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: Wrong Answer)

from collections import Counter

def cal(MC,m):
    n = len(MC)
    for i in range(n):
        if MC[i][0] != i:
            #歩ささiと深さi-1又は深さiと深さi+1を繋げない
            return 0
    if MC[0][1] != 1:
        #根が1以外にある
        return 0

    output = 1
    for i in range(n-1):
        output = output * (MC[i][1]**MC[i+1][1] % m)
        output %= m
    return output

def main():
    N = int(input())
    D = list(map(int,input().split()))
    M = 998244353
    #頂点1が根じゃないときも連結の仕方を変えずに見方を変えることで
    #頂点1を根にできる.
    #従ってD[i]は頂点iの深さと見なせる
    C = Counter(D)
    MC = C.most_common()
    for i in range(len(MC)):
        MC[i] = list(MC[i])
    MC.sort(key = lambda x:x[0])
    ans = cal(MC,M)
    print(ans)

if __name__ == "__main__":
    main()