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 Kotlin (1.3.71) (judge verdict: Wrong Answer)

// https://atcoder.jp/contests/nikkei2019-2-qual/tasks/nikkei2019_2_qual_b

import kotlin.math.pow

fun main(args: Array<String>) {
    nikkei20192qualb()
}

fun nikkei20192qualb() {
    // TODO: 頭が真っ白になったら再挑戦するのもいいかもね
    val n = readLine()!!.toInt()
    val d = readLine()!!.split(" ").map { it.toInt() }
    val appear = d.groupingBy { it }.eachCount()
    val destinationMax = appear.keys.max() ?: 0
    if (d.first() != 0 || appear[0] != 1) {
        println("0")
        return
    }
    var pattern = 1L
    for (i in 1..destinationMax) {
        val p = appear[i - 1] ?: 0
        val c = appear[i] ?: 0
        if (p == 0 || c == 0) {
            println(0)
            return
        }
        val f = p.toDouble().pow(c).toLong()
        pattern *= f % 998244353L
    }
    println(pattern)
}