Max Sum Array

2000ms

262144K

Description:

You are given an array $$$c = [c_1, c_2, \dots, c_m]$$$. An array $$$a = [a_1, a_2, \dots, a_n]$$$ is constructed in such a way that it consists of integers $$$1, 2, \dots, m$$$, and for each $$$i \in [1,m]$$$, there are exactly $$$c_i$$$ occurrences of integer $$$i$$$ in $$$a$$$. So, the number of elements in $$$a$$$ is exactly $$$\sum\limits_{i=1}^{m} c_i$$$.

Let's define for such array $$$a$$$ the value $$$f(a)$$$ as $$$$$$f(a) = \sum_{\substack{1 \le i < j \le n\\ a_i = a_j}}{j - i}.$$$$$$

In other words, $$$f(a)$$$ is the total sum of distances between all pairs of equal elements.

Your task is to calculate the maximum possible value of $$$f(a)$$$ and the number of arrays yielding the maximum possible value of $$$f(a)$$$. Two arrays are considered different, if elements at some position differ.

Input:

The first line contains a single integer $$$m$$$ ($$$1 \le m \le 5 \cdot 10^5$$$) — the size of the array $$$c$$$.

The second line contains $$$m$$$ integers $$$c_1, c_2, \dots, c_m$$$ ($$$1 \le c_i \le 10^6$$$) — the array $$$c$$$.

Output:

Print two integers — the maximum possible value of $$$f(a)$$$ and the number of arrays $$$a$$$ with such value. Since both answers may be too large, print them modulo $$$10^9 + 7$$$.

Sample Input:

6
1 1 1 1 1 1

Sample Output:

0 720

Sample Input:

1
1000000

Sample Output:

499833345 1

Sample Input:

7
123 451 234 512 345 123 451

Sample Output:

339854850 882811119

Note:

In the first example, all possible arrays $$$a$$$ are permutations of $$$[1, 2, 3, 4, 5, 6]$$$. Since each array $$$a$$$ will have $$$f(a) = 0$$$, so maximum value is $$$f(a) = 0$$$ and there are $$$6! = 720$$$ such arrays.

In the second example, the only possible array consists of $$$10^6$$$ ones and its $$$f(a) = \sum\limits_{1 \le i < j \le 10^6}{j - i} = 166\,666\,666\,666\,500\,000$$$ and $$$166\,666\,666\,666\,500\,000 \bmod{10^9 + 7} = 499\,833\,345$$$.

Informação

Provedor Codeforces

Código CF1612G