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 Go (1.6) (judge verdict: Accepted)

package main
import (
    "bufio"
    "fmt"
    "os"
    "strconv"
)
var sc *bufio.Scanner
func nextStr() string {
    sc.Scan()
    return sc.Text()
}
func nextInt() int {
    sc.Scan()
    i, e := strconv.Atoi(sc.Text())
    if e != nil {
        panic(e)
    }
    return i
}
const MOD = 998244353
func powInt(x, y int) int {
    ret := 1
    for i := 0; i < y; i++ {
        ret *= x
        ret %= MOD
    }
    return ret
}
func main() {
    sc = bufio.NewScanner(os.Stdin)
    sc.Buffer(make([]byte, 0), 1000000001*3)
    sc.Split(bufio.ScanWords)
    n := nextInt()
    depth := make([]int, n)
    d := nextInt()
    if d != 0 {
        fmt.Println(0)
        return
    }
    depth[d]++
    for i := 1; i < n; i++ {
        d := nextInt()
        depth[d]++
    }
    if depth[0] > 1 {
        fmt.Println(0)
        return
    }
    ans := 1
    dPrev := 1
    for _, d := range depth {
        switch {
        case d == 0:
            dPrev = 0
            continue
        case dPrev == 0:
            fmt.Println(0)
            return
        default:
            ans *= powInt(dPrev, d)
            ans %= MOD
            dPrev = d
        }
    }
    fmt.Println(ans)
}