Buggy python submission (verdict: Time Limit Exceeded):
a,b,c,d=map(int,input().split())
mod=998244353
dct=[[[-1]*3 for _ in range(d+1)] for _ in range(c+1)]

import sys
sys.setrecursionlimit(10**7)
def dp(i,j,bl):
  #print(i,j,bl)
  if dct[i][j][bl]>0:
    return dct[i][j][bl]
  if (i,j)==(a,b):
    return 1 if bl==0 else 0
  if i<a or j<b:
    return 0
  if i==a:
    if bl==2:
      ret=0 if j-b<2 else pow(i,j-b,mod)-pow(i-1,j-b-1,mod)*(j-b)-pow(i-1,j-b,mod)
    elif bl==1:
      ret=0 if j-b<1 else pow(i-1,j-b-1,mod)*(j-b)
    elif bl==0:
      ret=pow(i-1,j-b,mod)
  elif bl==2:
    ret=dp(i,j-1,1)+dp(i,j-1,2)*i
  elif bl==1:
    ret=(dp(i-1,j,0)+dp(i-1,j,1)+dp(i-1,j,2))*j
  else:
    ret=dp(i,j-1,0)*(i-1)
  ret%=mod
  dct[i][j][bl]=ret
  return ret

print((dp(c,d,0)+dp(c,d,1)+dp(c,d,2))%mod)


Fixed submission:
a,b,c,d=map(int,input().split())
mod=998244353

dp1=[[0]*(d+1) for _ in range(c+1)]
dp2=[[0]*(d+1) for _ in range(c+1)]

for i in range(b+1,d+1):
  if a==1:
    dp1[a][i]=1 if i==b+1 else 0
    dp2[a][i]=0 if i==b+1 else 1
  else:
    dp1[a][i]=(pow(a-1,i-b-1,mod)*(i-b))%mod
    dp2[a][i]=(pow(a,i-b,mod)-dp1[a][i])%mod
for i in range(a+1,c+1):
  dp1[i][b]=pow(b,i-a,mod)
  dp2[i][b]=0
dp2[a][b]=1
for i in range(a+1,c+1):
  for j in range(b+1,d+1):
    dp1[i][j]=(j*(dp1[i-1][j]+dp2[i-1][j]))%mod
    dp2[i][j]=(dp1[i][j-1]+dp2[i][j-1]*i)%mod
print((dp1[c][d]+dp2[c][d])%mod)

"""
print('dp1')
for x in dp1:
  print(x)
print('dp2')
for x in dp2:
  print(x)
"""

Buggy python submission (verdict: Time Limit Exceeded):
a,b,c,d=map(int,input().split())
mod=998244353
dct=[[[-1]*3 for _ in range(d+1)] for _ in range(c+1)]

import sys
sys.setrecursionlimit(10**7)
def dp(i,j,bl):
  #print(i,j,bl)
  if dct[i][j][bl]>0:
    return dct[i][j][bl]
  if (i,j)==(a,b):
    return 1 if bl==0 else 0
  if i<a or j<b:
    return 0
  if i==a:
    if bl==2:
      ret=0 if j-b<2 else pow(i,j-b,mod)-pow(i-1,j-b-1,mod)*(j-b)-pow(i-1,j-b,mod)
    elif bl==1:
      ret=0 if j-b<1 else pow(i-1,j-b-1,mod)*(j-b)
    elif bl==0:
      ret=pow(i-1,j-b,mod)
  elif bl==2:
    ret=dp(i,j-1,1)+dp(i,j-1,2)*i
  elif bl==1:
    ret=(dp(i-1,j,0)+dp(i-1,j,1)+dp(i-1,j,2))*j
  else:
    ret=dp(i,j-1,0)*(i-1)
  ret%=mod
  dct[i][j][bl]=ret
  return ret

print((dp(c,d,0)+dp(c,d,1)+dp(c,d,2))%mod)


Fixed submission:
a,b,c,d=map(int,input().split())
mod=998244353

dp1=[[0]*(d+1) for _ in range(c+1)]
dp2=[[0]*(d+1) for _ in range(c+1)]

for i in range(b+1,d+1):
  if a==1:
    dp1[a][i]=1 if i==b+1 else 0
    dp2[a][i]=0 if i==b+1 else 1
  else:
    dp1[a][i]=(pow(a-1,i-b-1,mod)*(i-b))%mod
    dp2[a][i]=(pow(a,i-b,mod)-dp1[a][i])%mod
for i in range(a+1,c+1):
  dp1[i][b]=pow(b,i-a,mod)
  dp2[i][b]=0
dp2[a][b]=1
for i in range(a+1,c+1):
  for j in range(b+1,d+1):
    dp1[i][j]=(j*(dp1[i-1][j]+dp2[i-1][j]))%mod
    dp2[i][j]=(dp1[i][j-1]+dp2[i][j-1]*i)%mod
print((dp1[c][d]+dp2[c][d])%mod)

"""
print('dp1')
for x in dp1:
  print(x)
print('dp2')
for x in dp2:
  print(x)
"""

Buggy python submission (verdict: Time Limit Exceeded):
a,b,c,d=map(int,input().split())
mod=998244353
dct=[[[-1]*3 for _ in range(d+1)] for _ in range(c+1)]
#from collections import defaultdict

import sys
sys.setrecursionlimit(10**7)
def dp(i,j,bl):
  #print(i,j,bl)
  if dct[i][j][bl]>0:
    return dct[i][j][bl]
  if (i,j)==(a,b):
    return 1 if bl==0 else 0
  if i<a or j<b:
    return 0
  if i==a:
    if bl==2:
      ret=0 if j-b<2 else pow(i,j-b)-pow(i-1,j-b-1)*(j-b)-pow(i-1,j-b)
    elif bl==1:
      ret=0 if j-b<1 else pow(i-1,j-b-1)*(j-b)
    elif bl==0:
      ret=pow(i-1,j-b)
  elif bl==2:
    ret=dp(i,j-1,1)+dp(i,j-1,2)*i
  elif bl==1:
    ret=(dp(i-1,j,0)+dp(i-1,j,1)+dp(i-1,j,2))*j
  else:
    ret=dp(i,j-1,0)*(i-1)
  ret%=mod
  dct[i][j][bl]=ret
  return ret

print((dp(c,d,0)+dp(c,d,1)+dp(c,d,2))%mod)
#print(dp(c,d,0),dp(c,d,1),dp(c,d,2))
#print(a,b,c,d)


Fixed submission:
a,b,c,d=map(int,input().split())
mod=998244353

dp1=[[0]*(d+1) for _ in range(c+1)]
dp2=[[0]*(d+1) for _ in range(c+1)]

for i in range(b+1,d+1):
  if a==1:
    dp1[a][i]=1 if i==b+1 else 0
    dp2[a][i]=0 if i==b+1 else 1
  else:
    dp1[a][i]=(pow(a-1,i-b-1,mod)*(i-b))%mod
    dp2[a][i]=(pow(a,i-b,mod)-dp1[a][i])%mod
for i in range(a+1,c+1):
  dp1[i][b]=pow(b,i-a,mod)
  dp2[i][b]=0
dp2[a][b]=1
for i in range(a+1,c+1):
  for j in range(b+1,d+1):
    dp1[i][j]=(j*(dp1[i-1][j]+dp2[i-1][j]))%mod
    dp2[i][j]=(dp1[i][j-1]+dp2[i][j-1]*i)%mod
print((dp1[c][d]+dp2[c][d])%mod)

"""
print('dp1')
for x in dp1:
  print(x)
print('dp2')
for x in dp2:
  print(x)
"""