POJ 1091 扩展欧拉函数
poj 3270 置换,循环节

Burnside引理 Polya定理

scturtle posted @ 2010年7月31日 03:36 in algorithm , 5076 阅读

Burnside引理:

设置换群作用于有限集合上,则在G作用下的等价类的数目为:

 

其中(g)为g在上的不动点个数,即满足g(a)=a的个数。

 Polya定理:

设G是p个对象的一个置换群,用k种颜色涂染这p个对象,在置换群作用下的不同的染色方案数为

其中m(f)为置换f的循环节数。

poj 2409 Let it Bead

#include <iostream>
#include <cstdio>
using namespace std;
typedef long long u64;

int gcd(int a,int b){return b?gcd(b,a%b):a;}

u64 pow(u64 a,u64 b)
{
    u64 ans=1;u64 t=a;
    while(b)
    {
        if(b&1) ans*=t;
        t*=t;
        b>>=1;
    }
    return ans;
}

u64 polya(int c,int n)
{
    if(n==0) return 0;
    u64 ans=0;
    for(int i=0;i<n;i++)
        ans+=pow(c,gcd(i,n));
    if(n&1)
        ans+=pow(c,n/2+1)*n;
    else
        ans+=pow(c,n/2)*n/2+pow(c,n/2+1)*n/2;
    return ans/(2*n);
}

int main()
{
    //freopen("in.txt","r",stdin);
    int c,n;
    while(cin>>c>>n&&(c+n)!=0)
        cout<<polya(c,n)<<endl;
}

 

 poj 2154 Color

 

#include <iostream>
#include <cstdio>
#include <cstring>
using namespace std;
typedef unsigned long long u64;
typedef unsigned long long u64 ;
#define maxn 35000
int p [ maxn ];
int pnum = 0 ;
void getprimtable () //筛法求出素数
{
    bool flag [ maxn ] = { false };
    for ( long i = 2 ; i < maxn ; i ++)
    {
        if (! flag [ i ])
        {
            p [ pnum ++] = i ;
            for ( u64 j = i * i ; j < maxn ; j += i ) flag [ j ] = true ;
        }
    }
}

int eular(int n)
{
    int sum=n;
    for(int i=0;i<pnum && p[i]*p[i]<=n;i++)
    {
        if(n%p[i]==0)
        {
            while(n%p[i]==0) n/=p[i];
            sum=sum/p[i]*(p[i]-1);
        }
    }
    if(n!=1)
    {
        sum=sum/n*(n-1);
    }
    return sum;
}int pow(int a,int b,int m)
{
    int ans=1;int t=a%m;
    while(b)
    {
        if(b&1) ans=(ans*t)%m;
        t=(t*t)%m;
        b>>=1;
    }
    return ans;
}

int polya(int n,int m)
{
    if(n==0) return 0;
    int ans=0;
    for(int i=1;i*i<=n;i++)
    {
        if(n%i) continue;
        ans=ans+eular(n/i)%m*pow(n,i-1,m);
        if(i*i!=n) 
            ans=ans+eular(i)%m*pow(n,n/i-1,m);
        ans%=m;
    }
    return ans;
}

int main()
{
    //freopen("in.txt","r",stdin);
    getprimtable();
    int cas;scanf("%d",&cas);
    int n,m;
    while(cas--)
    {
        scanf("%d%d",&n,&m);
        printf("%d\n",polya(n,m));
    }
}

 

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