#include <bits/stdc++.h>
using namespace std;
using ll = long long;
const int MOD = 1e9 + 7;
ll n, k, m;
map<vector<int>, int> memo;
ll mod_inv(ll x) {
ll res = 1, p = MOD - 2;
ll base = x;
while (p) {
if (p & 1) res = (ll)res * base % MOD;
base = base * base % MOD;
p >>= 1;
}
return res;
}
ll sum(initializer_list<ll> nums) {
ll s = 0;
for (ll x : nums) s = ((s + (x % MOD)) % MOD + MOD) % MOD;
return s;
}
ll mul(initializer_list<ll> nums) {
ll s = 1;
for (ll x : nums) s = ((s * (x % MOD)) % MOD + MOD) % MOD;
return s;
}
int solve(vector<int> state) {
sort(state.begin(), state.end());
if (state.back() >= m) return 0;
if (memo.count(state)) return memo[state];
ll inv_k = mod_inv(k);
ll res = 1;
ll future = 0;
for (int dice = 1; dice <= k && state[0] + dice < m; dice++) {
vector<int> next = state;
next[0] += dice;
future += solve(next);
future %= MOD;
}
future = future * inv_k % MOD;
res = (res + future) % MOD;
return memo[state] = res;
}
// double solve_double(vector<int>& state) {
// sort(state.begin(), state.end());
// if (state.back() >= m) return 0;
// double res = 1; // current move
// double future = 0;
// double cnt = 0;
// for (int dice = 1; dice <= k && state[0] + dice < m; dice++) {
// vector<int> next = state;
// next[0] += dice;
// future += solve_double(next);
// cnt++;
// }
// if (cnt > 0) {
// future = future / (double) k;
// }
// res = (res + future);
// return res;
// }
ll dfs(ll target) {
if (target == 0) {
return 0;
}
vector<ll> dp(target + k + 2, 0);
vector<ll> dp_sum(target + 5, 0);
ll inv_k = mod_inv(k);
dp[target - 1] = 1;
for (ll s = target - 2; s >= 0; --s) {
ll added = dp[s + 1];
ll dropped = (s + k + 1 < (ll) dp.size()) ? dp[s + k + 1] : 0;
dp_sum[s] = sum({dp_sum[s + 1], added, -dropped});
dp[s] = sum({1, mul({dp_sum[s], inv_k})});
}
return dp[0];
}
// double probability_not_reaching(int q, int k, int target) {
// vector<double> dp(target, 0.0);
// dp[0] = 1.0;
// for (int roll = 1; roll <= q; roll++) {
// vector<double> next_dp(target, 0.0);
// double window_sum = 0.0;
// for (int s = 1; s < target; s++) {
// window_sum += dp[s - 1];
// if (s - k - 1 >= 0)
// window_sum -= dp[s - k - 1];
// next_dp[s] = window_sum / k;
// }
// dp = next_dp;
// }
// // suma prawdopodobieństw dla s < target
// double res = 0.0;
// for (int s = 0; s < target; s++)
// res += dp[s];
// return res;
// }
// double probability_not_reaching(int n, int q, int k, int target) {
// double res = 0;
// for (int s = 0; s < q; s++)
// res += probability_not_reaching(n - 1, s, k, target);
// return res;
// }
// double dfs_double(ll target) {
// vector<double> dp(target + k + 1, 0);
// for (int s = target; s >= 0; --s) {
// if (s == target) dp[s] = 0;
// else {
// double sum = 0;
// for (int i = 1; i <= k; i++) sum = sum + dp[s+i];
// dp[s] = 1.0 + sum / (double) k;
// }
// }
// return dp[0];
// }
int main() {
cin >> n >> k >> m;
if (n == 1) {
cout << dfs(m) << endl;
return 0;
}
vector<int> state(n, 0);
cout << solve(state) << "\n";
}
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 | #include <bits/stdc++.h> using namespace std; using ll = long long; const int MOD = 1e9 + 7; ll n, k, m; map<vector<int>, int> memo; ll mod_inv(ll x) { ll res = 1, p = MOD - 2; ll base = x; while (p) { if (p & 1) res = (ll)res * base % MOD; base = base * base % MOD; p >>= 1; } return res; } ll sum(initializer_list<ll> nums) { ll s = 0; for (ll x : nums) s = ((s + (x % MOD)) % MOD + MOD) % MOD; return s; } ll mul(initializer_list<ll> nums) { ll s = 1; for (ll x : nums) s = ((s * (x % MOD)) % MOD + MOD) % MOD; return s; } int solve(vector<int> state) { sort(state.begin(), state.end()); if (state.back() >= m) return 0; if (memo.count(state)) return memo[state]; ll inv_k = mod_inv(k); ll res = 1; ll future = 0; for (int dice = 1; dice <= k && state[0] + dice < m; dice++) { vector<int> next = state; next[0] += dice; future += solve(next); future %= MOD; } future = future * inv_k % MOD; res = (res + future) % MOD; return memo[state] = res; } // double solve_double(vector<int>& state) { // sort(state.begin(), state.end()); // if (state.back() >= m) return 0; // double res = 1; // current move // double future = 0; // double cnt = 0; // for (int dice = 1; dice <= k && state[0] + dice < m; dice++) { // vector<int> next = state; // next[0] += dice; // future += solve_double(next); // cnt++; // } // if (cnt > 0) { // future = future / (double) k; // } // res = (res + future); // return res; // } ll dfs(ll target) { if (target == 0) { return 0; } vector<ll> dp(target + k + 2, 0); vector<ll> dp_sum(target + 5, 0); ll inv_k = mod_inv(k); dp[target - 1] = 1; for (ll s = target - 2; s >= 0; --s) { ll added = dp[s + 1]; ll dropped = (s + k + 1 < (ll) dp.size()) ? dp[s + k + 1] : 0; dp_sum[s] = sum({dp_sum[s + 1], added, -dropped}); dp[s] = sum({1, mul({dp_sum[s], inv_k})}); } return dp[0]; } // double probability_not_reaching(int q, int k, int target) { // vector<double> dp(target, 0.0); // dp[0] = 1.0; // for (int roll = 1; roll <= q; roll++) { // vector<double> next_dp(target, 0.0); // double window_sum = 0.0; // for (int s = 1; s < target; s++) { // window_sum += dp[s - 1]; // if (s - k - 1 >= 0) // window_sum -= dp[s - k - 1]; // next_dp[s] = window_sum / k; // } // dp = next_dp; // } // // suma prawdopodobieństw dla s < target // double res = 0.0; // for (int s = 0; s < target; s++) // res += dp[s]; // return res; // } // double probability_not_reaching(int n, int q, int k, int target) { // double res = 0; // for (int s = 0; s < q; s++) // res += probability_not_reaching(n - 1, s, k, target); // return res; // } // double dfs_double(ll target) { // vector<double> dp(target + k + 1, 0); // for (int s = target; s >= 0; --s) { // if (s == target) dp[s] = 0; // else { // double sum = 0; // for (int i = 1; i <= k; i++) sum = sum + dp[s+i]; // dp[s] = 1.0 + sum / (double) k; // } // } // return dp[0]; // } int main() { cin >> n >> k >> m; if (n == 1) { cout << dfs(m) << endl; return 0; } vector<int> state(n, 0); cout << solve(state) << "\n"; } |
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