}, yd[4] = {0,1,0,-1}; 

template<class T> bool ckmin(T& a, const T& b) { 
	return a > b ? a = b, 1 : 0; }
template<class T> bool ckmax(T& a, const T& b) { 
	return a < b ? a = b, 1 : 0; }
int pc(int x) { return __builtin_popcount(x); } //
mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count());

#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
using namespace __gnu_pbds;

template <class T> using Tree = tree<T, null_type, less<T>, 
	rb_tree_tag, tree_order_statistics_node_update>; 
// change null_type for map
#define ook order_of_key
#define fbo find_by_order

void treeExample() {
	Tree<int> t, t2; t.insert(8);
	auto it = t.insert(10).f; assert(it == t.lb(9));
	assert(t.ook(10) == 1); assert(t.ook(11) == 2);
	assert(*t.fbo(0) == 8);
	t.join(t2); // assuming T < T2 or T > T2, merge t2 into t
}

namespace input {
	template<class T> void re(complex<T>& x);
	template<class T1, class T2> void re(pair<T1,T2>& p);
	template<class T> void re(vector<T>& a);
	template<class T, size_t SZ> void re(array<T,SZ>& a);

	template<class T> void re(T& x) { cin >> x; }
	void re(double& x) { string t; re(t); x = stod(t); }
	void re(ld& x) { string t; re(t); x = stold(t); }
	template<class T, class... Ts> void re(T& t, Ts&... ts) { 
		re(t); re(ts...); 
	}

	template<class T> void re(complex<T>& x) { T a,b; re(a,b); x = cd(a,b); }
	template<class T1, class T2> void re(pair<T1,T2>& p) { re(p.f,p.s); }
	template<class T> void re(vector<T>& a) { F0R(i,sz(a)) re(a[i]); }
	template<class T, size_t SZ> void re(array<T,SZ>& a) { F0R(i,SZ) re(a[i]); }
}

using namespace input;

namespace output {
	void pr(int x) { cout << x; }
	void pr(long x) { cout << x; }
	void pr(ll x) { cout << x; }
	void pr(unsigned x) { cout << x; }
	void pr(unsigned long x) { cout << x; }
	void pr(unsigned long long x) { cout << x; }
	void pr(float x) { cout << x; }
	void pr(double x) { cout << x; }
	void pr(ld x) { cout << x; }
	void pr(char x) { cout << x; }
	void pr(const char* x) { cout << x; }
	void pr(const string& x) { cout << x; }
	void pr(bool x) { pr(x ? "true" : "false"); }
	template<class T> void pr(const complex<T>& x) { cout << x; }
	
	template<class T1, class T2> void pr(const pair<T1,T2>& x);
	template<class T> void pr(const T& x);
	
	template<class T, class... Ts> void pr(const T& t, const Ts&... ts) { 
		pr(t); pr(ts...); 
	}
	template<class T1, class T2> void pr(const pair<T1,T2>& x) { 
		pr("{",x.f,", ",x.s,"}"); 
	}
	template<class T> void pr(const T& x) { 
		pr("{"); // const iterator needed for vector<bool>
		bool fst = 1; for (const auto& a: x) pr(!fst?", ":"",a), fst = 0; 
		pr("}");
	}
	
	void ps() { pr("\n"); } // print w/ spaces
	template<class T, class... Ts> void ps(const T& t, const Ts&... ts) { 
		pr(t); if (sizeof...(ts)) pr(" "); ps(ts...); 
	}
	
	void pc() { pr("]\n"); } // debug w/ commas
	template<class T, class... Ts> void pc(const T& t, const Ts&... ts) { 
		pr(t); if (sizeof...(ts)) pr(", "); pc(ts...); 
	}
	#define dbg(x...) pr("[",#x,"] = ["), pc(x);
}

using namespace output;

namespace io {
	void setIn(string s) { freopen(s.c_str(),"r",stdin); }
	void setOut(string s) { freopen(s.c_str(),"w",stdout); }
	void setIO(string s = "") {
		ios_base::sync_with_stdio(0); cin.tie(0); // fast I/O
		// cin.exceptions(cin.failbit); // ex. throws exception when you try to read letter into int
		if (sz(s)) { setIn(s+".in"), setOut(s+".out"); } // for USACO
	}
}

using namespace io;

struct mi {
	typedef decay<decltype(MOD)>::type T;
	T val; 
	explicit operator T() const { return val; }
	mi() { val = 0; }
	mi(ll v) { 
		val = (-MOD <= v && v <= MOD) ? v : v % MOD;
		if (val < 0) val += MOD;
	}
	friend bool operator==(const mi& a, const mi& b) { 
		return a.val == b.val; }
	friend bool operator!=(const mi& a, const mi& b) { 
		return !(a == b); }
	friend bool operator<(const mi& a, const mi& b) { 
		return a.val < b.val; }
	friend void re(mi& a) { ll x; re(x); a = mi(x); }
	friend void pr(const mi& a) { pr(a.val); }
	friend ostream& operator<<(ostream& os, const mi& a) { 
		return os << a.val; }
   
	mi operator-() const { return mi(-val); }
	mi& operator+=(const mi& m) { 
		if ((val += m.val) >= MOD) val -= MOD; 
		return *this; }
	mi& operator-=(const mi& m) { 
		if ((val -= m.val) < 0) val += MOD; 
		return *this; }
	mi& operator++() { return *this += 1; }
	mi& operator--() { return *this -= 1; }
	friend mi operator+(mi a, const mi& b) { return a += b; }
	friend mi operator-(mi a, const mi& b) { return a -= b; }

	mi& operator*=(const mi& m) { 
		val = (ll)val*m.val%MOD; return *this; }
	friend mi pow(mi a, ll p) {
		mi ans = 1; assert(p >= 0);
		for (; p; p /= 2, a *= a) if (p&1) ans *= a;
		return ans;
	}
	friend mi inv(const mi& a) { 
		assert(!(a == 0)); return pow(a,MOD-2); }
	mi& operator/=(const mi& m) { return (*this) *= inv(m); }
	friend mi operator*(mi a, const mi& b) { return a *= b; }
	friend mi operator/(mi a, const mi& b) { return a /= b; }
};
typedef pair<mi,mi> pmi;
typedef vector<mi> vmi;
typedef vector<pmi> vpmi;

mi comb[11][11], icomb[11][11];
mi p2[11];

void genComb() {
	comb[0][0] = icomb[0][0] = 1;
	FOR(i,1,11) { 
		F0R(j,i+1) {
			comb[i][j] = comb[i-1][j];
			if (j) comb[i][j] += comb[i-1][j-1];
			icomb[i][j] = 1/comb[i][j];
		}
	}
}

int H,W,ROW[10],COL[10];
mi row[11][11][3], col[11][11][3];

int main() {
	ios_base::sync_with_stdio(0); cin.tie(0);
	p2[0] = 1; FOR(i,1,11) p2[i] = 2*p2[i-1];
	genComb();
	re(H,W);
	F0R(i,H) {
		str a; re(a);
		F0R(j,W) {
			if (a[j] == '.') {
				ROW[i] |= 1<<j;
				COL[j] |= 1<<i;
			}
		}
	}
	FOR(j,1,W+1) col[H][j][2] = comb[W][j]*p2[W-j];
	mi ans = p2[W];
	ROF(i,1,H+1) ROF(j,1,W+1) {
		col[i][j][0] += col[i][j][2]+col[i][j][1];
		FOR(I,1,i) {
			int dif = i-I;
			row[I][j][1] += comb[i][dif]*(2*col[i][j][2]+col[i][j][1]);
			row[I][j][2] += comb[i][dif]*(p2[dif]-2)*col[i][j][2];
		} 
		ans += col[i][j][2]*(p2[i]-2);
		row[i][j][0] = row[i][j][2]+row[i][j][1];
		F0R(J,j) {
			int dif = j-J;
			col[i][J][1] += comb[j][dif]*(2*row[i][j][2]+row[i][j][1]);
			col[i][J][2] += comb[j][dif]*(p2[dif]-2)*row[i][j][2];
		}
		ans += row[i][j][2]*(p2[j]-2);
	}
	FOR(i,1,1<<H) FOR(j,1,1<<W) {
		bool bad = 0;
		F0R(k,H) if (i&(1<<k)) {
			int lef = ROW[k]&j;
			if (lef == 0 || lef == j) bad = 1;
		}
		F0R(k,W) if (j&(1<<k)) {
			int lef = COL[k]&i;
			if (lef == 0 || lef == i) bad = 1;
		}
		if (bad) continue;
		int x = pc(i), y = pc(j);
		ans += (row[x][y][0]+col[x][y][0])*icomb[H][x]*icomb[W][y];
	}
	ps(ans);
	/*mi ans = 0;
	FOR(i,1,H+1) ans += col[i][0][2]+col[i][0][1];
	FOR(j,1,W+1) ans += row[0][j][2]+row[0][j][1];
	ps(ans);*/
	// you should actually read the stuff at the bottom
}

/* stuff you should look for
	* int overflow, array bounds
	* special cases (n=1?), slow multiset operations
	* do smth instead of nothing and stay organized
*/