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grid_lines.cpp
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174 lines (145 loc) · 3.57 KB
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#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
using namespace std;
typedef long long LL;
const int MAXN = 3033, MOD = 1000000007;
int N, M, K, C_cache[MAXN][MAXN], C_visited[MAXN][MAXN];
int gcd_cache[MAXN][MAXN], gcd_visited[MAXN][MAXN];
int C(int n, int k) {
if (k < 0 || k > n) return 0;
if (k == 0 || k == n) return 1;
int &res = C_cache[n][k];
if (C_visited[n][k]) return res;
C_visited[n][k] = true;
res = C(n - 1, k) + C(n - 1, k - 1);
if (res >= MOD) res -= MOD;
return res;
}
void C_init() {
for (int i = 0; i <= max(N, M); i++){
C_cache[i][K - 2] = C(i, K - 2);
C_cache[i][K - 2] %= MOD;
C_visited[i][K - 2] = true;
}
}
int gcd(int a, int b){
int &res = gcd_cache[a][b];
if (gcd_visited[a][b]) return res;
gcd_visited[a][b] = true;
return res = b == 0? a: gcd(b, a % b);
}
void gcd_init(){
for (int i = 0; i < MAXN; i++){
for (int j = 0; j < MAXN; j++){
gcd_cache[i][j] = gcd(i, j);
gcd_visited[i][j] = true;
}
}
}
int F_cache[MAXN][MAXN];
int F_visited[MAXN][MAXN];
int F(int n, int m){
if (n < 0 || m < 0) return 0;
int &res = F_cache[n][m];
if (F_visited[n][m]) return res;
F_visited[n][m] = true;
int t = gcd(n, m);
res = C(t - 1, K - 2);
return res;
}
void F_init(){
for (int i = 0; i <= N; i++){
for (int j = 0; j <= M; j++){
int tmp = gcd_cache[i][j] - 1;
if (tmp >= 0){
F_cache[i][j] = C_cache[tmp][K - 2];
F_cache[i][j] %= MOD;
}
F_visited[i][j] = true;
}
}
}
int F1_cache[MAXN][MAXN];
int F1_visited[MAXN][MAXN];
int F1(int n, int m){
if (n < 0 || m < 0) return 0;
int &res = F1_cache[n][m];
if (F1_visited[n][m]) return res;
F1_visited[n][m] = true;
res = F1(n, m - 1) + F(n, m);
if (res >= MOD) res -= MOD;
return res;
}
void F1_init(){
for (int i = 0; i <= N; i++){
for (int j = 0; j <= M; j++){
F1_cache[i][j] = (j - 1 >= 0 ? F1_cache[i][j - 1] : 0) + F_cache[i][j];
F1_cache[i][j] %= MOD;
F1_visited[i][j] = true;
}
}
}
int F2_cache[MAXN][MAXN];
int F2_visited[MAXN][MAXN];
int F2(int n, int m){
if (n < 0 || m < 0) return 0;
int &res = F2_cache[n][m];
if (F2_visited[n][m]) return res;
F2_visited[n][m] = true;
res = F2(n - 1, m) + F1(n, m);
if (res >= MOD) res -= MOD;
return res;
}
void F2_init(){
for (int i = 0; i <= N; i++){
for (int j = 0; j <= M; j++){
F2_cache[i][j] = (i - 1 >= 0 ? F2_cache[i - 1][j] : 0) + F1_cache[i][j];
F2_cache[i][j] %= MOD;
F2_visited[i][j] = true;
}
}
}
int solve(){
LL total = 0;
for (int i = 0; i < M; i++){
total += F2(N, i) + F2(N, M - i) - F2(N, 0);
total %= MOD;
}
for (int i = 0; i < N; i++){
total += F2(i, M) + F2(N - i, M) - F2(0, M);
total %= MOD;
}
total *= 2;
total %= MOD;
LL total1 = 0;
for (int i = 0; i < N; i++){
total1 += F2(M, i) - F2(M - 1, i - 1);
total1 += F2(M, N - i) - F2(M - 1, N - i - 1);
total1 -= F(0, M);
total1 += F1(0, i - 1);
total1 += F1(0, N - i - 1);
total1 %= MOD;
}
for (int i = 0; i < M; i++){
total1 += F2(N, i) - F2(N - 1, i - 1);
total1 += F2(N, M - i) - F2(N - 1, M - i - 1);
total1 -= F(0, N);
total1 += F1(0, i - 1);
total1 += F1(0, M - i - 1);
total1 %= MOD;
}
return ((total - total1)%MOD + MOD)%MOD;
}
int main() {
cin >> N >> M >> K;
C_init();
gcd_init();
F_init();
F1_init();
F2_init();
cout << solve() << endl;
return 0;
}