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 C (GCC 5.4.1) (judge verdict: Accepted)

#include<stdio.h>
#include<stdlib.h>
#include<string.h>
#include<stdint.h>
#include<stdbool.h>
#include<tgmath.h>
#include<sys/queue.h>
#include<time.h>
#define ll long long
#define min(a, b) ((a) > (b) ? (b) : (a))
#define max(a, b) ((a) > (b) ? (a) : (b))
#define miin(a, b, c) min((a), min((b), (c)))
#define maax(a, b, c) max((a), max((b), (c)))

#define MOD 998244353

ll hoge(int x, int y){
    ll res = 1;
    for(int i=0; i<y; i++){
        res *= x;
        res %= MOD;
    }
    return res;
}

int n, d, t[111111] = {}, maxi = 0;
ll ans = 1;

main(){
    scanf("%d", &n);
    for(int i=0; i<n; i++){
        scanf("%d", &d);
        if(!i){if(d){
            puts("0");
            return 0;
        }}else{
            if(!d){
                puts("0");
                return 0;
            }
        }
        t[d]++;
        maxi = max(maxi, d);
    }
    for(int i=1; i<=maxi; i++){
        ans *= hoge(t[i-1], t[i]);
        ans %= MOD;
    }
    printf("%lld\n", ans);
    return 0;
}