Cost Equilibrium

题意翻译

求将输入的长度为 $n$ 的数组 $a$ 重新排列得到的满足以下条件的不同数组 $b$ 的数量。 - 进行任意次如下操作可以使所有元素相同。 - 找到不同下标 $i$ 和 $j$，选择一整数 $x$，花费 $x \cdot |i-j|$ 的代价将 $a_i$ 变成 $a_i-x$，将 $a_j$ 变成 $a_j + x$。 - $a_i$ 减少后不能再被增加（可以再被减少），$a_j$ 增加后不能再被减少 。 - 进行任意次操作使所有元素相同的所有方案代价相同。

题目描述

An array is called beautiful if all the elements in the array are equal. You can transform an array using the following steps any number of times: 1. Choose two indices $ i $ and $ j $ ( $ 1 \leq i,j \leq n $ ), and an integer $ x $ ( $ 1 \leq x \leq a_i $ ). Let $ i $ be the source index and $ j $ be the sink index. 2. Decrease the $ i $ -th element by $ x $ , and increase the $ j $ -th element by $ x $ . The resulting values at $ i $ -th and $ j $ -th index are $ a_i-x $ and $ a_j+x $ respectively. 3. The cost of this operation is $ x \cdot |j-i| $ . 4. Now the $ i $ -th index can no longer be the sink and the $ j $ -th index can no longer be the source. The total cost of a transformation is the sum of all the costs in step $ 3 $ .For example, array $ [0, 2, 3, 3] $ can be transformed into a beautiful array $ [2, 2, 2, 2] $ with total cost $ 1 \cdot |1-3| + 1 \cdot |1-4| = 5 $ . An array is called balanced, if it can be transformed into a beautiful array, and the cost of such transformation is uniquely defined. In other words, the minimum cost of transformation into a beautiful array equals the maximum cost. You are given an array $ a_1, a_2, \ldots, a_n $ of length $ n $ , consisting of non-negative integers. Your task is to find the number of balanced arrays which are permutations of the given array. Two arrays are considered different, if elements at some position differ. Since the answer can be large, output it modulo $ 10^9 + 7 $ .

输入输出格式

输入格式

The first line contains a single integer $ n $ ( $ 1 \leq n \leq 10^5 $ ) — the size of the array. The second line contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 0 \le a_i \le 10^9 $ ).

输出格式

Output a single integer — the number of balanced permutations modulo $ 10^9+7 $ .

输入输出样例

输入样例 #1

3
1 2 3

输出样例 #1

6

输入样例 #2

4
0 4 0 4

输出样例 #2

2

输入样例 #3

5
0 11 12 13 14

输出样例 #3

120

说明

In the first example, $ [1, 2, 3] $ is a valid permutation as we can consider the index with value $ 3 $ as the source and index with value $ 1 $ as the sink. Thus, after conversion we get a beautiful array $ [2, 2, 2] $ , and the total cost would be $ 2 $ . We can show that this is the only transformation of this array that leads to a beautiful array. Similarly, we can check for other permutations too. In the second example, $ [0, 0, 4, 4] $ and $ [4, 4, 0, 0] $ are balanced permutations. In the third example, all permutations are balanced.