Problem:
Score : 300 points

Problem Statement

Given is an integer sequence D_1,...,D_N of N elements. Find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition:

For every integer i from 1 to N, the distance between Vertex 1 and Vertex i is D_i.

Notes

A tree of N vertices is a connected undirected graph with N vertices and N-1 edges, and the distance between two vertices are the number of edges in the shortest path between them.

Two trees are considered different if and only if there are two vertices x and y such that there is an edge between x and y in one of those trees and not in the other.

Constraints

1 \leq N \leq 10^5

0 \leq D_i \leq N-1

Input

Input is given from Standard Input in the following format:

N
D_1 D_2 ... D_N

Output

Print the answer.

Sample Input 1

4
0 1 1 2

Sample Output 1

2

For example, a tree with edges (1,2), (1,3), and (2,4) satisfies the condition.

Sample Input 2

4
1 1 1 1

Sample Output 2

0

Sample Input 3

7
0 3 2 1 2 2 1

Sample Output 3

24

Solution in Python (3.4.3) (judge verdict: Accepted)

#B
import collections

N = int(input())
D = list(map(int, input().split()))
is_first_zero = True
if D[0] != 0:
    is_first_zero = False
if is_first_zero:
    c = sorted(collections.Counter(D).items(), key=lambda x:x[0])
    if len(c) == 2:
        res = 1
    for index, i in enumerate(c):
        ind = i[0]
        num = i[1]
        if ind == 0:
            if num != 1:
                is_first_zero = False
                break
        else:
            if ind != pre_ind + 1:
                is_first_zero = False
                break
            else:
                if index == 2:
                    res = pre_num ** num
                elif index > 2:
                    res = res * (pre_num ** num)
                #print(res)
        pre_ind = ind
        pre_num = num
                
if is_first_zero:
    print(res % 998244353)
else:
    print("0")