![Codeforces 809E Surprise me! [莫比乌斯反演]](https://image.shishitao.com:8440/aHR0cHM6Ly9ia3FzaW1nLmlrYWZhbi5jb20vdXBsb2FkL2NoYXRncHQtcy5wbmc%2FIQ%3D%3D.png?!?w=700&webp=1 "Codeforces 809E Surprise me! [莫比乌斯反演]")
非常套路的一道题，很适合我在陷入低谷时提升信心……
思路
显然我们需要大力推式子。
设\(p_{a_i}=i\)，则有
\[\begin{align*}
n(n-1)ans&=\sum_i \sum_j \varphi(ij)dis(p_i,p_j)\\
&=\sum_i \sum_j \frac{\varphi(i)\varphi(j)\gcd(i,j)}{\varphi(\gcd(i,j))} dis(i,j)\\
&=\sum_d \frac{d}{\varphi(d)} \sum_i\sum_j [\gcd(i,j)=d]\varphi(i)\varphi(j)dis(i,j)\\
&=\sum_d \frac{d}{\varphi(d)} \sum_{k=1}^{n/d} \mu(k) \sum_{i=1}^{n/dk} \sum_{j=1}^{n/dk} \varphi(idk)\varphi(jdk)dis(p_{idk},p_{jdk})\\
&=\sum_{T=1}^n (\sum_{d|T} \frac{d\mu(T/d)}{\varphi(d)})\sum_{i=1}^{n/T} \sum_{j=1}^{n/T} \varphi(iT)\varphi(jT)dis(p_{iT},p_{jT})\\
\end{align*}
\]
n(n-1)ans&=\sum_i \sum_j \varphi(ij)dis(p_i,p_j)\\
&=\sum_i \sum_j \frac{\varphi(i)\varphi(j)\gcd(i,j)}{\varphi(\gcd(i,j))} dis(i,j)\\
&=\sum_d \frac{d}{\varphi(d)} \sum_i\sum_j [\gcd(i,j)=d]\varphi(i)\varphi(j)dis(i,j)\\
&=\sum_d \frac{d}{\varphi(d)} \sum_{k=1}^{n/d} \mu(k) \sum_{i=1}^{n/dk} \sum_{j=1}^{n/dk} \varphi(idk)\varphi(jdk)dis(p_{idk},p_{jdk})\\
&=\sum_{T=1}^n (\sum_{d|T} \frac{d\mu(T/d)}{\varphi(d)})\sum_{i=1}^{n/T} \sum_{j=1}^{n/T} \varphi(iT)\varphi(jT)dis(p_{iT},p_{jT})\\
\end{align*}
\]
现在思路已经非常清晰了。
\(f(T)=\sum_{d|T}\frac{d\mu(T/d)}{\varphi(d)}\)：这显然可以调和级数\(O(n\log n)\)得到。
现在就是要求
\[X=\sum_{i\in S} \sum_{j\in S} \varphi(i)\varphi(j)dis(p_i,p_j)
\]
\]
显然通过调和级数我们发现\(\sum|S|\)是\(O(n\log n)\)级别的。
考虑把\(dis(p_i,p_j)\)拆开，我们得到
\[\begin{align*}
X&=\sum_{i\in S}\sum_{j\in S} \varphi(i)\varphi(j)(dep_{p_i}+dep_{p_j}-2dep_{lca(p_i,p_j)})\\
&=M-2\sum_{i\in S}\varphi(i)\sum_{j\in S} \varphi(j)dep_{lca(p_i,p_j)}
\end{align*}
\]
X&=\sum_{i\in S}\sum_{j\in S} \varphi(i)\varphi(j)(dep_{p_i}+dep_{p_j}-2dep_{lca(p_i,p_j)})\\
&=M-2\sum_{i\in S}\varphi(i)\sum_{j\in S} \varphi(j)dep_{lca(p_i,p_j)}
\end{align*}
\]
其中\(M\)可以很容易求出，只需要求出后面。
后面其实是个经典套路，可以用大众化的DP或是用“ [HNOI2015]开店”相同的方法（链上加，求到根的距离）做。
由于我是个SB，我选了后者，写了个树上差分。
然后各种SB错误调了1个小时……
代码
#include<bits/stdc++.h>
clock_t t=clock();
namespace my_std{
using namespace std;
#define pii pair<int,int>
#define fir first
#define sec second
#define MP make_pair
#define rep(i,x,y) for (int i=(x);i<=(y);i++)
#define drep(i,x,y) for (int i=(x);i>=(y);i--)
#define go(x) for (int i=head[x];i;i=edge[i].nxt)
#define templ template<typename T>
#define sz 202020
#define mod 1000000007ll
typedef long long ll;
typedef double db;
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
templ inline T rnd(T l,T r) {return uniform_int_distribution<T>(l,r)(rng);}
templ inline bool chkmax(T &x,T y){return x<y?x=y,1:0;}
templ inline bool chkmin(T &x,T y){return x>y?x=y,1:0;}
templ inline void read(T& t)
{
t=0;char f=0,ch=getchar();double d=0.1;
while(ch>'9'||ch<'0') f|=(ch=='-'),ch=getchar();
while(ch<='9'&&ch>='0') t=t*10+ch-48,ch=getchar();
if(ch=='.'){ch=getchar();while(ch<='9'&&ch>='0') t+=d*(ch^48),d*=0.1,ch=getchar();}
t=(f?-t:t);
}
template<typename T,typename... Args>inline void read(T& t,Args&... args){read(t); read(args...);}
char __sr[1<<21],__z[20];int __C=-1,__zz=0;
inline void Ot(){fwrite(__sr,1,__C+1,stdout),__C=-1;}
inline void print(register int x)
{
if(__C>1<<20)Ot();if(x<0)__sr[++__C]='-',x=-x;
while(__z[++__zz]=x%10+48,x/=10);
while(__sr[++__C]=__z[__zz],--__zz);__sr[++__C]='\n';
}
void file()
{
#ifndef ONLINE_JUDGE
freopen("a.in","r",stdin);
#endif
}
inline void chktime()
{
#ifndef ONLINE_JUDGE
cout<<(clock()-t)/1000.0<<'\n';
#endif
}
#ifdef mod
ll ksm(ll x,int y){ll ret=1;for (;y;y>>=1,x=x*x%mod) if (y&1) ret=ret*x%mod;return ret;}
ll inv(ll x){return ksm(x,mod-2);}
#else
ll ksm(ll x,int y){ll ret=1;for (;y;y>>=1,x=x*x) if (y&1) ret=ret*x;return ret;}
#endif
// inline ll mul(ll a,ll b){ll d=(ll)(a*(double)b/mod+0.5);ll ret=a*b-d*mod;if (ret<0) ret+=mod;return ret;}
}
using namespace my_std;
int pri[sz],mu[sz],phi[sz],cnt;
bool npri[sz];
ll f[sz];
void init()
{
mu[1]=phi[1]=1;
#define x i*pri[j]
rep(i,2,sz-1)
{
if (!npri[i]) pri[++cnt]=i,mu[i]=-1,phi[i]=i-1;
for (int j=1;j<=cnt&&x<sz;j++)
{
npri[x]=1;
if (i%pri[j]) mu[x]=-mu[i],phi[x]=(pri[j]-1)*phi[i];
else { phi[x]=pri[j]*phi[i]; break; }
}
}
#undef x
rep(i,1,sz-1) for (int j=i;j<sz;j+=i) (f[j]+=1ll*i*mu[j/i]*inv(phi[i])%mod)%=mod;
}
int n;
int p[sz];
struct hh{int t,nxt;}edge[sz<<1];
int head[sz],ecnt;
void make_edge(int f,int t)
{
edge[++ecnt]=(hh){t,head[f]};
head[f]=ecnt;
edge[++ecnt]=(hh){f,head[t]};
head[t]=ecnt;
}
int fa[sz][30],dep[sz],dfn[sz],c;
void dfs(int x,int f)
{
dep[x]=dep[fa[x][0]=f]+1;dfn[x]=++c;
rep(i,1,20) fa[x][i]=fa[fa[x][i-1]][i-1];
#define v edge[i].t
go(x) if (v!=f) dfs(v,x);
#undef v
}
int lca(int x,int y)
{
if (dep[x]<dep[y]) swap(x,y);
drep(i,20,0)
if (fa[x][i]&&dep[fa[x][i]]>=dep[y])
x=fa[x][i];
if (x==y) return x;
drep(i,20,0)
if (fa[x][i]!=fa[y][i])
x=fa[x][i],y=fa[y][i];
return fa[x][0];
}
namespace Solve
{
int m;
struct hhh{int id,p;}S[sz];
struct hh{int t,nxt;}edge[sz<<1];
int head[sz],ecnt;
void make_edge(int f,int t)
{
edge[++ecnt]=(hh){t,head[f]};
head[f]=ecnt;
edge[++ecnt]=(hh){f,head[t]};
head[t]=ecnt;
}
int st[sz],top;
void insert(int x)
{
if (top<=1) return void(st[++top]=x);
int lca=::lca(x,st[top]);
if (lca==st[top]) return void(st[++top]=x);
while (top>1)
{
int q=st[top-1],&p=st[top];
if (dfn[q]==dfn[lca]) { make_edge(q,p); --top; break; }
if (dfn[q]<dfn[lca]) { make_edge(lca,p); p=lca; break; }
make_edge(p,q),--top;
}
st[++top]=x;
}
ll D[sz];
#define v edge[i].t
void dfs1(int x,int fa)
{
go(x)
if (v!=fa)
dfs1(v,x),(D[x]+=D[v])%=mod;
}
void dfs2(int x,int fa){go(x) if (v!=fa) dfs2(v,x);D[x]=D[x]*(dep[x]-dep[fa])%mod;}
void dfs3(int x,int fa){(D[x]+=D[fa])%=mod;go(x) if (v!=fa) dfs3(v,x);}
void del(int x,int fa){D[x]=0;go(x) if (v!=fa) del(v,x);head[x]=0;}
#undef v
ll solve()
{
bool flg=0;
rep(i,1,m) flg|=(S[i].p==1);
if (!flg) S[++m]=(hhh){0,1};
sort(S+1,S+m+1,[](const hhh &x,const hhh &y){return dfn[x.p]<dfn[y.p];});
rep(i,1,m) insert(S[i].p);
while (top>1) make_edge(st[top],st[top-1]),--top;
top=0;
rep(i,1,m) (D[S[i].p]+=phi[S[i].id])%=mod;
dfs1(1,0);D[1]=0;dfs2(1,0);dfs3(1,0);
ll ret=0;
rep(i,1,m) (ret+=phi[S[i].id]*D[S[i].p]%mod)%=mod;
rep(i,1,m) S[i]=(hhh){0,0};
del(1,0);
ecnt=m=0;
return ret;
}
}
ll solve(int T)
{
ll ret,S1=0,S2=0;
for (int i=T;i<=n;i+=T)
Solve::S[++Solve::m]=(Solve::hhh){i,p[i]},
(S1+=phi[i])%=mod,
(S2+=1ll*(dep[p[i]]-1)*phi[i]%mod)%=mod;
ret=S1*S2*2%mod;
ll t=Solve::solve();
ret=(ret-2ll*t+mod+mod)%mod;
return ret*f[T]%mod;
}
int main()
{
file();
init();
read(n);
int x,y;
rep(i,1,n) read(x),p[x]=i;
rep(i,1,n-1) read(x,y),make_edge(x,y);
dfs(1,0);
ll ans=0;
rep(T,1,n) (ans+=solve(T)+mod)%=mod;
cout<<ans*inv(1ll*n*(n-1)%mod)%mod;
return 0;
}