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 (
	"fmt"
)

func main() {
	var N int
	fmt.Scan(&N)

	V := make(map[int]int)
	var d int
	l := 0
	for i := 0; i < N; i++ {
		fmt.Scan(&d)
		if i == 0 && d != 0 {
			fmt.Println(0)
			return
		}

		V[d]++
		if l < d {
			l = d
		}
	}

	if V[0] != 1 {
		fmt.Println(0)
		return
	}

	ans := 1
	p := 1
	for i := 0; i <= l; i++ {
		if v, ok := V[i]; ok {
			for j := 0; j < v; j++ {
				ans = (ans * p) % 998244353
			}
			p = v
		} else {
			fmt.Println(0)
			return
		}
	}
	fmt.Println(ans)
}