結果

問題 No.2747 Permutation Adjacent Sum
コンテスト
ユーザー PNJ
提出日時 2024-11-01 22:56:04
言語 PyPy3
(7.3.15)
結果
TLE  
実行時間 -
コード長 14,495 bytes
コンパイル時間 421 ms
コンパイル使用メモリ 82,252 KB
実行使用メモリ 362,936 KB
最終ジャッジ日時 2024-11-01 22:57:46
合計ジャッジ時間 89,077 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge3
このコードへのチャレンジ
（要ログイン）
ファイルパターン 結果
sample AC * 2
other AC * 32 TLE * 8
権限があれば一括ダウンロードができます

ソースコード

diff # 

mod = 998244353

n = 2000000

Inv = [1 for j in range(n+1)]
for a in range(2,n+1):
  # ax + py = 1 <=> rx + p(-x-qy) = -q => x = -(inv[r]) * (p//a)  (r = p % a)
  res = (mod - Inv[mod%a]) * (mod // a)
  Inv[a] = res % mod

def inv(x):
  x %= mod
  if x <= 2*10**6:
    return Inv[x]
  else:
    res = pow(x,mod-2,mod)
    return res

fact = [1 for i in range(n + 1)]
for i in range(1,n + 1):
  fact[i] = fact[i - 1] * i % mod

fact_inv = [1 for i in range(n + 1)]
fact_inv[-1] = pow(fact[-1],mod - 2,mod)
for i in range(n,0,-1):
  fact_inv[i - 1] = fact_inv[i] * i % mod

def binom(n,r):
  if n < r or n < 0 or r < 0:
    return 0
  res = fact_inv[n - r] * fact_inv[r] % mod
  res *= fact[n]
  res %= mod
  return res

NTT_friend = [120586241,167772161,469762049,754974721,880803841,924844033,943718401,998244353,1045430273,1051721729,1053818881]
NTT_dict = {}
for i in range(len(NTT_friend)):
  NTT_dict[NTT_friend[i]] = i
NTT_info = [[20,74066978],[25,17],[26,30],[24,362],[23,211],[21,44009197],[22,663003469],[23,31],[20,363],[20,330],[20,2789]]

def popcount(n):
  c = (n&0x5555555555555555) + ((n>>1)&0x5555555555555555)
  c = (c&0x3333333333333333) + ((c>>2)&0x3333333333333333)
  c = (c&0x0f0f0f0f0f0f0f0f) + ((c>>4)&0x0f0f0f0f0f0f0f0f)
  c = (c&0x00ff00ff00ff00ff) + ((c>>8)&0x00ff00ff00ff00ff)
  c = (c&0x0000ffff0000ffff) + ((c>>16)&0x0000ffff0000ffff)
  c = (c&0x00000000ffffffff) + ((c>>32)&0x00000000ffffffff)
  return c

def topbit(n):
  h = n.bit_length()
  h -= 1
  return h

def prepared_fft(mod = 998244353):
  rank2 = NTT_info[NTT_dict[mod]][0]
  root,iroot = [0] * 30,[0] * 30
  rate2,irate2= [0] * 30,[0] * 30
  rate3,irate3= [0] * 30,[0] * 30

  root[rank2] = NTT_info[NTT_dict[mod]][1]
  iroot[rank2] = pow(root[rank2],mod - 2,mod)
  for i in range(rank2 - 1,-1,-1):
    root[i] = root[i + 1] * root[i + 1] % mod
    iroot[i] = iroot[i + 1] * iroot[i + 1] % mod

  prod,iprod = 1,1
  for i in range(rank2-1):
    rate2[i] = root[i + 2] * prod % mod
    irate2[i] = iroot[i + 2] * iprod % mod
    prod = prod * iroot[i + 2] % mod
    iprod = iprod * root[i + 2] % mod
  
  prod,iprod = 1,1
  for i in range(rank2-2):
    rate3[i] = root[i + 3] * prod % mod
    irate3[i] = iroot[i + 3] * iprod % mod
    prod = prod * iroot[i + 3] % mod
    iprod = iprod * root[i + 3] % mod
  
  return root,iroot,rate2,irate2,rate3,irate3

root,iroot,rate2,irate2,rate3,irate3 = prepared_fft()

def ntt(a):
  n = len(a)
  h = topbit(n)
  assert (n == 1 << h)
  le = 0
  while le < h:
    if h - le == 1:
      p = 1 << (h - le - 1)
      rot = 1
      for s in range(1 << le):
        offset = s << (h - le)
        for i in range(p):
          l = a[i + offset]
          r = a[i + offset + p] * rot % mod
          a[i + offset] = (l + r) % mod
          a[i + offset + p] = (l - r) % mod
        rot = rot * rate2[topbit(~s & -~s)] % mod
      le += 1
    else:
      p = 1 << (h - le - 2)
      rot,imag = 1,root[2]
      for s in range(1 << le):
        rot2 = rot * rot % mod
        rot3 = rot2 * rot % mod
        offset = s << (h - le)
        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) % mod * imag
          a[i + offset] = (a0 + a2 + a1 + a3) % mod
          a[i + offset + p] = (a0 + a2 - a1 - a3) % mod
          a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % mod
          a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % mod
        rot = rot * rate3[topbit(~s & -~s)] % mod
      le += 2

def intt(a):
  n = len(a)
  h = topbit(n)
  assert (n == 1 << h)
  coef = pow(n,mod - 2,mod)
  for i in range(n):
    a[i] = a[i] * coef % mod
  le = h
  while le:
    if le == 1:
      p = 1 << (h - le)
      irot = 1
      for s in range(1 << (le - 1)):
        offset = s << (h - le + 1)
        for i in range(p):
          l = a[i + offset]
          r = a[i + offset + p]
          a[i + offset] = (l + r) % mod
          a[i + offset + p] = (l - r) * irot % mod
        irot = irot * irate2[topbit(~s & -~s)] % mod
      le -= 1
    else:
      p = 1 << (h - le)
      irot,iimag = 1,iroot[2]
      for s in range(1 << (le - 2)):
        irot2 = irot * irot % mod
        irot3 = irot2 * irot % mod
        offset = s << (h - le + 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) * iimag % mod
          a[i + offset] = (a0 + a1 + a2 + a3) % mod
          a[i + offset + p] = (a0 - a1 + a2na3iimag) * irot % mod
          a[i + offset + p * 2] = (a0 + a1 - a2 - a3) * irot2 % mod
          a[i + offset + p * 3] = (a0 - a1 - a2na3iimag) * irot3 % mod
        irot *= irate3[topbit(~s & -~s)]
        irot %= mod
      le -= 2

def convolute_naive(a,b):
  res = [0] * (len(a) + len(b) - 1)
  for i in range(len(a)):
    for j in range(len(b)):
      res[i + j] = (res[i + j] + a[i] * b[j] % mod) % mod
  return res

def convolute(a,b):
  s = a[:]
  t = b[:]
  n = len(s)
  m = len(t)
  if min(n,m) <= 60:
    return convolute_naive(s,t)
  le = 1
  while le < n + m - 1:
    le *= 2
  s += [0] * (le - n)
  t += [0] * (le - m)
  ntt(s)
  ntt(t)
  for i in range(le):
    s[i] = s[i] * t[i] % mod
  intt(s)
  s = s[:n + m - 1]
  return s

def fps_inv(f,deg = -1):
  assert (f[0] != 0)
  if deg == -1:
    deg = len(f)
  res = [0] * deg
  res[0] = pow(f[0],mod-2,mod)
  d = 1
  while d < deg:
    a = [0] * (d << 1)
    tmp = min(len(f),d << 1)
    a[:tmp] = f[:tmp]
    b = [0] * (d << 1)
    b[:d] = res[:d]
    ntt(a)
    ntt(b)
    for i in range(d << 1):
      a[i] = a[i] * b[i] % mod
    intt(a)
    a[:d] = [0] * d
    ntt(a)
    for i in range(d << 1):
      a[i] = a[i] * b[i] % mod
    intt(a)
    for j in range(d,min(d << 1,deg)):
      if a[j]:
        res[j] = mod - a[j]
      else:
        res[j] = 0
    d <<= 1
  return res

def fps_div(f,g):
  n,m = len(f),len(g)
  if n < m:
    return [],f
  rev_f = f[:]
  rev_f = rev_f[::-1]
  rev_g = g[:]
  rev_g = rev_g[::-1]
  rev_q = convolute(rev_f,fps_inv(rev_g,n-m+1))[:n-m+1]
  q = rev_q[:]
  q = q[::-1]
  p = convolute(g,q)
  r = f[:]
  for i in range(min(len(p),len(r))):
    r[i] -= p[i]
    r[i] %= mod
  while len(r):
    if r[-1] != 0:
      break
    r.pop()
  return q,r

def fps_add(f,g):
  n = max(len(f),len(g))
  res = [0] * n
  for i in range(len(f)):
    res[i] = f[i]
  for i in range(len(g)):
    res[i] = (res[i] + g[i]) % mod
  return res

def fps_diff(f):
  if len(f) <= 1:
    return [0]
  res = []
  for i in range(1,len(f)):
    res.append(i * f[i] % mod)
  return res

def fps_integrate(f):
  n = len(f)
  res = [0] * (n + 1)
  for i in range(n):
    res[i+1] = pow(i + 1,mod-2,mod) * f[i] % mod
  return res

def fps_log(f,deg = -1):
  assert (f[0] == 1)
  if deg == -1:
    deg = len(f)
  res = convolute(fps_diff(f),fps_inv(f,deg))
  res = fps_integrate(res)
  return res[:deg]

def fps_exp(f,deg = -1):
  assert (f[0] == 0)
  if deg == -1:
    deg = len(f)
  res = [1,0]
  if len(f) > 1:
    res[1] = f[1]
  g = [1]
  p = []
  q = [1,1]
  m = 2
  while m < deg:
    y = res + [0]*m
    ntt(y)
    p = q[:]
    z = [y[i] * p[i] for i in range(len(p))]
    intt(z)
    z[:m >> 1] = [0] * (m >> 1)
    ntt(z)
    for i in range(len(p)):
      z[i] = z[i] * (-p[i]) % mod
    intt(z)
    g[m >> 1:] = z[m >> 1:]
    q = g + [0] * m
    ntt(q)
    tmp = min(len(f),m)
    x = f[:tmp] + [0] * (m - tmp)
    x = fps_diff(x)
    x.append(0)
    ntt(x)
    for i in range(len(x)):
      x[i] = x[i] * y[i] % mod
    intt(x)
    for i in range(len(res)):
      if i == 0:
        continue
      x[i-1] -= res[i] * i % mod
    x += [0] * m
    for i in range(m-1):
      x[m+i],x[i] = x[i],0
    ntt(x)
    for i in range(len(q)):
      x[i] = x[i] * q[i] % mod
    intt(x)
    x.pop()
    x = fps_integrate(x)
    x[:m] = [0] * m
    for i in range(m,min(len(f),m << 1)):
      x[i] += f[i]
    ntt(x)
    for i in range(len(y)):
      x[i] = x[i] * y[i] % mod
    intt(x)
    res[m:] = x[m:]
    m <<= 1
  return res[:deg]

def fps_pow(f,k,deg = -1):
  if deg == -1:
    deg = len(f)
  if k == 0:
    return [1] + [0] * (deg - 1)
  while len(f) < deg:
    f.append(0)
  p = 0
  while p < deg:
    if f[p]:
      break
    p += 1
  if p * k >= deg:
    return [0] * deg
  a = f[p]
  g = [0 for _ in range(deg - p)]
  a_inv = pow(a,mod-2,mod)
  for i in range(deg - p):
    g[i] = f[i + p] * a_inv % mod
  g = fps_log(g)
  for i in range(deg-p):
    g[i] = g[i] * k % mod
  g = fps_exp(g)
  a = pow(a,k,mod)
  res = [0] * deg
  for i in range(deg):
    j = i + p * k
    if j >= deg:
      break
    res[j] = g[i] * a % mod
  return res

def transposed_ntt(a):
  b = a[:]
  intt(b)
  b = [b[0]] + b[1:][::-1]
  for i in range(len(a)):
    a[i] = b[i] * len(a) % mod
  return a

def transposed_ntt_inv(a):
  b = [a[0]] + a[1:][::-1]
  ntt(b)
  n = len(b)
  n_inv = pow(n,mod - 2,mod)
  for i in range(len(b)):
    a[i] = b[i] * n_inv % mod
  return

def ntt_doubling(a,flag = 1):
  root,iroot,rate2,irate2,rate3,irate3 = prepared_fft(mod)

  if flag == 0:
    M = len(a) // 2
    tmp = a[:M]
    aa = a[M:]
    transposed_ntt(aa)
    r = 1
    zeta = root[topbit(2*M)]
    for i in range(M):
      aa[i] = aa[i] * r % mod
      r = r * zeta % mod
    transposed_ntt_inv(aa)
    for i in range(M):
      aa[i] = (aa[i] + tmp[i]) % mod
    while len(a) > M:
      a.pop()
    for i in range(M):
      a[i] = aa[i]
    return

  M = len(a)
  b = a[:]
  intt(b)
  r = 1
  zeta = root[topbit(2*M)]
  for i in range(M):
    b[i] = b[i] * r % mod
    r = r * zeta % mod
  ntt(b)
  a += b
  return
 
def middle_product(a,b):
  assert (len(a) >= len(b))
  # naive
  if min(len(b), len(a) - len(b) + 1) <= 60:
    res = [0] * (len(a) - len(b) + 1)
    for i in range(len(res)):
      for j in range(len(b)):
        res[i] = (res[i] + b[j] * a[i + j] % mod) % mod
    return res
  n = 1 << (len(a) - 1).bit_length()
  fa = [0] * n
  fb = [0] * n
  for i in range(len(a)):
    fa[i] = a[i]
  for i in range(len(b)):
    fb[i] = b[~i]
  ntt(fa)
  ntt(fb)
  for i in range(n):
    fa[i] = fa[i] * fb[i] % mod
  intt(fa)
  fa = fa[len(b) - 1:len(a)]
  return fa

def multipoint_evaluation(f,point):
  n = 1
  while n < len(point):
    n <<= 1
  k = topbit(n)
  F = [[0 for _ in range(n)] for _ in range(k + 1)]
  F2 = [[0 for _ in range(n)] for _ in range(k + 1)]
  G = [[0 for _ in range(n)] for _ in range(k + 1)]
  for i in range(len(point)):
    F[0][i] = (-point[i]) % mod
  
  for d in range(k):
    b = 1 << d
    L = 0
    while L < n:
      f1 = F[d][L:L+b]
      f2 = F[d][L+b:L+2*b]
      ntt_doubling(f1)
      ntt_doubling(f2)
      for i in range(b):
        f1[i] = (f1[i] + 1) % mod
        f2[i] = (f2[i] + 1) % mod
      for i in range(b,2*b):
        f1[i] = (f1[i] - 1) % mod
        f2[i] = (f2[i] - 1) % mod
      for i in range(2 * b):
        F[d][L + i] = f1[i]
        F2[d][L + i] = f2[i]
        F[d + 1][L + i] = (f1[i] * f2[i] % mod - 1) % mod
      L += 2 * b
  
  P = F[k][:]
  intt(P)
  P.append(1)
  P = P[::-1]
  P = P[:len(f)]
  while len(P) < len(f):
    P.append(0)
  P = fps_inv(P)

  f = f[:n + len(P) - 1]
  while len(f) < n + len(P) - 1:
    f.append(0)
  f = middle_product(f,P)
  f = f[::-1]
  transposed_ntt_inv(f)
  G[k] = f

  for d in range(k - 1,-1,-1):
    b = 1 << d
    L = 0
    while L < n:
      g1 = [0] * (2 * b)
      g2 = [0] * (2 * b)
      for i in range(2 * b):
        g1[i] = G[d + 1][L + i] * F2[d][L + i] % mod
        g2[i] = G[d + 1][L + i] * F[d][L + i] % mod
      ntt_doubling(g1,0)
      ntt_doubling(g2,0)
      for i in range(b):
        G[d][L + i] = g1[i]
        G[d][L + b + i] = g2[i]
      L += 2 * b
  res = G[0][:len(point)]
  return res

def online_convolute(F):
  N = len(F)
  def f(l,r):
    if l + 1 == r:
      return F[l]
    else:
      m = (l + r) // 2
      res = convolute(f(l,m),f(m,r))
      return res
  return f(0,N)

def sum_of_rationals(W,A):
  # sum (W[i] / (x - A[i]))
  assert (len(W) == len(A))
  N = len(W)
  def calc(l,r):
    if l + 1 == r:
      return ([W[l]],[-A[l],1])
    m = (l + r) // 2
    f,ff = calc(l,m)
    g,gg = calc(m,r)
    h = fps_add(convolute(f,gg),convolute(ff,g))
    hh = convolute(ff,gg)
    return (h,hh)
  return calc(0,N)

def polynominal_interpolation(X,Y):
  assert (len(X) == len(Y))
  N = len(X)
  G = [[-X[i],1] for i in range(N)]
  g = online_convolute(G)
  gg = fps_diff(g)
  YY = multipoint_evaluation(gg,X)
  for i in range(N):
    Y[i] = Y[i] * pow(YY[i],mod - 2,mod) % mod
  return sum_of_rationals(Y,X)[0]

def shift_of_sampling_points(Y,M,c):
  # https://suisen-cp.github.io/cp-library-cpp/library/polynomial/shift_of_sampling_points.hpp
  N = len(Y)
  # step1
  A = [Y[j] * fact_inv[j] % mod for j in range(N)]
  B = [fact_inv[i] * pow(-1,i) % mod for i in range(N)]
  f = convolute(A,B)[:N]
  if M == 1:
    d = 1
    res = 0
    for i in range(N):
      res += f[i] * d % mod
      res %= mod
      d = d * (c - i) % mod
    return [res]
  # step2
  A = [f[i] * fact[i] % mod for i in range(N)]
  A = A[::-1]
  B = [fact_inv[j] for j in range(N)]
  b = 1
  for i in range(N):
    B[i] = B[i] * b % mod
    b = b * (c - i) % mod
  B = convolute(A,B)[:N]
  A = [B[N - 1 - j] * fact_inv[j] % mod for j in range(N)]
  B = [fact_inv[i] for i in range(M)]
  res = convolute(A,B)[:M]
  for i in range(M):
    res[i] = res[i] * fact[i] % mod
  return res

K = 9
B = 1 << K
P = mod

i = 1
point = [1,3]
while i < K:
  t = 1 << i
  f = point + shift_of_sampling_points(point,3 * t,t)
  point = [0 for j in range(2 * t)]
  for j in range(2 * t):
    point[j] = (f[2 * j] * f[2 * j + 1] % mod) * (t * (2 * j + 1) % mod) % mod
  i += 1
point = shift_of_sampling_points(point,P // B,0)
T = [1] + point
for i in range(1,len(T)):
  T[i] = T[i] * (i * B) % mod

for i in range(len(T) - 1):
  T[i + 1] = T[i + 1] * T[i] % mod

def get_fact(n):
  r = n % B
  q = n // B
  res = T[q]
  for i in range(1,r + 1):
    res = res * (q * B + i) % mod
  return res

N,K = map(int,input().split())


# K乗和
y = [0]
r = 0
for i in range(1,K+2):
  r += pow(i,K,mod)
  r %= mod
  y.append(r)
p = shift_of_sampling_points(y,1,N)[0]
# (K + 1) 乗和
y = [0]
r = 0
for i in range(1,K+3):
  r += pow(i,K+1,mod)
  r %= mod
  y.append(r)
q = shift_of_sampling_points(y,1,N)[0]

ans = N*p - q
ans %= mod
ans *= (N - 1)
ans %= mod
ans *= 2
ans = ans * get_fact(N - 2) % mod
print(ans)
0