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 JavaScript (node.js v5.12) (judge verdict: Wrong Answer)

"use strict";
    
function factorial(num) {
  if (num == 1) {
    return num;
  } else {
    return num * factorial(num - 1);
  }
}

const main = arg => {
    arg = arg.trim().split("\n");
    const N = parseInt(arg[0]);
    const D = arg[1].split(" ").map(n=>parseInt(n));
    
    const max = Math.max(...D);
    
    if(D[0] !== 0) {
        console.log(0);
        return;
    }
    
    let answer = 1;
    let list = {0: 1};
    
    for(let i=1; i<N; i++) {
        if(list[D[i]]) {
           list[D[i]]++; 
        } else {
           list[D[i]] = 1;
        }
    }
    
    // console.log(JSON.stringify(list))
    
    for(let i=1; i<=max; i++) {
        // 親としてありえる個数
        const parents = list[i-1];
        // console.log("親の候補個数:" + parents);
        
        const temp = Math.pow(parents, list[i])
        // console.log(i + ":" + temp)
        
        answer *= temp;
    }
    
    console.log(answer % 998244353);
}
main(require('fs').readFileSync('/dev/stdin', 'utf8'));