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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

/**
 * Created at 21:02 on 2019-11-09
 */

import java.io.*;
import java.util.*;

public class Main {

  static FastScanner sc = new FastScanner();
  static Output out = new Output(System.out);

  static final int[] dx = {0, 1, 0, -1};
  static final int[] dy = {-1, 0, 1, 0};

  static final long MOD = (long)998244353;
  static final long INF = Long.MAX_VALUE / 2;

  static final int e5 = (int) 1e5;

  public static class Solver {
    public Solver() {

      int N = sc.nextInt();

      long[] count = new long[(int)1e5+1];

      for (int i=0; i<N; i++) {
        count[sc.nextInt()]++;
      }

      long ans = 1;

      if (count[0] != 1) ans = 0;

      for (int i=1; i<=1e5; i++) {
        if (count[i] > 0) {
          ans = Mod.mul(ans, Mod.power(count[i-1], count[i]));
        }
      }

      out.println(ans);

    }

    public static class Mod {

      static final long MOD = (long)998244353;

      public static long[] fact;

      public Mod(int n) {
        fact = new long[n+1];
      }

      public Mod(long n) {
        fact = new long[(int)n+1];
      }

      public static long comb(long n, long r) {
        if (r < 0 || r > n) return 0;
        return (((factorial(n) * factorialInverse(r)) % MOD) * factorialInverse(n - r)) % MOD;
      }

      public static long factorial(long n) {
        if (n == 0) return 1;
        if (fact[(int) n] == 0) {
          return fact[(int) n] = (n * factorial(n - 1)) % MOD;
        } else {
          return fact[(int) n];
        }
      }

      public static long factorialInverse(long n) {
        return inverse(factorial(n));
      }

      public static long inverse(long n) {
        return power(n, MOD - 2);
      }

      public static long power(long a, long b) {
        if (b == 0) return 1;

        if (b % 2 == 0) {
          long p = power(a, b / 2);
          return (p * p) % MOD;
        } else {
          return (a * power(a, b - 1)) % MOD;
        }
      }

      public static long add(long... a) {
        long ans = 0;
        for (int i=0; i<a.length; i++) {
          ans = (ans + a[i]) % MOD;
        }
        return ans;
      }

      public static long sub(long a, long b) {
        return (a - b + MOD) % MOD;
      }

      public static long mul(long... a) {
        long ans = 1;
        for (int i=0; i<a.length; i++) {
          ans = (ans * a[i]) % MOD;
        }
        return ans;
      }

      public static long div(long a, long b) {
        return (a * Mod.inverse(b)) % MOD;
      }

      public static long piano(long u, long d) {
        long a = add(u, 1);
        long b = d;
        long sum = add(a, b);
        long dif = sub(a, b);

        return mul(div(dif, sum), comb(sum, a));
      }

      public static long catalan(long n) {
        return piano(n, n);
      }

    }
  }

  public static void main(String[] args) {
    new Solver();
    out.flush();
  }

  static class FastScanner {
    private InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;


    public void load() {
      try {
        in = new FileInputStream(next());
      } catch (Exception e) {
        e.printStackTrace();
      }
    }

    private boolean hasNextByte() {
      if (ptr < buflen) {
        return true;
      } else {
        ptr = 0;
        try {
          buflen = in.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (buflen <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) return buffer[ptr++];
      else return -1;
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    private void skipUnprintable() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
    }

    public boolean hasNext() {
      skipUnprintable();
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) throw new NoSuchElementException();
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    public long nextLong() {
      if (!hasNext()) throw new NoSuchElementException();
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }

    public int nextInt() {
      return (int) nextLong();
    }

    public int[] nextIntArray(int N, boolean oneBased) {
      if (oneBased) {
        int[] array = new int[N + 1];
        for (int i = 1; i <= N; i++) {
          array[i] = sc.nextInt();
        }
        return array;
      } else {
        int[] array = new int[N];
        for (int i = 0; i < N; i++) {
          array[i] = sc.nextInt();
        }
        return array;
      }
    }

    public long[] nextLongArray(int N, boolean oneBased) {
      if (oneBased) {
        long[] array = new long[N + 1];
        for (int i = 1; i <= N; i++) {
          array[i] = sc.nextLong();
        }
        return array;
      } else {
        long[] array = new long[N];
        for (int i = 0; i < N; i++) {
          array[i] = sc.nextLong();
        }
        return array;
      }
    }
  }

  static class Output extends PrintWriter {

    private long startTime;

    public Output(PrintStream ps) {
      super(ps);
    }

    public void print(int[] a, String separator) {
      for (int i = 0; i < a.length; i++) {
        if (i == 0) print(a[i]);
        else print(separator + a[i]);
      }
      println();
    }

    public void print(long[] a, String separator) {
      for (int i = 0; i < a.length; i++) {
        if (i == 0) print(a[i]);
        else print(separator + a[i]);
      }
      println();
    }

    public void print(String[] a, String separator) {
      for (int i = 0; i < a.length; i++) {
        if (i == 0) print(a[i]);
        else print(separator + a[i]);
      }
      println();
    }

    public void print(ArrayList a, String separator) {
      for (int i = 0; i < a.size(); i++) {
        if (i == 0) print(a.get(i));
        else print(separator + a.get(i));
      }
      println();
    }

    public void start() {
      startTime = System.currentTimeMillis();
    }

    public void time(String s) {
      long time = System.currentTimeMillis() - startTime;
      println(s + "(" + time + ")");
    }

  }

}