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 Go (1.14.1) (judge verdict: Accepted)

package main

import (
	"bufio"
	"fmt"
	"os"
	"sort"
	"strconv"
)

func out(x ...interface{}) {
	fmt.Println(x...)
}

var sc = bufio.NewScanner(os.Stdin)

func getInt() int {
	sc.Scan()
	i, e := strconv.Atoi(sc.Text())
	if e != nil {
		panic(e)
	}
	return i
}

func getInts(N int) []int {
	ret := make([]int, N)
	for i := 0; i < N; i++ {
		ret[i] = getInt()
	}
	return ret
}

func getString() string {
	sc.Scan()
	return sc.Text()
}

// min, max, asub, absなど基本関数
func max(a, b int) int {
	if a > b {
		return a
	}
	return b
}

func min(a, b int) int {
	if a < b {
		return a
	}
	return b
}

func asub(a, b int) int {
	if a > b {
		return a - b
	}
	return b - a
}

func abs(a int) int {
	if a >= 0 {
		return a
	}
	return -a
}

func lowerBound(a []int, x int) int {
	idx := sort.Search(len(a), func(i int) bool {
		return a[i] >= x
	})
	return idx
}

func upperBound(a []int, x int) int {
	idx := sort.Search(len(a), func(i int) bool {
		return a[i] > x
	})
	return idx
}

const mod = 998244353

func mpow(a, b int) int {
	if b == 0 {
		return 1
	} else if b%2 == 0 {
		tmp := mpow(a, b/2)
		return tmp * tmp % mod
	}
	return mpow(a, b-1) * a % mod
}

var fracMemo = []int{1, 1}

func mfrac(n int) int {
	if len(fracMemo) > n {
		return fracMemo[n]
	}
	if len(fracMemo) == 0 {
		fracMemo = append(fracMemo, 1)
	}
	for len(fracMemo) <= n {
		size := len(fracMemo)
		fracMemo = append(fracMemo, fracMemo[size-1]*size%mod)
	}
	return fracMemo[n]
}

var ifracMemo = []int{1, 1}

func mifrac(n int) int {
	if len(ifracMemo) > n {
		return ifracMemo[n]
	}
	if len(ifracMemo) == 0 {
		fracMemo = append(ifracMemo, 1)
	}
	for len(ifracMemo) <= n {
		size := len(ifracMemo)
		ifracMemo = append(ifracMemo, ifracMemo[size-1]*mpow(size, mod-2)%mod)
	}
	return ifracMemo[n]
}

func nCr(n, r int) int {
	if n == r {
		return 1
	}
	if n < r || r < 0 {
		return 0
	}
	ret := 1
	ret *= mfrac(n)
	ret %= mod
	ret *= mifrac(r)
	ret %= mod
	ret *= mifrac(n - r)
	ret %= mod
	return (ret)
}

func main() {
	sc.Split(bufio.ScanWords)
	A, B, C, D := getInt(), getInt(), getInt(), getInt()

	dp := make([][]int, C+1)
	for i := 0; i <= C; i++ {
		dp[i] = make([]int, D+1)
	}

	dp[A][B] = 1
	for i := A; i < C; i++ {
		dp[i+1][B] = (dp[i][B] * B) % mod
	}
	for i := B; i < D; i++ {
		dp[A][i+1] = (dp[A][i] * A) % mod
	}

	for i := A; i < C; i++ {
		for j := B; j < D; j++ {
			x := dp[i][j] * i * j
			dp[i+1][j+1] = (dp[i][j+1]*(j+1) +
				dp[i+1][j]*(i+1) - x) % mod
			if dp[i+1][j+1] < 0 {
				dp[i+1][j+1] += mod
			}
		}
	}
	// for i := 0; i <= C; i++ {
	// 	out(i, dp[i])
	// }
	out(dp[C][D])
}