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)

def calc(n, D):
    if D[0] != 0:
        return 0
    count = [0 for i in range(n)]
    for i in D:
        count[i] = count[i] + 1
    if count[0] != 1:
        return 0
    #for i in count:
    #    print("count",i)
    tmp = 1
    sum = 0
    for i in range(n-1):
    #    print(count[i],count[i+1])
        sum = sum + count[i]
        if sum == n:
            break
        if count[i] == 0:
            tmp = 0
            break
        tmp = tmp * (count[i]**count[i+1])
    return tmp

n = int(input())
#print("n=",n)
D = list(map(int, input().split()))

print(calc(n,D) % 998244353)