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 C (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>

int main(){
  int n;
  scanf("%d", &n);
  int d[n];
  int num[n];
  for(int i = 0; i < n; i++){
    scanf("%d", &d[i]);
    num[i] = 0;
  }
  int max = 0;
  int zr = 0;
  for(int i = 0; i < n; i++){
    num[d[i]]++;
    if(d[i] > max){
      max = d[i];
    }
    if(d[i] == 0){
      zr++;
    }
  }
  long long int ans = 1;
  for(int i = 0; i < max; i++){
    for(int j = 0; j < num[i + 1]; j++){
      ans = ans * num[i] % 998244353;
    }
  }
  if((zr == 1) && (d[0] == 0)){
    printf("%lld\n", ans);
  }else{
    printf("0\n");
  }

  return 0;
}