#include #include #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; } #define COLOR(s) ("\x1b[" s "m") //////////////////////////////////////////////////////////////////////////////// // Barrett struct ModInt { static unsigned M; static unsigned long long NEG_INV_M; static void setM(unsigned m) { M = m; NEG_INV_M = -1ULL / M; } unsigned x; ModInt() : x(0U) {} ModInt(unsigned x_) : x(x_ % M) {} ModInt(unsigned long long x_) : x(x_ % M) {} ModInt(int x_) : x(((x_ %= static_cast(M)) < 0) ? (x_ + static_cast(M)) : x_) {} 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) { const unsigned long long y = static_cast(x) * a.x; const unsigned long long q = static_cast((static_cast(NEG_INV_M) * y) >> 64); const unsigned long long r = y - M * q; x = r - M * (r >= 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; } }; unsigned ModInt::M; unsigned long long ModInt::NEG_INV_M; // !!!Use ModInt::setM!!! //////////////////////////////////////////////////////////////////////////////// using Mint = ModInt; vector pw; vector uf; vector> his; Mint now; void init(int n) { uf.assign(n, -1); his.clear(); now = n * pw[1]; } void update(int u, int p) { his.emplace_back(u, uf[u]); uf[u] = p; } int snapshot() { return his.size(); } void rollback(int stamp) { for (; (int)his.size() > stamp; his.pop_back()) uf[his.back().first] = his.back().second; } int root(int u) { for (; ; u = uf[u]) { if (uf[u] < 0) { return u; } } } bool connect(int u, int v) { u = root(u); v = root(v); if (u == v) return false; if (uf[u] > uf[v]) swap(u, v); now -= pw[-uf[u]]; now -= pw[-uf[v]]; now += pw[-(uf[u] + uf[v])]; update(u, uf[u] + uf[v]); update(v, u); return true; } int M, N, MO; Int K; int id(int x, int y) { return x * N + y; } Mint ans; bool a[30][30]; void dfs(int x, int y, int ign) { if (y == N) { ++x; y = 0; } if (x == M) { ans += (now - ign * pw[1]); return; } dfs(x, y + 1, ign + 1); a[x][y] = true; const auto stamp = snapshot(); const Mint save = now; if (x && a[x - 1][y]) connect(id(x - 1, y), id(x, y)); if (y && a[x][y - 1]) connect(id(x, y - 1), id(x, y)); dfs(x, y + 1, ign); a[x][y] = false; rollback(stamp); now = save; } int main() { for (; ~scanf("%d%d%lld%d", &M, &N, &K, &MO); ) { Mint::setM(MO); pw.resize(M*N + 1); for (int i = 0; i <= M*N; ++i) pw[i] = Mint(i).pow(K); init(M*N); ans = 0; memset(a, 0, sizeof(a)); dfs(0, 0, 0); ans /= Mint(2).pow(M*N); Mint har = 0; for (int i = 1; i <= M*N; ++i) har += Mint(i).inv(); ans *= har; printf("%u\n", ans.x); } return 0; }