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)

import sys

N = int(input())
#D = list(sorted(list(map(int, input().split()))))
D = list(map(int, input().split()))

check = 0
if not check == D[0]:
    print(0)
    sys.exit()

countl = [0]
D = list(sorted(D))
if not check == D[0]:
    print(0)
    sys.exit()
countl = [0]
for d in D:
    if not check == d:
        check += 1
        countl.append(0)
        if not check == d:
            print(0)
            sys.exit()
    countl[-1] += 1

if not countl[0] == 1:
    print(0)
    sys.exit()

ans = 1
for i in range(len(countl) - 1):
    ans *= pow(countl[i], countl[i + 1], 998244353)
print(ans % 998244353)