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: Accepted)

class Integer
  def combination(k)
    return 1 if k.zero?

    (self - k + 1..self).inject(:*) / k.factorial
  end

  def permutation(k)
    return 1 if k.zero?

    (self - k + 1..self).inject(:*)
  end

  def factorial
    return 1 if self.zero?

    (1..self).inject(:*)
  end
end

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

flag = 1

flag = 0 if d[0] != 0
ds = Hash.new(0)
d.each do |dn|
  ds[dn] += 1
end
prek = -1
ds.sort.each do |k,v|
  flag = 0 if k == 0 && 1 < v
  if prek != -1
    flag = 0 if prek+1 != k
  end
  prek = k
end

if flag == 1
  ans = 1
  prev = 1
  ds.sort.each do |k,v|
    if k >= 1
      ans *= prev**v
      prev = v
    end
  end
  puts ans % 998244353
else
  puts 0
end