import sys,random from itertools import permutations from collections import deque input = lambda :sys.stdin.readline().rstrip() mi = lambda :map(int,input().split()) li = lambda :list(mi()) mod = 998244353 omega = pow(3,119,mod) rev_omega = pow(omega,mod-2,mod) N = 10**6 g1 = [1]*(N+1) # 元テーブル g2 = [1]*(N+1) #逆元テーブル inv = [1]*(N+1) #逆元テーブル計算用テーブル for i in range( 2, N + 1 ): g1[i]=( ( g1[i-1] * i ) % mod ) inv[i]=( ( -inv[mod % i] * (mod//i) ) % mod ) g2[i]=( (g2[i-1] * inv[i]) % mod ) inv[0]=0 _fft_mod = 998244353 _fft_imag = 911660635 _fft_iimag = 86583718 _fft_rate2 = (911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601, 842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899) _fft_irate2 = (86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235) _fft_rate3 = (372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099, 183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204) _fft_irate3 = (509520358, 929031873, 170256584, 839780419, 282974284, 395914482, 444904435, 72135471, 638914820, 66769500, 771127074, 985925487, 262319669, 262341272, 625870173, 768022760, 859816005, 914661783, 430819711, 272774365, 530924681) def _butterfly(a): n = len(a) h = (n - 1).bit_length() len_ = 0 while len_ < h: if h - len_ == 1: p = 1 << (h - len_ - 1) rot = 1 for s in range(1 << len_): offset = s << (h - len_) for i in range(p): l = a[i + offset] r = a[i + offset + p] * rot % _fft_mod a[i + offset] = (l + r) % _fft_mod a[i + offset + p] = (l - r) % _fft_mod if s + 1 != (1 << len_): rot *= _fft_rate2[(~s & -~s).bit_length() - 1] rot %= _fft_mod len_ += 1 else: p = 1 << (h - len_ - 2) rot = 1 for s in range(1 << len_): rot2 = rot * rot % _fft_mod rot3 = rot2 * rot % _fft_mod offset = s << (h - len_) for i in range(p): a0 = a[i + offset] a1 = a[i + offset + p] * rot a2 = a[i + offset + p * 2] * rot2 a3 = a[i + offset + p * 3] * rot3 a1na3imag = (a1 - a3) % _fft_mod * _fft_imag a[i + offset] = (a0 + a2 + a1 + a3) % _fft_mod a[i + offset + p] = (a0 + a2 - a1 - a3) % _fft_mod a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % _fft_mod a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % _fft_mod if s + 1 != (1 << len_): rot *= _fft_rate3[(~s & -~s).bit_length() - 1] rot %= _fft_mod len_ += 2 def _butterfly_inv(a): n = len(a) h = (n - 1).bit_length() len_ = h while len_: if len_ == 1: p = 1 << (h - len_) irot = 1 for s in range(1 << (len_ - 1)): offset = s << (h - len_ + 1) for i in range(p): l = a[i + offset] r = a[i + offset + p] a[i + offset] = (l + r) % _fft_mod a[i + offset + p] = (l - r) * irot % _fft_mod if s + 1 != (1 << (len_ - 1)): irot *= _fft_irate2[(~s & -~s).bit_length() - 1] irot %= _fft_mod len_ -= 1 else: p = 1 << (h - len_) irot = 1 for s in range(1 << (len_ - 2)): irot2 = irot * irot % _fft_mod irot3 = irot2 * irot % _fft_mod offset = s << (h - len_ + 2) for i in range(p): a0 = a[i + offset] a1 = a[i + offset + p] a2 = a[i + offset + p * 2] a3 = a[i + offset + p * 3] a2na3iimag = (a2 - a3) * _fft_iimag % _fft_mod a[i + offset] = (a0 + a1 + a2 + a3) % _fft_mod a[i + offset + p] = (a0 - a1 + a2na3iimag) * irot % _fft_mod a[i + offset + p * 2] = (a0 + a1 - a2 - a3) * irot2 % _fft_mod a[i + offset + p * 3] = (a0 - a1 - a2na3iimag) * irot3 % _fft_mod if s + 1 != (1 << (len_ - 1)): irot *= _fft_irate3[(~s & -~s).bit_length() - 1] irot %= _fft_mod len_ -= 2 def _convolution_naive(a, b): n = len(a) m = len(b) ans = [0] * (n + m - 1) if n < m: for j in range(m): for i in range(n): ans[i + j] = (ans[i + j] + a[i] * b[j]) % _fft_mod else: for i in range(n): for j in range(m): ans[i + j] = (ans[i + j] + a[i] * b[j]) % _fft_mod return ans def _convolution_fft(a, b): a = a.copy() b = b.copy() n = len(a) m = len(b) z = 1 << (n + m - 2).bit_length() a += [0] * (z - n) _butterfly(a) b += [0] * (z - m) _butterfly(b) for i in range(z): a[i] = a[i] * b[i] % _fft_mod _butterfly_inv(a) a = a[:n + m - 1] iz = pow(z, _fft_mod - 2, _fft_mod) for i in range(n + m - 1): a[i] = a[i] * iz % _fft_mod return a def _convolution_square(a): a = a.copy() n = len(a) z = 1 << (2 * n - 2).bit_length() a += [0] * (z - n) _butterfly(a) for i in range(z): a[i] = a[i] * a[i] % _fft_mod _butterfly_inv(a) a = a[:2 * n - 1] iz = pow(z, _fft_mod - 2, _fft_mod) for i in range(2 * n - 1): a[i] = a[i] * iz % _fft_mod return a def convolution(a, b): """It calculates (+, x) convolution in mod 998244353. Given two arrays a[0], a[1], ..., a[n - 1] and b[0], b[1], ..., b[m - 1], it calculates the array c of length n + m - 1, defined by > c[i] = sum(a[j] * b[i - j] for j in range(i + 1)) % 998244353. It returns an empty list if at least one of a and b are empty. Constraints ----------- > len(a) + len(b) <= 8388609 Complexity ---------- > O(n log n), where n = len(a) + len(b). """ n = len(a) m = len(b) if n == 0 or m == 0: return [] if min(n, m) <= 0: return _convolution_naive(a, b) if a is b: return _convolution_square(a) return _convolution_fft(a, b) def bostan_mori(P,Q,N): """ [x^N]P(x)/Q(x)を求める """ d = len(Q) - 1 z = 1 << (2*d).bit_length() iz = pow(z, _fft_mod - 2, _fft_mod) while N: """ P(x)/Q(x) = P(x)Q(-x)/Q(x)Q(-x) """ P += [0] * (z-len(P)) Q += [0] * (z-len(Q)) _butterfly(P) _butterfly(Q) dft_t = Q.copy() for i in range(0,z,2): dft_t[i],dft_t[i^1] = dft_t[i^1],dft_t[i] P = [a*b % mod for a,b in zip(P,dft_t)] _butterfly_inv(P) Q = [a*b % mod for a,b in zip(Q,dft_t)] _butterfly_inv(Q) P = [a * iz % mod for a in P][N&1::2] Q = [a * iz % mod for a in Q][0::2] N >>= 1 res = P[0] * pow(Q[0],mod-2,mod) % mod return res def taylor_shift(f,a): g = [f[i]*g1[i]%mod for i in range(len(f))][::-1] e = [g2[i] for i in range(len(f))] t = 1 for i in range(1,len(f)): t = t * a % mod e[i] = e[i] * t % mod res = convolution(g,e)[:len(f)] return [res[len(f)-1-i]*g2[i]%mod for i in range(len(f))] def inverse(f,limit): assert(f[0]!=0) l = len(f) L = 1<<((l-1).bit_length()) n = L.bit_length()-1 f = f[:L] f+=[0]*(L-len(f)) res = [pow(f[0],mod-2,mod)] for i in range(1,n+1): h = convolution(res,f[:2**i])[:2**i] h = [(-h[i]) % mod for i in range(2**i)] h[0] = (h[0]+2) % mod res = convolution(res,h)[:2**i] return res[:limit] def cmb(n,r): if r < 0 or n < r: return 0 return g1[n] * (g2[r] * g2[n-r] % mod) % mod def brute(H,W,N,K): inv = pow((H-K+1)*(W-K+1),mod-2,mod) ans = 0 for i in range(H): for j in range(W): """ h <= i < h+K and 0 <= h < H-K+1 max(i-K+1,0) <= h <= min(i,H-K) """ h = min(i,H-K) - max(i-K+1,0) + 1 w = min(j,W-K) - max(j-K+1,0) + 1 ans += 1 - pow(1-h*w*inv % mod,N,mod) ans %= mod return ans def calc_const_h(H,W,N,K,h): inv = pow((H-K+1)*(W-K+1),mod-2,mod) ans = W cosnt_W = W for j in range(K-1): w = min(j,W-K) - max(j-K+1,0) + 1 cosnt_W -= 1 ans -= pow(1-h*w*inv,N,mod) ans %= mod if (W-1-j) < K-1: continue w = min(W-1-j,W-K) - max(W-1-j-K+1,0) + 1 cosnt_W -= 1 ans -= pow(1-h*w*inv,N,mod) ans %= mod w = K ans -= pow(1-h*w*inv,N,mod) * cosnt_W % mod ans %= mod return ans def solve_easy(H,W,N,K): inv = pow((H-K+1)*(W-K+1),mod-2,mod) ans = 0 const_H = H for i in range(K-1): h = min(i,H-K) - max(i-K+1,0) + 1 const_H -= 1 ans += calc_const_h(H,W,N,K,h) ans %= mod if H-1-i < K-1: continue h = min(H-1-i,H-K) - max(H-1-i-K+1,0) + 1 const_H -= 1 ans += calc_const_h(H,W,N,K,h) ans %= mod h = K ans += const_H * calc_const_h(H,W,N,K,h) % mod ans %= mod return ans def calc_pow_sum(N,K): """ k^i for k in range(1,K+1) を i = 1,2,...,N まで計算 """ f = [1] * (N+2) f[0] = 1 for n in range(1,N+2): f[n] = f[n-1] * ((K+1) * inv[n] % mod) % mod f[0] = 0 f = f[1:] g = [g2[i] for i in range(N+2)] g[0] = 0 g = g[1:] ig = inverse(g,N+1) h = convolution(f,ig) h = [h[i]*g1[i]%mod for i in range(N+1)] return [K] + h[1:] def solve_hard(H,W,N,K): all_inv = pow((H-K+1)*(W-K+1),mod-2,mod) ph = min(K-1,H-K) tmp_h = calc_pow_sum(N,ph) tmp_h = [2*x % mod for x in tmp_h] need = H - 2 * ph maxi = ph + 1 for i in range(N+1): tmp_h[i] += pow(maxi,i,mod) * need % mod tmp_h[i] %= mod pw = min(K-1,W-K) tmp_w = calc_pow_sum(N,pw) tmp_w = [2*x % mod for x in tmp_w] need = W - 2 * ph maxi = ph + 1 for i in range(N+1): tmp_w[i] += pow(maxi,i,mod) * (need) % mod tmp_w[i] %= mod #print(tmp_h,tmp_w) ans = H*W for i in range(N+1): if i & 1 == 0: ans -= (tmp_h[i] * tmp_w[i] % mod) * (cmb(N,i) * pow(all_inv,i,mod) % mod) % mod else: ans += (tmp_h[i] * tmp_w[i] % mod) * (cmb(N,i) * pow(all_inv,i,mod) % mod) % mod ans %= mod return ans H,W,N,K = mi() print(solve_hard(H,W,N,K))