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: Runtime Error)

mod = 998244353
N = int(input())
D = [int(i) for i in input().split()]
from collections import defaultdict as ddct
import numpy as np
def sum_matrix(Pre,Aft):
    matrix = np.zeros((Aft,Pre))
    matrix[0,:] = np.arange(Pre)+1
    matrix[:,0] = 1
    for pre in range(1,Pre):
        for aft in range(1,Aft):
            matrix[aft,pre] = matrix[aft-1,pre] + matrix[aft,pre-1]
    return int(matrix[Aft-1,Pre-1])
length_dict = ddct(int)
enable = True
for i,d in enumerate(D):
    if i==0 and d != 0:
        enable  *= False
        break
    elif  i==0 and d == 0:
        pass
    elif d == 0:
        enable  *= False
        break
        
    length_dict[d] += 1
ans = 1
for i in range(0,max(D)+1):
    if enable == False:
        break
    if i == 0:
        ans *= 1
    else:
        ans *= sum_matrix(length_dict[i-1],length_dict[i]) if length_dict[i-1] != 1 else length_dict[i]
        ans %=mod
print(ans) if enable == True else print(0)