P4827 [国家集训队] Crash 的文明世界

传送门：洛谷

题目大意：设$$S(i)=\sum_{j=1}^ndis(i,j)^k$$，求$S(1),S(2),\ldots,S(n)$。

数据范围：$n\leq 50000,k\leq 150$

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这道题，看见$k$次方和就直接上斯特林数。

$$S(x)=\sum_{i=0}^ki!S(k,i)\sum_{y=1}^nC_{dis(x,y)}^i$$

然后我们考虑求最后一项。

设$$up_{x,t}=\sum_{y\notin x}C_{dis(x,y)}^t,dn_{x,t}=\sum_{y\in x}C_{dis(x,y)}^t$$

我们先考虑$dn$。

$$dn_{x,t}=\sum_{(x,v)}\sum_{y\in v}C_{dis(v,y)+1}^t$$

$$=\sum_{(x,v)}\sum_{y\in v}(C_{dis(v,y)}^t+C_{dis(v,y)}^{t-1})$$

$$=\sum_{(x,v)}(dn_{v,t}+dn_{v,t-1})$$

然后考虑$up$

$$up_{x,t}=\sum_{v\notin fa}C_{dis(v,fa)+1}^t+\sum_{v\in fa}C_{dis(v,fa)+1}^t-\sum_{v\in x}C_{dis(v,x)+2}^t$$

$$=up_{fa,t}+up_{fa,t-1}+dn_{fa,t}+dn_{fa,t-1}-dn_{x,t}-2dn_{x,t-1}-dn_{x,t-2}$$

其中$up_{1,t}=0$

然后把式子直接输进去就可以了。

 #include<cstdio>
 #define Rint register int
 using namespace std;
 const int N = , K = , mod = ;
 int n, k, head[N], to[N << ], nxt[N << ], S[K][K], fac[K];
 inline void add(int a, int b){
     static int cnt = ;
     to[++ cnt] = b; nxt[cnt] = head[a]; head[a] = cnt;
 }
 int dn[N][K], up[N][K];
 inline void dfs1(int x, int f){
     dn[x][] = ;
     for(Rint i = head[x];i;i = nxt[i])
         if(to[i] != f){
             dfs1(to[i], x);
             dn[x][] = (dn[x][] + dn[to[i]][]) % mod;
             for(Rint t = ;t <= k;t ++)
                 dn[x][t] = (dn[to[i]][t] + dn[to[i]][t - ] + dn[x][t]) % mod;
         }
 }
 inline void dfs2(int x, int f){
     for(Rint i = head[x];i;i = nxt[i])
         if(to[i] != f){
             up[to[i]][] = (up[x][] + dn[x][] - dn[to[i]][] + mod) % mod;
             up[to[i]][] = (up[x][] + up[x][] + dn[x][] + dn[x][] - dn[to[i]][] -  * dn[to[i]][] +  * mod) % mod;
             for(Rint t = ;t <= k;t ++)
                 up[to[i]][t] = (up[x][t] + up[x][t - ] + dn[x][t] + dn[x][t - ] - dn[to[i]][t] -  * dn[to[i]][t - ] - dn[to[i]][t - ] +  * mod) % mod;
             dfs2(to[i], x);
         }
 }
 int main(){
     scanf("%d%d", &n, &k);
     S[][] = ;
     for(Rint i = ;i <= k;i ++)
         for(Rint j = ;j <= k;j ++)
             S[i][j] = (S[i - ][j - ] + S[i - ][j] * j) % mod;
     fac[] = ;
     for(Rint i = ;i <= k;i ++) fac[i] = i * fac[i - ] % mod;
     for(Rint i = ;i < n;i ++){
         int a, b;
         scanf("%d%d", &a, &b);
         add(a, b); add(b, a);
     }
     dfs1(, ); dfs2(, );
     for(Rint x = ;x <= n;x ++){
         int ans = ;
         for(Rint i = ;i <= k;i ++)
             ans = (ans + fac[i] * S[k][i] % mod * (up[x][i] + dn[x][i]) % mod) % mod;
         printf("%d\n", ans);
     }
 }