#define _USE_MATH_DEFINES #include using namespace std; #define FOR(i,m,n) for(int i=(m);i<(n);++i) #define REP(i,n) FOR(i,0,n) #define ALL(v) (v).begin(),(v).end() using ll = long long; constexpr int INF = 0x3f3f3f3f; constexpr long long LINF = 0x3f3f3f3f3f3f3f3fLL; constexpr double EPS = 1e-8; constexpr int MOD = 998244353; constexpr int dy[] = {1, 0, -1, 0}, dx[] = {0, -1, 0, 1}; constexpr int dy8[] = {1, 1, 0, -1, -1, -1, 0, 1}, dx8[] = {0, -1, -1, -1, 0, 1, 1, 1}; template inline bool chmax(T &a, U b) { return a < b ? (a = b, true) : false; } template inline bool chmin(T &a, U b) { return a > b ? (a = b, true) : false; } struct IOSetup { IOSetup() { std::cin.tie(nullptr); std::ios_base::sync_with_stdio(false); std::cout << fixed << setprecision(20); } } iosetup; template struct MInt { unsigned int val; MInt(): val(0) {} MInt(long long x) : val(x >= 0 ? x % MOD : x % MOD + MOD) {} static constexpr int get_mod() { return MOD; } static void set_mod(int divisor) { assert(divisor == MOD); } static void init(int x = 10000000) { inv(x, true); fact(x); fact_inv(x); } static MInt inv(int x, bool init = false) { // assert(0 <= x && x < MOD && std::__gcd(x, MOD) == 1); static std::vector inverse{0, 1}; int prev = inverse.size(); if (init && x >= prev) { // "x!" and "MOD" must be disjoint. inverse.resize(x + 1); for (int i = prev; i <= x; ++i) inverse[i] = -inverse[MOD % i] * (MOD / i); } if (x < inverse.size()) return inverse[x]; unsigned int a = x, b = MOD; int u = 1, v = 0; while (b) { unsigned int tmp = a / b; std::swap(a -= tmp * b, b); std::swap(u -= tmp * v, v); } return u; } static MInt fact(int x) { static std::vector f{1}; int prev = f.size(); if (x >= prev) { f.resize(x + 1); for (int i = prev; i <= x; ++i) f[i] = f[i - 1] * i; } return f[x]; } static MInt fact_inv(int x) { static std::vector finv{1}; int prev = finv.size(); if (x >= prev) { finv.resize(x + 1); finv[x] = inv(fact(x).val); for (int i = x; i > prev; --i) finv[i - 1] = finv[i] * i; } return finv[x]; } static MInt nCk(int n, int k) { if (n < 0 || n < k || k < 0) return 0; if (n - k > k) k = n - k; return fact(n) * fact_inv(k) * fact_inv(n - k); } static MInt nPk(int n, int k) { return n < 0 || n < k || k < 0 ? 0 : fact(n) * fact_inv(n - k); } static MInt nHk(int n, int k) { return n < 0 || k < 0 ? 0 : (k == 0 ? 1 : nCk(n + k - 1, k)); } MInt pow(long long exponent) const { MInt tmp = *this, res = 1; while (exponent > 0) { if (exponent & 1) res *= tmp; tmp *= tmp; exponent >>= 1; } return res; } MInt &operator+=(const MInt &x) { if((val += x.val) >= MOD) val -= MOD; return *this; } MInt &operator-=(const MInt &x) { if((val += MOD - x.val) >= MOD) val -= MOD; return *this; } MInt &operator*=(const MInt &x) { val = static_cast(val) * x.val % MOD; return *this; } MInt &operator/=(const MInt &x) { return *this *= inv(x.val); } bool operator==(const MInt &x) const { return val == x.val; } bool operator!=(const MInt &x) const { return val != x.val; } bool operator<(const MInt &x) const { return val < x.val; } bool operator<=(const MInt &x) const { return val <= x.val; } bool operator>(const MInt &x) const { return val > x.val; } bool operator>=(const MInt &x) const { return val >= x.val; } MInt &operator++() { if (++val == MOD) val = 0; return *this; } MInt operator++(int) { MInt res = *this; ++*this; return res; } MInt &operator--() { val = (val == 0 ? MOD : val) - 1; return *this; } MInt operator--(int) { MInt res = *this; --*this; return res; } MInt operator+() const { return *this; } MInt operator-() const { return MInt(val ? MOD - val : 0); } MInt operator+(const MInt &x) const { return MInt(*this) += x; } MInt operator-(const MInt &x) const { return MInt(*this) -= x; } MInt operator*(const MInt &x) const { return MInt(*this) *= x; } MInt operator/(const MInt &x) const { return MInt(*this) /= x; } friend std::ostream &operator<<(std::ostream &os, const MInt &x) { return os << x.val; } friend std::istream &operator>>(std::istream &is, MInt &x) { long long val; is >> val; x = MInt(val); return is; } }; namespace std { template MInt abs(const MInt &x) { return x; } } using ModInt = MInt; struct UnionFind { std::vector data; UnionFind(int n) : data(n, -1) {} int root(int ver) { return data[ver] < 0 ? ver : data[ver] = root(data[ver]); } bool unite(int u, int v) { u = root(u); v = root(v); if (u == v) return false; if (data[u] > data[v]) std::swap(u, v); data[u] += data[v]; data[v] = u; return true; } bool same(int u, int v) { return root(u) == root(v); } int size(int ver) { return -data[root(ver)]; } }; int c; map, pair, unordered_set>> memo1; pair, unordered_set> f(const string &v, char ub) { if (memo1.count({v, ub}) == 1) return memo1[{v, ub}]; vector> no(c + 1); UnionFind base(c * 2 + 1); REP(j, c) { if (v[j] == '0') { base.unite(j, c * 2); } else { no[v[j] - '0'].emplace_back(j); } } FOR(i, 1, c + 1) { FOR(j, 1, no[i].size()) base.unite(no[i][j - 1], no[i][j]); no[i].clear(); } unordered_set emp; FOR(j, 1, c - 1) { if ('1' <= v[j] && v[j] <= ub) emp.emplace(base.root(j)); } return memo1[{v, ub}] = {base.data, emp}; } int main() { int r, n; cin >> r >> c >> n; map, ModInt> dp{{{string(c, '0'), '0', 0}, 1}}; ModInt ans = 0; REP(ri, r + 1) { map, ModInt> nx; for (auto [fst, pat] : dp) { auto [v, ub, edge] = fst; auto [base, emp] = f(v, ub); REP(i, 1 << c) { if (ri == 0 && i == 0) continue; if (ri == r && i > 0) break; // cout << v << '\n'; // REP(j, c) cout << (i >> j & 1); // cout << '\n'; UnionFind uf(c * 2 + 1); REP(j, c * 2 + 1) uf.data[j] = base[j]; int e = edge; REP(j, c) { if (i >> j & 1) { if (v[j] > ub) uf.unite(j, c + j); if (j > 0 && (i >> (j - 1) & 1)) uf.unite(c + j - 1, c + j); } else { if (v[j] <= ub) uf.unite(j, c + j); if (j > 0 && !(i >> (j - 1) & 1)) uf.unite(c + j - 1, c + j); } } if (!(i & 1)) uf.unite(c, c * 2); if (!(i >> (c - 1) & 1)) uf.unite(c + c - 1, c * 2); unordered_set nx_emp{uf.root(c * 2)}; FOR(j, 1, c - 1) { if (!(i >> j & 1)) nx_emp.emplace(uf.root(c + j)); } bool illegal = false; for (int e : emp) { if (nx_emp.count(uf.root(e)) == 0) { illegal = true; break; } } if (illegal) { // cout << "hall\n"; continue; } for (int j = 0; j < c;) { if (!(i >> j & 1)) { ++j; continue; } e += v[j] <= ub || (j > 0 && v[j - 1] > ub); int k = j; while (k < c && (i >> k & 1)) { e += v[k] <= ub && (k == j || v[k - 1] > ub); ++k; } e += v[k - 1] <= ub || (k < c && v[k] > ub); j = k; } REP(j, c) e += v[j] > ub && !(i >> j & 1) && (j == 0 || v[j - 1] <= ub || (i >> (j - 1) & 1)); // cout << "edge: " << edge << " -> " << e << '\n'; if (e > n) continue; if (i == 0) { if (e < n) continue; unordered_set root; REP(j, c) { if (v[j] > ub) root.emplace(uf.root(j)); } // cout << "connected?: " << (root.size() == 1) << '\n'; if (root.size() == 1) ans += pat * (r - ri + 1); } else { unordered_set adj; REP(j, c) { if (i >> j & 1) adj.emplace(uf.root(c + j)); } REP(j, c) { if (v[j] > ub && adj.count(uf.root(j)) == 0) { illegal = true; break; } } if (illegal) { // cout << "separate\n"; continue; } string u = ""; unordered_map mp{{uf.root(c * 2), 0}}; REP(j, c) { if (!(i >> j & 1)) { int root = uf.root(c + j); if (mp.count(root) == 0) { int size = mp.size(); mp[root] = size; } } } char nx_ub = '0' + (mp.size() - 1); REP(j, c) { if (i >> j & 1) { int root = uf.root(c + j); if (mp.count(root) == 0) { int size = mp.size(); mp[root] = size; } } } REP(j, c) u += '0' + mp[uf.root(c + j)]; nx[{u, nx_ub, e}] += pat; // cout << "next: " << u << '\n'; } } } dp.swap(nx); // for (auto [fst, pat] : dp) { // auto [v, edge] = fst; // cout << v << ' ' << edge << " : " << pat << '\n'; // } // cout << '\n'; } cout << ans << '\n'; return 0; }