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: Wrong Answer)

#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <inttypes.h>
#include <math.h>

/* ソート関数 */
int int_sort( const void * a , const void * b ) {
  /* 引数はvoid*型と規定されているのでint型にcastする */
  if( *( uint32_t * )a < *( uint32_t * )b ) {
    return -1;
  }
  else
  if( *( uint32_t * )a == *( uint32_t * )b ) {
    return 0;
  }
  return 1;
}

int
main (void)
{
  uint32_t N, i, D[100000], distance_num[100000] = {0, };
  uint64_t ans = 1, distance;

  scanf("%d", &N);
  for (i = 0; i < N; i++) {
    scanf("%d", &D[i]);
  }

  if (D[0] != 0) {
    printf("0");
    return 0;
  }

  qsort(D, N, sizeof(uint32_t), int_sort);

  for (i = 0, distance = 0; i < N; i++) {
    if (D[i] == distance)
      distance_num[distance]++;
    else if (D[i] == distance + 1) {
      distance++;
      distance_num[distance]++;
    } else {
      printf("0");
      return 0;
    }
  }

  for (i = 0; i < distance; i++) {
    ans *= (uint64_t) pow(distance_num[i], distance_num[i+1]) % 998244353;
  }
  printf("%"PRIu64"\n", ans);

  return 0;
}