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 Perl (v5.18.2) (judge verdict: Wrong Answer)

#!/usr/bin/perl

my $md = 998244353;

my ($n) = map { $_ - 0 } split(/\s+/,<STDIN>);
my @d = map { $_ - 0 } split(/\s+/,<STDIN>);

if( $d[0] != 0 ){
  print "0\n";
  exit(0);
}

if( $n == 1 ){
  print "1\n";
  exit(0);
}

my $mx = 0;
my @ct = ();
$#ct = $n;
for(my $i=1;$i<$n;$i++){
  $ct[$d[$i]] ++;
  $mx = $d[$i] if $d[$i] > $mx;
}

my $r = 1;
my $c = 1;
for(my $i=1;$i<=$mx;$i++){
  my $c1 = $ct[$i] - 0;
  my $a = 1;
  for(my $j=0;$j<$c1;$j++){
    $a = ( $a * $c ) % $md;
  }
  $r = ( $r * $a ) % $md;
  $c = $c1;
}
print "$r\n";