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 Scala (2.11.7) (judge verdict: Wrong Answer)

import scala.io.StdIn._

object Main extends App {
  def pow(base: Long, exp: Int, result: Long = 1): Long =
    if (exp == 0) result
    else pow(base, exp - 1, result * base % 998244353)

  def count(l: List[(Int, Int)], cur: Int, nPre: Int, result: Long): Long = l match {
    case Nil => result
    case (`cur`, n) :: xs => count(xs, cur + 1, n, result * pow(nPre, n) % 998244353)
    case _ => 0
  }

  readLine
  val ds = readLine.split(" ").map(_.toInt)
  val sorted: List[(Int, Int)] = ds.groupBy(identity).mapValues(_.size).toList.sorted
  println(if (sorted.head._2 != 1) 0 else count(sorted, 0, 1, 1))
}