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2020-02-16 18:54:00
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f[i,j]表示用完了前i种油漆,有j对相邻且同色的木块

考虑怎么转移:

要将第i种颜色的木块分两种情况插入:

  1. 设其中a个插到之前弄好的木块中间(无序)

  2. 设另外b个插在同色木块中间

其中有a-b组满足插空放且不相邻

f[i,j]=\sum f[i-1,j-(c_i-a-b)]\times {c_i-1 \choose a-1} \times {j \choose b} \times{S_{i-1}+1-j \choose a-b}

{c_i-1 \choose a-1}表示a个的方案数

{j \choose b}表示b个的方案数

{S_{i-1}+1-j \choose a-b}表示a-b个插进去的方案

#include<bits/stdc++.h>
namespace ZDY{
    #pragma GCC optimize(3)
    #define il __inline__ __attribute__ ((always_inline))
    #define register
    #define ll long long
    #define ull unsigned long long
    #define db double
    #define sht short
    #define MB template <class T>il
    #define Fur(i,x,y) for(int i(x);i<=y;++i)
    #define Fdr(i,x,y) for(int i(x);i>=y;--i)
    #define fl(i,x) for(int i(head[x]),to;to=e[i].to,i;i=e[i].nxt)
    #define clr(x,y) memset(x,y,sizeof(x))
    #define cpy(x,y) memcpy(x,y,sizeof(x))
    #define fin(s) freopen(s".in","r",stdin)
    #define fout(s) freopen(s".out","w",stdout)
    #define fcin ios::sync_with_stdio(false)
    #define l2(n) ((int)(log2(n)))
    #define inf 2122219134
    MB T ABS(T x){return x>0?x:-x;}
    MB T MAX(T x,T y){return x>y?x:y;}
    MB T MIN(T x,T y){return x<y?x:y;}
    MB T GCD(T x,T y){return y?GCD(y,x%y):x;}
    MB void SWAP(T &x,T &y){T t=x;x=y;y=t;}
}using namespace ZDY;using namespace std;
namespace IO{const int str=1<<20;static char in_buf[str],*in_s,*in_t;bool __=0;il char gc(){return (in_s==in_t)&&(in_t=(in_s=in_buf)+fread(in_buf,1,str,stdin)),in_s==in_t?__=1,EOF:*in_s++;}il void in(string &ch){ch.clear();if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}ch+=c;while((c=gc())!=EOF&&!isspace(c))ch+=c;if(c==EOF)__=1;}il void in(char &ch){if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)__=1;else ch=c;}il void in(char *ch){*ch='\0';if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}*ch=c;ch++;while((c=gc())!=EOF&&!isspace(c))*ch=c,ch++;if(c==EOF)__=1;*ch='\0';}template<typename T>il void in(T &x){if(__)return;char c=gc();bool f=0;while(c!=EOF&&(c<'0'||c>'9'))f^=(c=='-'),c=gc();if(c==EOF){__=1;return;}x=0;while(c!=EOF&&'0'<=c&&c<='9')x=x*10+c-48,c=gc();if(c==EOF)__=1;if(f)x=-x;}template<typename T,typename ... arr>il void in(T &x,arr & ... y){in(x),in(y...);}const char ln='\n';static char out_buf[str],*out_s=out_buf,*out_t=out_buf+str;il void flush(){fwrite(out_buf,1,out_s-out_buf,stdout);out_s=out_buf;}il void pt(char c){(out_s==out_t)?(fwrite(out_s=out_buf,1,str,stdout),*out_s++=c):(*out_s++=c);}il void out(const char* s){while(*s)pt(*s++);}il void out(char* s){while(*s)pt(*s++);}il void out(char c){pt(c);}il void out(string s){for(int i=0;s[i];i++)pt(s[i]);}template<typename T>il void out(T x){if(!x){pt('0');return;}if(x<0)pt('-'),x=-x;char a[50],t=0;while(x)a[t++]=x%10,x/= 10;while(t--)pt(a[t]+'0');}template<typename T,typename ... arr>il void out(T x,arr & ... y){out(x),out(y...);}}using namespace IO;
int n,c[20],f[20][80],C[80][80],P=1000000007;
int main(){
    C[0][0]=1;
    Fur(i,1,75){
        C[i][0]=C[i][i]=1;
        Fur(j,1,i-1)
            C[i][j]=(C[i-1][j]+C[i-1][j-1])%P;
    }
    in(n);
    Fur(i,1,n)in(c[i]);
    f[1][c[1]-1]=1;
    int s=0;
    Fur(i,1,n-1){
        s+=c[i];
        Fur(j,0,s-1)if(f[i][j])
            Fur(a,1,c[i+1])
                Fur(b,0,MIN(a,j))
                f[i+1][j+c[i+1]-a-b]=(f[i+1][j+c[i+1]-a-b]+
                1ll*f[i][j]*C[c[i+1]-1][a-1]%P*C[j][b]%P*C[s+1-j][a-b]%P)%P;
    }
    printf("%d\n",f[n][0]);
}
LG 2476 [SCOI2008]着色方案
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