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: Runtime Error)

#include<stdio.h>
#include<stdlib.h>

typedef long long ll;

int a[100000];
int D[100000];

int comp(const void* x, const void* y){
    return D[*(int *)x] - D[*(int *)y];
}

int main(void){
    int N;
    ll ans = 1;
    ll node[10];
    ll via = 1;
    int cnt = 1;
    int node_sum = 1;
    node[0] = 1;


    scanf("%d", &N);
    int i;
    for(i = 0; i< N; i++){
        scanf("%d", &D[i]);
        a[i] = i;
    }
    if(D[0] != 0){
        printf("0");
        return 0;
    }
    qsort(a, N, sizeof(int), comp);
    if(D[a[1]] != 1){
        printf("0");
        return 0;
    }
    int j;
    for(i = 1; i < N; i++){
        if(cnt != D[a[i]] && cnt != 0){
            node[cnt] = i-node_sum;
            node_sum += node[cnt];
            cnt++;
        }
    }
    node[cnt] = i-node_sum;

    for(i = 1; i < N; i++){
        if(D[a[i-1]] != D[a[i]]){
            for(j = 0; j < node[D[a[i]]]; j++){
                via *= node[D[a[i-1]]];
            }
            ans *= via;
            via = 1;
        }

    }
    
    printf("%lld", ans);
    return 0;
}