A.Monotonic Matrix

题意：

问有多少个矩阵A满足如下性质
* Ai, j ∈ {0, 1, 2} for all 1 ≤ i ≤ n, 1 ≤ j ≤ m.
* Ai, j ≤ Ai + 1, j for all 1 ≤ i < n, 1 ≤ j ≤ m.
* Ai, j ≤ Ai, j + 1 for all 1 ≤ i ≤ n, 1 ≤ j < m.

思路：找0,1；1,2的分割线，求路线不重合的路径数（即严格不相交路径）。套Lindström–Gessel–Viennot引理

#include<bits/stdc++.h>
#define PI acos(-1.0)
#define pb push_back
#define F first
#define S second
using namespace std;
typedef long long ll;
const int N=2005;
const int MOD=1e9+7;
//ll a[N],sum[N],dis[N];
int a[200][200];
 
ll c[N][N];
void init(){
    c[0][0]=c[1][0]=c[1][1]=1;
    for(int i=2;i<=2000;i++){
        c[i][0]=1;
        for(int j=1;j<=i;j++)   c[i][j]=c[i-1][j]+c[i-1][j-1],c[i][j]%=MOD;
    }
}
int main(void){
    ios::sync_with_stdio(false);
    cin.tie(0);cout.tie(0);
 
    init();
    ll n,m;
    while(cin>>n>>m){
       ll ans=c[n+m][n]*c[n+m][m]%MOD-c[n+m][n-1]*c[n+m][m-1]%MOD;
       ans%=MOD;
       ans+=MOD;
       cout << ans%MOD << endl;
    }
 
    return 0;
}