【洛谷P4128】【SHOI2006】—有色图（Burnside）

传送门

看起来像要用$polya$
但是仔细分析发现用不了

仔细想想发现和图的同构那道题没任何区别
只是把颜色数从$2$变成了$m$

#include<bits/stdc++.h>
using namespace std;
const int RLEN=1<<20|1;
inline char gc(){
    static char ibuf[RLEN],*ib,*ob;
    (ob==ib)&&(ob=(ib=ibuf)+fread(ibuf,1,RLEN,stdin));
    return (ob==ib)?EOF:*ib++;
}
inline int read(){
    char ch=gc();
    int res=0,f=1;
    while(!isdigit(ch))f^=ch=='-',ch=gc();
    while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
    return f?res:-res;
}
#define ll long long
#define re register
#define pii pair<int,int>
#define fi first
#define se second
#define pb push_back
#define cs const
#define bg begin
#define poly vector<int>  
int mod;
inline int add(int a,int b){return (a+=b)>=mod?a-mod:a;}
inline void Add(int &a,int b){(a+=b)>=mod?a-=mod:0;}
inline int dec(int a,int b){return (a-=b)<0?a+mod:a;}
inline void Dec(int &a,int b){(a-=b)<0?a+=mod:0;}
inline int mul(int a,int b){return 1ll*a*b%mod;}
inline void Mul(int &a,int b){a=1ll*a*b%mod;}
inline int ksm(int a,int b,int res=1){for(;b;b>>=1,Mul(a,a))(b&1)&&(Mul(res,a),1);return res;}
inline int Inv(int x){return ksm(x,mod-2);}
#define chemx(a,b) ((a)<(b)?(a)=(b):0)
#define chemn(a,b) ((a)>(b)?(a)=(b):0)
cs int N=55;
int n,m,ans;
int G[N][N],cnt[N],fac[N],ifac[N],inv[N];
vector<int> divi;
inline int gcd(int a,int b){
	return b?gcd(b,a%b):a;
}
inline void calc(){
	int coef=fac[n];
	for(int i=1;i<=n;i++)
		Mul(coef,ifac[cnt[i]]);
	for(int &x:divi)
		Mul(coef,inv[x]);
	int tot=0;
	for(int &x:divi)tot+=x/2;
	for(int i=0;i<divi.size();i++)
	for(int j=i+1;j<divi.size();j++)
	tot+=G[divi[i]][divi[j]];
	Add(ans,mul(coef,ksm(m,tot)));
}
void dfs(int res,int last){
	if(!res)return calc();
	if(res<last)return;
	for(int i=last;i<=res;i++)
		cnt[i]++,divi.pb(i),dfs(res-i,i),divi.pop_back(),cnt[i]--;
}
inline void init(){
	fac[0]=ifac[0]=inv[0]=inv[1]=1;
	for(int i=1;i<=n;i++)fac[i]=mul(fac[i-1],i);
	ifac[n]=Inv(fac[n]);
	for(int i=n-1;i;i--)ifac[i]=mul(ifac[i+1],i+1);
	for(int i=2;i<=n;i++)inv[i]=mul(mod-mod/i,inv[mod%i]);
}
int main(){
	#ifdef Stargazer
	freopen("lx.cpp","r",stdin);
	#endif
	n=read(),m=read(),mod=read();
	init();
	for(int i=1;i<=n;i++)
	for(int j=1;j<=i;j++)
	G[i][j]=G[j][i]=gcd(i,j);
	dfs(n,1);
	cout<<mul(ans,ifac[n]);
}