[BEST定理 矩阵树定理] BZOJ 3659 Which Dreamed It

BEST theorem
一个证明？

注意区分下题目中要求的“欧拉回路”的条数和定理中欧拉回路的条数
欧拉回路是个回路 所以存在循环同构
题中要求起点是1 实际上还要乘上1的度数 因为从1的任一边出发在题中都算作一种不同方案

#include
#include
#include
using namespace std;
typedef long long ll;

const int N=105;
const int P=1000003;

int n;
char s[N][N];
ll fac[200005];

inline ll Pow(ll a,int b){
  ll ret=1;
  for (;b;b>>=1,a=a*a%P) if (b&1) ret=ret*a%P;
  return ret;
}
inline ll Inv(ll a){
  return Pow(a,P-2);
}

int a[N][N];
inline int det(int n){
  int f=0;
  for (int i=1;i<=n;i++){
    int k=0;
    for (int j=i;j<=n;j++) if (a[j][i]) {k=j; break; }
    if (i^k) { for (int j=i;j<=n;j++) swap(a[i][j],a[k][j]); f^=1; }
    for (int j=i+1;j<=n;j++){
      ll t=(ll)Inv(a[i][i])*a[j][i]%P;
      for (int k=i;k<=n;k++)
    (a[j][k]+=P-(ll)t*a[i][k]%P)%=P;
    }
  }
  ll ret=1;
  for (int i=1;i<=n;i++) ret=ret*a[i][i]%P;
  return f?(P-ret)%P:ret;
}

#define read(x) scanf("%d",&(x))

int deg[N];

int main(){
  int x,k;
  freopen("t.in","r",stdin);
  freopen("t.out","w",stdout);
  fac[0]=1; for (int i=1;i<=200000;i++) fac[i]=fac[i-1]*i%P;
  while (1){
    read(n); if (!n) break;
    for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) a[i][j]=0;
    for (int i=1;i<=n;i++){
      read(k); deg[i]=k;
      while (k--){
    read(x);
    if (i^x) a[n-i+1][n-x+1]--,a[n-x+1][n-x+1]++;
      }
    }
    if (n==1) { printf("%lld\n",fac[deg[1]]); continue; }
    for (int i=1;i<=n;i++) for (int j=1;j<=n;j++) a[i][j]=(P+a[i][j])%P;
    ll ans=(ll)det(n-1)*deg[1]%P;
    for (int i=1;i<=n;i++) ans=ans*fac[deg[i]-1]%P;
    printf("%lld\n",ans);
  }
  return 0;
}