2015
09-17

# Cut Pieces

Suppose we have a sequence of n blocks. Then we paint the blocks. Each block should be painted a single color and block i can have color 1 to color ai. So there are a total of prod(ai) different ways to color the blocks.
Consider one way to color the blocks. We call a consecutive sequence of blocks with the same color a "piece". For example, sequence "Yellow Yellow Red" has two pieces and sequence "Yellow Red Blue Blue Yellow" has four pieces. What is S, the total number of pieces of all possible ways to color the blocks?

This is not your task. Your task is to permute the blocks (together with its corresponding ai) so that S is maximized.

First line, number of test cases, T.
Following are 2*T lines. For every two lines, the first line is n, length of sequence; the second line contains n numbers, a1, …, an.

Sum of all n <= 106.
All numbers in the input are positive integers no larger than 109.

First line, number of test cases, T.
Following are 2*T lines. For every two lines, the first line is n, length of sequence; the second line contains n numbers, a1, …, an.

Sum of all n <= 106.
All numbers in the input are positive integers no larger than 109.

1
3
1 2 3

14
Hint
Both sequence 1 3 2 and sequence 2 3 1 result in an S of 14.


#include <stdio.h>
#include <string.h>
#include <algorithm>
#include <iostream>
using namespace std;
#define LL __int64
const LL mod=1e9+7;
LL a[1000001],b[1000001];
LL pows(LL a,LL b)
{
LL s=1;
a=a%mod;
while(b)
{
if(b&1)
s=(s*a)%mod;
a=(a*a)%mod;
b=b>>1;
}
return s;
}
int main()
{
LL T;
scanf("%I64d",&T);
while(T--)
{
LL n,i,j,k,mul=1,p;
scanf("%I64d",&n);
for(i=0;i<n;i++)
{
scanf("%I64d",&a[i]);
mul=(a[i]*mul)%mod;
}
sort(a,a+n);
for(i=0,j=0;i<n;i=i+2,j++)
b[i]=a[j];
for(i=(n%2==0?n-1:n-2),k=j;i>=0;k++,i=i-2)
b[i]=a[k];
LL s=0;
for(i=1;i<n;i++)
{
p=(mul*pows(b[i]*b[i-1],mod-2))%mod;
s=(s+min(b[i],b[i-1])*p%mod)%mod;
}
s=(mul*n-s+mod)%mod;
printf("%I64d\n",s);
}
return 0;
}
/*

输入一组n长的序列ai，表示n个格子的各自能涂的颜色数，颜色为1~ai。求找出ai的一种

1,3,2
颜色组合共6种，(1,1,1),(1,1,2),(1,2,1),(1,2,2),(1,3,1),(1,3,2)
各自的片数 为1,2,3,2,3,3，共14片

对于序列：a1,a2,a3,a4,a5，总的颜色组合有s=a1*a2*a3*a4*a5
如果不计颜色连续，有s*5片
其中重复的有p=min(a1,a2)*s/(a1*a2)+min(a2,a3)*s/(a2*a3)+...
要让s*5-p最大，p要尽可能小，所以将序列排序，然后排成一小一大一小。。顺序
如a1<a3<a5<a4<a2

*/