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 Ruby (2.3.3) (judge verdict: Wrong Answer)


MOD = 998244353

def solve

  n = gets.to_i
  dists = gets.split.map(&:to_i)


  dists.sort!

  orig = dists[0]
  if dists.count(0) != 1
    puts "0"
    return
  end

  current_dist = 0
  current_dist_num = 0
  parent_dist_num = 1
  combs = []
  res = 0


  dists.each do |d|
    if current_dist != d
      if d - current_dist > 1
        puts "0"
        return
      end
      combs << parent_dist_num ** current_dist_num
      parent_dist_num = current_dist_num
      current_dist_num = 0
      # puts "dist: #{d - 1}: #{parent_dist_num} nodes"
      current_dist = d 
    end
    # puts "current: #{current_dist}"
    current_dist_num += 1
  end

  combs << parent_dist_num ** current_dist_num


  # p combs
  puts combs.inject(:*) % MOD
end

solve()