#line 1 "library/Template/template.hpp" #include using namespace std; #define rep(i, a, b) for (int i = (int)(a); i < (int)(b); i++) #define rrep(i, a, b) for (int i = (int)(b)-1; i >= (int)(a); i--) #define ALL(v) (v).begin(), (v).end() #define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end()) #define SZ(v) (int)v.size() #define MIN(v) *min_element(ALL(v)) #define MAX(v) *max_element(ALL(v)) #define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin()) #define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin()) using uint = unsigned int; using ll = long long int; using ull = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; const int inf = 0x3fffffff; const ll INF = 0x1fffffffffffffff; template inline bool chmax(T &a, T b) { if (a < b) { a = b; return 1; } return 0; } template inline bool chmin(T &a, T b) { if (a > b) { a = b; return 1; } return 0; } template T ceil(T x, U y) { assert(y != 0); if (y < 0) x = -x, y = -y; return (x > 0 ? (x + y - 1) / y : x / y); } template T floor(T x, U y) { assert(y != 0); if (y < 0) x = -x, y = -y; return (x > 0 ? x / y : (x - y + 1) / y); } template int popcnt(T x) { return __builtin_popcountll(x); } template int topbit(T x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } template int lowbit(T x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template ostream &operator<<(ostream &os, const pair &p) { os << "P(" << p.first << ", " << p.second << ")"; return os; } template ostream &operator<<(ostream &os, const vector &vec) { os << "{"; for (int i = 0; i < vec.size(); i++) { os << vec[i] << (i + 1 == vec.size() ? "" : ", "); } os << "}"; return os; } template ostream &operator<<(ostream &os, const map &map_var) { os << "{"; for (auto itr = map_var.begin(); itr != map_var.end(); itr++) { os << "(" << itr->first << ", " << itr->second << ")"; itr++; if (itr != map_var.end()) os << ", "; itr--; } os << "}"; return os; } template ostream &operator<<(ostream &os, const set &set_var) { os << "{"; for (auto itr = set_var.begin(); itr != set_var.end(); itr++) { os << *itr; ++itr; if (itr != set_var.end()) os << ", "; itr--; } os << "}"; return os; } #ifdef LOCAL #define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__) #else #define show(...) true #endif template void _show(int i, T name) { cerr << '\n'; } template void _show(int i, const T1 &a, const T2 &b, const T3 &...c) { for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i]; cerr << ":" << b << " "; _show(i + 1, a, c...); } #line 2 "library/Utility/fastio.hpp" #include namespace fastio { static constexpr uint32_t SZ = 1 << 17; char ibuf[SZ]; char obuf[SZ]; char out[100]; // pointer of ibuf, obuf uint32_t pil = 0, pir = 0, por = 0; struct Pre { char num[10000][4]; constexpr Pre() : num() { for (int i = 0; i < 10000; i++) { int n = i; for (int j = 3; j >= 0; j--) { num[i][j] = n % 10 | '0'; n /= 10; } } } } constexpr pre; inline void load() { memmove(ibuf, ibuf + pil, pir - pil); pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin); pil = 0; if (pir < SZ) ibuf[pir++] = '\n'; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } void rd(char &c) { do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); } void rd(string &x) { x.clear(); char c; do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); do { x += c; if (pil == pir) load(); c = ibuf[pil++]; } while (!isspace(c)); } template void rd_real(T &x) { string s; rd(s); x = stod(s); } template void rd_integer(T &x) { if (pil + 100 > pir) load(); char c; do c = ibuf[pil++]; while (c < '-'); bool minus = 0; if constexpr (is_signed::value || is_same_v) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed::value || is_same_v) { if (minus) x = -x; } } void rd(int &x) { rd_integer(x); } void rd(ll &x) { rd_integer(x); } void rd(i128 &x) { rd_integer(x); } void rd(uint &x) { rd_integer(x); } void rd(ull &x) { rd_integer(x); } void rd(u128 &x) { rd_integer(x); } void rd(double &x) { rd_real(x); } void rd(long double &x) { rd_real(x); } template void rd(pair &p) { return rd(p.first), rd(p.second); } template void rd_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); rd(x); rd_tuple(t); } } template void rd(tuple &tpl) { rd_tuple(tpl); } template void rd(array &x) { for (auto &d : x) rd(d); } template void rd(vector &x) { for (auto &d : x) rd(d); } void read() {} template void read(H &h, T &...t) { rd(h), read(t...); } void wt(const char c) { if (por == SZ) flush(); obuf[por++] = c; } void wt(const string s) { for (char c : s) wt(c); } void wt(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) wt(s[i]); } template void wt_integer(T x) { if (por > SZ - 100) flush(); if (x < 0) { obuf[por++] = '-', x = -x; } int outi; for (outi = 96; x >= 10000; outi -= 4) { memcpy(out + outi, pre.num[x % 10000], 4); x /= 10000; } if (x >= 1000) { memcpy(obuf + por, pre.num[x], 4); por += 4; } else if (x >= 100) { memcpy(obuf + por, pre.num[x] + 1, 3); por += 3; } else if (x >= 10) { int q = (x * 103) >> 10; obuf[por] = q | '0'; obuf[por + 1] = (x - q * 10) | '0'; por += 2; } else obuf[por++] = x | '0'; memcpy(obuf + por, out + outi + 4, 96 - outi); por += 96 - outi; } template void wt_real(T x) { ostringstream oss; oss << fixed << setprecision(15) << double(x); string s = oss.str(); wt(s); } void wt(int x) { wt_integer(x); } void wt(ll x) { wt_integer(x); } void wt(i128 x) { wt_integer(x); } void wt(uint x) { wt_integer(x); } void wt(ull x) { wt_integer(x); } void wt(u128 x) { wt_integer(x); } void wt(double x) { wt_real(x); } void wt(long double x) { wt_real(x); } template void wt(const pair val) { wt(val.first); wt(' '); wt(val.second); } template void wt_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get(t); wt(x); wt_tuple(t); } } template void wt(tuple tpl) { wt_tuple(tpl); } template void wt(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template void wt(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template void print(Head &&head, Tail &&...tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward(tail)...); } void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::flush; using fastio::print; using fastio::read; inline void first(bool i = true) { print(i ? "first" : "second"); } inline void Alice(bool i = true) { print(i ? "Alice" : "Bob"); } inline void Takahashi(bool i = true) { print(i ? "Takahashi" : "Aoki"); } inline void yes(bool i = true) { print(i ? "yes" : "no"); } inline void Yes(bool i = true) { print(i ? "Yes" : "No"); } inline void No() { print("No"); } inline void YES(bool i = true) { print(i ? "YES" : "NO"); } inline void NO() { print("NO"); } inline void Yay(bool i = true) { print(i ? "Yay!" : ":("); } inline void Possible(bool i = true) { print(i ? "Possible" : "Impossible"); } inline void POSSIBLE(bool i = true) { print(i ? "POSSIBLE" : "IMPOSSIBLE"); } /** * @brief Fast IO */ #line 3 "sol.cpp" #line 2 "library/Math/fastdiv.hpp" struct FastDiv{ using u64=uint64_t; using u128=__uint128_t; constexpr FastDiv():m(),s(),x(){} constexpr FastDiv(int _m) :m(_m),s(__lg(m-1)),x(((u128(1)<<(s+64))+m-1)/m){} constexpr int get(){return m;} constexpr friend u64 operator/(u64 n,const FastDiv& d){ return (u128(n)*d.x>>d.s)>>64; } constexpr friend int operator%(u64 n,const FastDiv& d){ return n-n/d*d.m; } constexpr pair divmod(u64 n)const{ u64 q=n/(*this); return {q,n-q*m}; } int m,s; u64 x; }; /** * @brief Fast Division */ #line 3 "library/Math/dynamic.hpp" struct Fp { using u64 = uint64_t; uint v; static int get_mod() { return _getmod(); } static void set_mod(int _m) { bar = FastDiv(_m); } Fp inv() const { int tmp, a = v, b = get_mod(), x = 1, y = 0; while (b) { tmp = a / b, a -= tmp * b; swap(a, b); x -= tmp * y; swap(x, y); } if (x < 0) { x += get_mod(); } return x; } Fp() : v(0) {} Fp(ll x) { v = x % get_mod(); if (v < 0) v += get_mod(); } Fp operator-() const { return Fp() - *this; } Fp pow(ll t) { assert(t >= 0); Fp res = 1, b = *this; while (t) { if (t & 1) res *= b; b *= b; t >>= 1; } return res; } Fp &operator+=(const Fp &x) { v += x.v; if (v >= get_mod()) v -= get_mod(); return *this; } Fp &operator-=(const Fp &x) { v += get_mod() - x.v; if (v >= get_mod()) v -= get_mod(); return *this; } Fp &operator*=(const Fp &x) { v = (u64(v) * x.v) % bar; return *this; } Fp &operator/=(const Fp &x) { (*this) *= x.inv(); return *this; } Fp operator+(const Fp &x) const { return Fp(*this) += x; } Fp operator-(const Fp &x) const { return Fp(*this) -= x; } Fp operator*(const Fp &x) const { return Fp(*this) *= x; } Fp operator/(const Fp &x) const { return Fp(*this) /= x; } bool operator==(const Fp &x) const { return v == x.v; } bool operator!=(const Fp &x) const { return v != x.v; } friend istream &operator>>(istream &is, Fp &x) { return is >> x.v; } friend ostream &operator<<(ostream &os, const Fp &x) { return os << x.v; } private: static FastDiv bar; static int _getmod() { return bar.get(); } }; FastDiv Fp::bar(998244353); void rd(Fp &x) { fastio::rd(x.v); } void wt(Fp x) { fastio::wt(x.v); } /** * @brief Dynamic Modint */ #line 2 "library/Math/comb.hpp" template T Inv(ll n) { static int md; static vector buf({0, 1}); if (md != T::get_mod()) { md = T::get_mod(); buf = vector({0, 1}); } assert(n > 0); n %= md; while (SZ(buf) <= n) { int k = SZ(buf), q = (md + k - 1) / k; buf.push_back(buf[k * q - md] * q); } return buf[n]; } template T Fact(ll n, bool inv = 0) { static int md; static vector buf({1, 1}), ibuf({1, 1}); if (md != T::get_mod()) { md = T::get_mod(); buf = ibuf = vector({1, 1}); } assert(n >= 0 and n < md); while (SZ(buf) <= n) { buf.push_back(buf.back() * SZ(buf)); ibuf.push_back(ibuf.back() * Inv(SZ(ibuf))); } return inv ? ibuf[n] : buf[n]; } template T nPr(int n, int r, bool inv = 0) { if (n < 0 || n < r || r < 0) return 0; return Fact(n, inv) * Fact(n - r, inv ^ 1); } template T nCr(int n, int r, bool inv = 0) { if (n < 0 || n < r || r < 0) return 0; return Fact(n, inv) * Fact(r, inv ^ 1) * Fact(n - r, inv ^ 1); } // sum = n, r tuples template T nHr(int n, int r, bool inv = 0) { return nCr(n + r - 1, r - 1, inv); } // sum = n, a nonzero tuples and b tuples template T choose(int n, int a, int b) { if (n == 0) return !a; return nCr(n + b - 1, a + b - 1); } /** * @brief Combination */ #line 2 "library/DataStructure/unionfind.hpp" struct UnionFind{ vector par; int n; UnionFind(){} UnionFind(int _n):par(_n,-1),n(_n){} int root(int x){return par[x]<0?x:par[x]=root(par[x]);} bool same(int x,int y){return root(x)==root(y);} int size(int x){return -par[root(x)];} bool unite(int x,int y){ x=root(x),y=root(y); if(x==y)return false; if(size(x)>size(y))swap(x,y); par[y]+=par[x]; par[x]=y; n--; return true; } }; /** * @brief Union Find */ #line 7 "sol.cpp" using P = pair; map> mp = { {{1, 1}, {0, 1}}, {{1, 2}, {0, 2, 1}}, {{1, 3}, {0, 5, 2, 1}}, {{1, 4}, {0, 12, 5, 2, 1}}, {{1, 5}, {0, 28, 12, 5, 2, 1}}, {{1, 6}, {0, 64, 28, 12, 5, 2, 1}}, {{1, 7}, {0, 144, 64, 28, 12, 5, 2, 1}}, {{1, 8}, {0, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 9}, {0, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 10}, {0, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 11}, {0, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 12}, {0, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 13}, {0, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 14}, {0, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 15}, {0, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 16}, {0, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 17}, {0, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 18}, {0, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 19}, {0, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 20}, {0, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 21}, {0, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 22}, {0, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 23}, {0, 26214400, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 24}, {0, 54525952, 26214400, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{1, 25}, {0, 113246208, 54525952, 26214400, 12582912, 6029312, 2883584, 1376256, 655360, 311296, 147456, 69632, 32768, 15360, 7168, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1}}, {{2, 2}, {0, 8, 4, 4, 1}}, {{2, 3}, {0, 40, 17, 14, 10, 6, 1}}, {{2, 4}, {0, 192, 80, 64, 39, 28, 20, 8, 1}}, {{2, 5}, {0, 896, 368, 296, 184, 120, 81, 58, 34, 10, 1}}, {{2, 6}, {0, 4096, 1664, 1344, 856, 560, 368, 240, 167, 112, 52, 12, 1}}, {{2, 7}, {0, 18432, 7424, 6016, 3904, 2568, 1712, 1112, 728, 488, 337, 198, 74, 14, 1}}, {{2, 8}, {0, 81920, 32768, 26624, 17536, 11584, 7816, 5136, 3384, 2208, 1456, 992, 647, 324, 100, 16, 1}}, {{2, 9}, {0, 360448, 143360, 116736, 77824, 51584, 35136, 23304, 15520, 10232, 6704, 4392, 2936, 1976, 1169, 498, 130, 18, 1}}, {{2, 10}, {0, 1572864, 622592, 507904, 342016, 227328, 156032, 104256, 70024, 46640, 30888, 20320, 13304, 8784, 5904, 3792, 1991, 728, 164, 20, 1}}, {{2, 11}, {0, 6815744, 2686976, 2195456, 1490944, 993280, 686080, 461184, 311872, 209416, 139968, 93048, 61440, 40328, 26480, 17624, 11672, 6952, 3217, 1022, 202, 22, 1}}, {{2, 12}, {0, 29360128, 11534336, 9437184, 6455296, 4308992, 2992128, 2021376, 1374592, 929088, 625544, 419408, 279656, 185376, 122088, 80112, 52888, 35200, 22416, 12160, 4967, 1388, 244, 24, 1}}, {{3, 3}, {0, 400, 144, 106, 84, 69, 52, 32, 9, 1}}, {{3, 4}, {0, 3840, 1360, 952, 705, 550, 445, 360, 270, 164, 62, 12, 1}}, {{3, 5}, {0, 35840, 12544, 8720, 6408, 4897, 3772, 2972, 2371, 1873, 1385, 854, 376, 101, 15, 1}}, {{3, 6}, {0, 327680, 113664, 78592, 57936, 44456, 34161, 26334, 20329, 15846, 12510, 9758, 7131, 4478, 2164, 714, 149, 18, 1}}, {{3, 7}, {0, 2949120, 1015808, 699392, 516864, 398928, 308296, 238385, 183352, 140476, 108244, 84130, 65926, 50898, 36946, 23558, 12123, 4602, 1205, 206, 21, 1}}, {{3, 8}, {0, 26214400, 8978432, 6160384, 4561920, 3537664, 2748496, 2136040, 1652625, 1269794, 973997, 746362, 574842, 446028, 347449, 266040, 192322, 124150, 66909, 28072, 8654, 1876, 272, 24, 1}}, {{4, 4}, {0, 73728, 25600, 17152, 11728, 9040, 7504, 6288, 5340, 4436, 3432, 2360, 1296, 492, 116, 16, 1}}, {{4, 5}, {0, 1376256, 471040, 314368, 213248, 158976, 128272, 105936, 89480, 75998, 64363, 53486, 42473, 31078, 19942, 10358, 3972, 1054, 186, 20, 1}}, {{4, 6}, {0, 25165824, 8519680, 5668864, 3856384, 2863104, 2288896, 1868032, 1553240, 1299184, 1090650, 919360, 774295, 643000, 517972, 394544, 274893, 167464, 83894, 32492, 9316, 1920, 272, 24, 1}}, {{5, 5}, {0, 51380224, 17301504, 11501568, 7749632, 5599232, 4361216, 3564416, 3023760, 2603088, 2265120, 1971974, 1703398, 1448460, 1194986, 939036, 687581, 454452, 258314, 119182, 42437, 11295, 2192, 296, 25, 1}}}; int main() { int H, W; ll K, mod; read(H, W, K, mod); if(H>W)swap(H,W); Fp::set_mod(mod); vector pw(H * W + 2); rep(x, 1, H * W + 1) pw[x] = Fp(x).pow(K); Fp ret; auto cnt=mp[{H,W}]; rep(x,1,H*W+1)ret+=pw[x]*cnt[x]; Fp sub; rep(x, 1, H * W + 1) sub += Inv(x); ret *= sub; ret /= Fp(2).pow(H * W); print(ret); return 0; }