POJ 2888 Magic Bracelet (Burnside lemma + matrix)

Test instructions: A necklace of length n, each bead dyed in M color. Some restrictions give some color beads that cannot be adjacent. The same is considered the same after rotation. How many different kinds of necklaces are there?

Idea: This problem is a bit comprehensive, first of all, we consider the factor I of each n, the number of invariant permutations under this factor I, and then multiplied by (n/i) Euler function added to the ANS, and then the ANS multiplied by N in the inverse of modulo p. As to how to find the number of invariant permutations under factor I, I believe we have done no restrictions, as for the restrictions, we can consider this: the initial array a[m][m] are 1, for each limit x, Y, make a[x][y]=a[y][x]=0, we have a series: B1,B2,B3, B4...BM, so for each i∈[1,m-1], there are a[bi][bi+1]=1, so think of what? is the matrix multiplication, for just the matrix A We let s=a^i, and then put each a[i][i] add up, so that means: each I start and return to I the number of methods, so it becomes the number of invariant permutation under factor i!

1#include <algorithm>2#include <cstdio>3#include <cmath>4#include <cstring>5#include <iostream>6 Const intMod=9973;7 inta[ the][ the],b[ the][ the],c[ the][ the],s[ the][ the],n,m,k;8 intRead () {9     CharCh=GetChar ();Ten     intt=0, f=1; One      while(ch<'0'|| Ch>'9'){ A         if(ch=='-') f=-1; -Ch=GetChar (); -     } the      while('0'<=ch&&ch<='9'){ -t=t*Ten+ch-'0'; -Ch=GetChar (); -     } +     returnt*F; - } + voidEXGCD (intAintBint&x,int&y) { A     if(b==0){ atx=1; -y=0; -         return; -     } -EXGCD (b,a%b,x,y); -     intt=x; inx=y; -Y= (t-a/b*y); to } + intInvintN) { -     inta,b,x,y; theA=n; b=Mod; * EXGCD (a,b,x,y); $     return(x+mod)%Mod;Panax Notoginseng } - intEulerintN) { the     intRes=n,a=N; +      for(intI=2; i*i<=n;i++) A         if(a%i==0){ theres=res/i* (I-1); +              while(a%i==0) a/=i; -         } $     if(a>1) res=res/a* (A-1); $     returnres%Mod; - } - voidmult_matrix1 () { the      for(intI=1; i<=m;i++) -       for(intj=1; j<=m;j++)Wuyic[i][j]=0; the      for(intI=1; i<=m;i++) -       for(intj=1; j<=m;j++) Wu        for(intk=1; k<=m;k++) -C[i][j]= (C[i][j]+s[i][k]*b[k][j])%Mod; About      for(intI=1; i<=m;i++) $       for(intj=1; j<=m;j++) -s[i][j]=C[i][j];  - } - voidmult_matrix2 () { A      for(intI=1; i<=m;i++) +       for(intj=1; j<=m;j++) thec[i][j]=0; -      for(intI=1; i<=m;i++) $       for(intj=1; j<=m;j++) the        for(intk=1; k<=m;k++) theC[i][j]= (C[i][j]+b[i][k]*b[k][j])%Mod; the      for(intI=1; i<=m;i++) the       for(intj=1; j<=m;j++) -b[i][j]=C[i][j];  in } the intWorkintx) { the      for(intI=1; i<=m;i++) About       for(intj=1; j<=m;j++) theb[i][j]=a[i][j],s[i][j]=0; the     intans=0;  the      for(intI=1; i<=m;i++) +s[i][i]=1;  -      while(x) { the         if(%2) mult_matrix1 ();Bayi mult_matrix2 (); theX/=2; the     }   -      for(intI=1; i<=m;i++) ans= (Ans+s[i][i])%Mod; -     returnans; the } the intMain () { the     intt=read (); the      while(t--){ -N=read (); M=read (); k=read (); the         for(intI=1; i<=m;i++) the           for(intj=1; j<=m;j++) thea[i][j]=1;94         for(intI=1; i<=k;i++){ the             intX=read (), y=read (); thea[x][y]=a[y][x]=0; the        }98        intans=0; About         for(intI=1; i*i<=n;i++) -         if(n%i==0){101Ans= (Ans+work (i) *euler (n/i))%Mod;102             if(i*i!=N)103Ans= (Ans+work (n/i) *euler (i))%Mod;104         } theAns= (ANS*INV (n%mod))%Mod;106printf"%d\n", ans);107     }108}

POJ 2888 Magic Bracelet (Burnside lemma + matrix)