Problem:
Score : 600 points

Problem Statement

We have a grid with A horizontal rows and B vertical columns, with the squares painted white. On this grid, we will repeatedly apply the following operation:

Assume that the grid currently has a horizontal rows and b vertical columns. Choose "vertical" or "horizontal".

If we choose "vertical", insert one row at the top of the grid, resulting in an (a+1) \times b grid.

If we choose "horizontal", insert one column at the right end of the grid, resulting in an a \times (b+1) grid.

Then, paint one of the added squares black, and the other squares white.

Assume the grid eventually has C horizontal rows and D vertical columns. Find the number of ways in which the squares can be painted in the end, modulo 998244353.

Constraints

1 \leq A \leq C \leq 3000

1 \leq B \leq D \leq 3000

A, B, C, and D are integers.

Input

Input is given from Standard Input in the following format:

A B C D

Output

Print the number of ways in which the squares can be painted in the end, modulo 998244353.

Sample Input 1

1 1 2 2

Sample Output 1

3

Any two of the three squares other than the bottom-left square can be painted black.

Sample Input 2

2 1 3 4

Sample Output 2

65

Sample Input 3

31 41 59 265

Sample Output 3

387222020

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;
using static System.Console;
using static System.Math;
using static MyIO;
using static MyUtil;

public class B { public static void Main() => (new Solver()).Solve(); }

public class Solver
{
	public void Solve()
	{
		long A = GetLong();
		long B = GetLong();
		long C = GetLong();
		long D = GetLong();

		var dp = new MInt[C + 1, D + 1];
		dp[A, B] = new MInt(1);
		for(long i = A; i <= C; i++)
		{
			for(long j = B; j <= D; j++)
			{
				if(i > A)
					dp[i, j] += dp[i - 1, j] * j;
				
				if(j > B)
					dp[i, j] += dp[i, j - 1] * i;

				if(i > A && j > B)
					dp[i, j] -= dp[i - 1, j - 1] * (i - 1) * (j - 1);
			}
		}
		WriteLine(dp[C, D].Val);
	}
}

public struct MInt
{
	private const long MOD = 998244353;
	private static MInt[] f = null;
	private static MInt[] g = null;
	public long Val { get; private set; }

	public MInt(long v)
	{
		Val = (v + MOD) % MOD;
	}

	public static void InitTable(int N) => InitTable((long)N);
	public static void InitTable(long N)
	{
		f = new MInt[N + 1];
		f[0] = new MInt(1);
		for(long i = 1; i <= N; i++)
			f[i] = f[i - 1] * i;

		g = new MInt[N + 1];
		g[N] = MPow(f[N], MOD - 2);
		for(long i = N; i > 0; i--)
			g[i - 1] = g[i] * i;
	}

	public static MInt MComb(long n, long k)
	{
		if(n < 0 || n < k)
			return new MInt(0);
			
		return f[n] * g[k] * g[n - k];
	}

	public static MInt operator+(MInt x, long y) => x + new MInt(y);	
	public static MInt operator+(long x, MInt y) => new MInt(x) + y;
	public static MInt operator+(MInt x, MInt y)
	{
		long tmp = x.Val + y.Val;
		if(tmp >= MOD)
			tmp -= MOD;
		return new MInt(tmp);
	}

	public static MInt operator-(MInt x, long y) => x - new MInt(y);
	public static MInt operator-(long x, MInt y) => new MInt(x) - y;
	public static MInt operator-(MInt x, MInt y)
	{
		long tmp = x.Val;
		if(tmp < y.Val)
			tmp += MOD;
		tmp -= y.Val;
		return new MInt(tmp);
	}

	public static MInt operator*(MInt x, MInt y) => new MInt((x.Val * y.Val) % MOD);
	public static MInt operator*(MInt x, long y) => x * new MInt(y);
	public static MInt operator*(long x, MInt y) => new MInt(x) * y;

	public static MInt operator/(MInt x, MInt y) => x * MPow(y, MOD - 2);
	public static MInt operator/(MInt x, long y) => x * MPow(y, MOD - 2);
	public static MInt operator/(long x, MInt y) => x * MPow(y, MOD - 2);

	public static MInt MPow(MInt x, MInt y) => MPow(x.Val, y.Val);
	public static MInt MPow(MInt x, long y) => MPow(x.Val, y);
	public static MInt MPow(long x, MInt y) => MPow(x, y.Val);
	public static MInt MPow(long x, long y)
	{
		long exp = y;
		long xx = x;
		long res = 1;
		while(exp > 0)
		{
			if((exp & 1) == 1)
				res = (res * xx) % MOD;

			xx = (xx * xx) % MOD;
			exp >>= 1;
		}
		return new MInt(res);
	}
}

public static class MyIO
{
	private static string[] args = new string[0];
	private static int cnt = 0;
	private static IEnumerable<string> getArgs(long N) => Range(0, N).Select(_ => getArg());
	private static string getArg()
	{
		if(cnt == args.Length) { args = Console.ReadLine().Split(); cnt = 0; }
		return args[cnt++];
	}

	public static int GetInt() => int.Parse(getArg());
	public static long GetLong() => long.Parse(getArg());
	public static double GetDouble() => double.Parse(getArg());
	public static decimal GetDecimal() => decimal.Parse(getArg());
	public static string GetString() => getArg();
	public static char GetChar() => getArg()[0];
	public static int[] GetInts(long N) => getArgs(N).Select(int.Parse).ToArray();
	public static long[] GetLongs(long N) => getArgs(N).Select(long.Parse).ToArray();
	public static double[] GetDoubles(long N) => getArgs(N).Select(double.Parse).ToArray();
	public static decimal[] GetDecimals(long N) => getArgs(N).Select(decimal.Parse).ToArray();
	public static string[] GetStrings(long N) => getArgs(N).ToArray();
}

public static class MyUtil
{
	public static void chmax<T>(ref T x, T y) where T : IComparable<T> { if(x.CompareTo(y) < 0) x = y; }
	public static void chmin<T>(ref T x, T y) where T : IComparable<T> { if(x.CompareTo(y) > 0)	x = y; }
	public static void swap<T>(ref T x, ref T y) { T tmp = x; x = y; y = tmp; }
	public static long rightShift(long x, long y) => (x >> (int)y);
	public static long leftShift(long x, long y) => (x << (int)y);
	public static bool contains(long s, long x) => (rightShift(s, x) & 1L) == 1L;
	public static long gcd(long x, long y) => (y == 0) ? x : gcd(y, x % y);
	public static long lcm(long x, long y) => (x / gcd(x,y)) * y;	
	public static IEnumerable<int> Range(long start, long count) => Enumerable.Range((int)start, (int)count);
	public static void Sort(this List<string> source) => source.Sort(StringComparer.Ordinal);
}