#line 2 "/Users/noya2/Desktop/Noya2_library/template/template.hpp" using namespace std; #include #line 1 "/Users/noya2/Desktop/Noya2_library/template/inout_old.hpp" namespace noya2 { template ostream &operator<<(ostream &os, const pair &p){ os << p.first << " " << p.second; return os; } template istream &operator>>(istream &is, pair &p){ is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const vector &v){ int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template istream &operator>>(istream &is, vector &v){ for (auto &x : v) is >> x; return is; } void in() {} template void in(T &t, U &...u){ cin >> t; in(u...); } void out() { cout << "\n"; } template void out(const T &t, const U &...u){ cout << t; if (sizeof...(u)) cout << sep; out(u...); } template void out(const vector> &vv){ int s = (int)vv.size(); for (int i = 0; i < s; i++) out(vv[i]); } struct IoSetup { IoSetup(){ cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetup_noya2; } // namespace noya2 #line 1 "/Users/noya2/Desktop/Noya2_library/template/const.hpp" namespace noya2{ const int iinf = 1'000'000'007; const long long linf = 2'000'000'000'000'000'000LL; const long long mod998 = 998244353; const long long mod107 = 1000000007; const long double pi = 3.14159265358979323; const vector dx = {0,1,0,-1,1,1,-1,-1}; const vector dy = {1,0,-1,0,1,-1,-1,1}; const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"; const string alp = "abcdefghijklmnopqrstuvwxyz"; const string NUM = "0123456789"; void yes(){ cout << "Yes\n"; } void no(){ cout << "No\n"; } void YES(){ cout << "YES\n"; } void NO(){ cout << "NO\n"; } void yn(bool t){ t ? yes() : no(); } void YN(bool t){ t ? YES() : NO(); } } // namespace noya2 #line 2 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp" #line 6 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp" namespace noya2{ unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){ if (a == 0 || b == 0) return a + b; int n = __builtin_ctzll(a); a >>= n; int m = __builtin_ctzll(b); b >>= m; while (a != b) { int mm = __builtin_ctzll(a - b); bool f = a > b; unsigned long long c = f ? a : b; b = f ? b : a; a = (c - b) >> mm; } return a << std::min(n, m); } template T gcd_fast(T a, T b){ return static_cast(inner_binary_gcd(std::abs(a),std::abs(b))); } long long sqrt_fast(long long n) { if (n <= 0) return 0; long long x = sqrt(n); while ((x + 1) * (x + 1) <= n) x++; while (x * x > n) x--; return x; } template T floor_div(const T n, const T d) { assert(d != 0); return n / d - static_cast((n ^ d) < 0 && n % d != 0); } template T ceil_div(const T n, const T d) { assert(d != 0); return n / d + static_cast((n ^ d) >= 0 && n % d != 0); } template void uniq(std::vector &v){ std::sort(v.begin(),v.end()); v.erase(unique(v.begin(),v.end()),v.end()); } template inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template inline bool range(T l, T x, T r){ return l <= x && x < r; } } // namespace noya2 #line 8 "/Users/noya2/Desktop/Noya2_library/template/template.hpp" #define rep(i,n) for (int i = 0; i < (int)(n); i++) #define repp(i,m,n) for (int i = (m); i < (int)(n); i++) #define reb(i,n) for (int i = (int)(n-1); i >= 0; i--) #define all(v) (v).begin(),(v).end() using ll = long long; using ld = long double; using uint = unsigned int; using ull = unsigned long long; using pii = pair; using pll = pair; using pil = pair; using pli = pair; namespace noya2{ /* ~ (. _________ . /) */ } using namespace noya2; #line 2 "c.cpp" #line 2 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp" #line 4 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp" #line 2 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp" #line 4 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp" namespace noya2 { constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } constexpr long long pow_mod_constexpr(long long x, long long n, int m) { if (m == 1) return 0; unsigned int _m = (unsigned int)(m); unsigned long long r = 1; unsigned long long y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; long long d = n - 1; while (d % 2 == 0) d /= 2; constexpr long long bases[3] = {2, 7, 61}; for (long long a : bases) { long long t = d; long long y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template constexpr bool is_prime_flag = is_prime_constexpr(n); constexpr std::pair inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; long long s = b, t = a; long long m0 = 0, m1 = 1; while (t) { long long u = s / t; s -= t * u; m0 -= m1 * u; auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } if (m0 < 0) m0 += b / s; return {s, m0}; } constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (long long)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template constexpr int primitive_root_flag = primitive_root_constexpr(m); // constexpr long long primitive_root_constexpr(long long m){ // if (m == (1LL << 47) - (1LL << 24) + 1) return 3; // return primitive_root_constexpr(static_cast(m)); // } } // namespace noya2 #line 6 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp" namespace noya2{ struct barrett { unsigned int _m; unsigned long long im; explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {} unsigned int umod() const { return _m; } unsigned int mul(unsigned int a, unsigned int b) const { unsigned long long z = a; z *= b; unsigned long long x = (unsigned long long)((__uint128_t(z) * im) >> 64); unsigned int v = (unsigned int)(z - x * _m); if (_m <= v) v += _m; return v; } }; template struct static_modint { using mint = static_modint; public: static constexpr int mod() { return m; } static mint raw(int v) { mint x; x._v = v; return x; } constexpr static_modint() : _v(0) {} template constexpr static_modint(T v){ long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template constexpr static_modint(T v){ _v = (unsigned int)(v % umod()); } constexpr unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } constexpr mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } constexpr mint& operator-=(const mint& rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } constexpr mint& operator*=(const mint& rhs) { unsigned long long z = _v; z *= rhs._v; _v = (uint)(z % umod()); return *this; } constexpr mint& operator/=(const mint& rhs) { return *this = *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); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } constexpr mint inv() const { if (prime) { assert(_v); return pow(umod() - 2); } else { auto eg = inv_gcd(_v, m); assert(eg.first == 1); return eg.second; } } friend constexpr mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend constexpr mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend constexpr mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend constexpr mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend constexpr bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend constexpr bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } friend std::ostream &operator<<(std::ostream &os, const mint& p) { return os << p.val(); } friend std::istream &operator>>(std::istream &is, mint &a) { long long t; is >> t; a = mint(t); return (is); } private: unsigned int _v; static constexpr unsigned int umod() { return m; } static constexpr bool prime = is_prime_flag; }; template struct dynamic_modint { using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template dynamic_modint(T v){ long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template dynamic_modint(T v){ _v = (unsigned int)(v % umod()); } uint val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v += mod() - rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator*=(const mint& rhs) { _v = bt.mul(_v, rhs._v); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { auto eg = noya2::inv_gcd(_v, mod()); assert(eg.first == 1); return eg.second; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } friend std::ostream &operator<<(std::ostream &os, const mint& p) { return os << p.val(); } friend std::istream &operator>>(std::istream &is, mint &a) { long long t; is >> t; a = mint(t); return (is); } private: unsigned int _v; static barrett bt; static unsigned int umod() { return bt.umod(); } }; template noya2::barrett dynamic_modint::bt(998244353); using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; using modint = dynamic_modint<-1>; template concept Modint = requires (T &a){ T::mod(); a.inv(); a.val(); a.pow(declval()); }; } // namespace noya2 #line 4 "c.cpp" using mint = modint; #line 2 "/Users/noya2/Desktop/Noya2_library/math/binomial.hpp" #line 4 "/Users/noya2/Desktop/Noya2_library/math/binomial.hpp" namespace noya2 { template struct binomial { binomial(int len = 300000){ extend(len); } static mint fact(int n){ if (n < 0) return 0; while (n >= (int)_fact.size()) extend(); return _fact[n]; } static mint ifact(int n){ if (n < 0) return 0; while (n >= (int)_fact.size()) extend(); return _ifact[n]; } static mint inv(int n){ return ifact(n) * fact(n-1); } static mint C(int n, int r){ if (!(0 <= r && r <= n)) return 0; return fact(n) * ifact(r) * ifact(n-r); } static mint P(int n, int r){ if (!(0 <= r && r <= n)) return 0; return fact(n) * ifact(n-r); } inline mint operator()(int n, int r) { return C(n, r); } template static mint M(const Cnts&... cnts){ return multinomial(0,1,cnts...); } static void initialize(int len = 2){ _fact.clear(); _ifact.clear(); extend(len); } private: static mint multinomial(const int& sum, const mint& div_prod){ if (sum < 0) return 0; return fact(sum) * div_prod; } template static mint multinomial(const int& sum, const mint& div_prod, const int& n1, const Tail&... tail){ if (n1 < 0) return 0; return multinomial(sum+n1,div_prod*ifact(n1),tail...); } static inline std::vector _fact, _ifact; static void extend(int len = -1){ if (_fact.empty()){ _fact = _ifact = {1,1}; } int siz = _fact.size(); if (len == -1) len = siz * 2; len = (int)min(len, mint::mod() - 1); if (len < siz) return ; _fact.resize(len+1), _ifact.resize(len+1); for (int i = siz; i <= len; i++) _fact[i] = _fact[i-1] * i; _ifact[len] = _fact[len].inv(); for (int i = len; i > siz; i--) _ifact[i-1] = _ifact[i] * i; } }; } // namespace noya2 #line 2 "/Users/noya2/Desktop/Noya2_library/graph/grid.hpp" #line 5 "/Users/noya2/Desktop/Noya2_library/graph/grid.hpp" #include namespace noya2 { struct grid { int h, w; grid (int _h = 0, int _w = 0) : h(_h), w(_w) {} int idx(int x, int y) const { return x * w + y; } std::pair pos(int id) const { return {id / w, id % w}; } bool operator()(int x, int y) const { return 0 <= x && x < h && 0 <= y && y < w; } bool operator()(int id) const { return operator()(id / w, id % w); } static constexpr std::array, 4> dxys = {std::pair{0, 1}, {-1, 0}, {0, -1}, {1, 0}}; auto adj4(int x, int y) const { return dxys | std::views::transform([x, y](std::pair dxy){ return std::pair(dxy.first + x, dxy.second + y); }) | std::views::filter([this](std::pair xy){ return operator()(xy.first, xy.second); }); } auto adj4(int id) const { return adj4(id / w, id % w) | std::ranges::views::transform([this](std::pair xy){ return idx(xy.first, xy.second); }); } }; } // namespace noya2 #line 7 "c.cpp" namespace noya2{ struct dsu { public: dsu() : _n(0) {} dsu(int n) : _n(n) { fill(all(parent_or_size),-1); } int8_t merge(int8_t a, int8_t b) { assert(0 <= a && a < _n); assert(0 <= b && b < _n); int8_t x = leader(a), y = leader(b); if (x == y) return x; if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y); parent_or_size[x] += parent_or_size[y]; parent_or_size[y] = x; return x; } bool same(int8_t a, int8_t b) { assert(0 <= a && a < _n); assert(0 <= b && b < _n); return leader(a) == leader(b); } int8_t leader(int8_t a) { assert(0 <= a && a < _n); if (parent_or_size[a] < 0) return a; return parent_or_size[a] = leader(parent_or_size[a]); } int8_t size(int8_t a) { assert(0 <= a && a < _n); return -parent_or_size[leader(a)]; } private: int _n; // root node: -1 * component size // otherwise: parent std::array parent_or_size; }; } // namespace noya2 void solve(){ int h, w; in(h,w); ll kk; in(kk); { int p; in(p); mint::set_mod(p); } binomial bnm(2); bnm.initialize(100); const int hw = h*w; mint coef = 0; const mint i2 = mint(2).inv(); for (int i = 1; i <= hw; i++){ coef += bnm.inv(hw-i+1); } coef *= i2.pow(hw); vector pows(hw+1); for (int c = 1; c <= hw; c++){ pows[c] = mint(c).pow(kk); } vector p2(hw+1); rep(i,hw+1){ p2[i] = mint(2).pow(i); } if (h < w) swap(h,w); grid g(h,w); auto connected = [&](dsu &d, int s) -> bool { if (s == 0) return false; int cnt = popcount(s); rep(id,hw){ if (d.size(id) == cnt){ return true; } } return false; }; using pdi = pair; auto proc = [&](dsu d, int s, int add, int i) -> pdi { s ^= (add << (i*w)); bool lastempty = true; rep(j,w){ int id = g.idx(i,j); if (~s >> id & 1) continue; lastempty = false; if (i > 0){ int jd = g.idx(i-1,j); if (s >> jd & 1){ d.merge(id,jd); } } if (j > 0){ int jd = g.idx(i,j-1); if (s >> jd & 1){ d.merge(id,jd); } } } if (lastempty){ if (s == 0){ return {d, s}; } return {d, ~s}; } rep(id,i*w){ if (~s >> id & 1) continue; bool con = false; rep(j,w){ int jd = g.idx(i,j); if (d.same(id,jd)){ con = true; } } if (!con){ return {d, INT_MAX}; } } return {d, s}; }; vector a = {{dsu(hw),0}}; vector adone; rep(i,h){ vector b; for (auto [d, s] : a){ rep(add,1< ex = {}; rep(i,h) rep(j,w){ int id = g.idx(i, j); if (~s >> id & 1) continue; ex[id] = true; cnt++; for (auto [ni, nj] : g.adj4(i, j)){ ex[g.idx(ni,nj)] = true; } } int other = h*w; rep(i,h) rep(j,w){ if (ex[g.idx(i,j)]){ other--; } } // out(s); ans += p2[other] * pows[cnt]; } // out(tot); // out(ans); ans *= coef; out(ans); } int main(){ int t = 1; //in(t); while (t--) { solve(); } }