結果
| 問題 | No.2747 Permutation Adjacent Sum |
| ユーザー |
 PNJ
|
| 提出日時 | 2024-11-01 22:46:54 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 14,937 bytes |
| コンパイル時間 | 467 ms |
| コンパイル使用メモリ | 82,580 KB |
| 実行使用メモリ | 426,844 KB |
| 最終ジャッジ日時 | 2024-11-01 22:47:18 |
| 合計ジャッジ時間 | 22,955 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge5 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | TLE | - |
| testcase_01 | AC | 2,132 ms
408,224 KB |
| testcase_02 | AC | 2,787 ms
417,956 KB |
| testcase_03 | AC | 2,160 ms
408,392 KB |
| testcase_04 | TLE | - |
| testcase_05 | TLE | - |
| testcase_06 | -- | - |
| testcase_07 | -- | - |
| testcase_08 | -- | - |
| testcase_09 | -- | - |
| testcase_10 | -- | - |
| testcase_11 | -- | - |
| testcase_12 | -- | - |
| testcase_13 | -- | - |
| testcase_14 | -- | - |
| testcase_15 | -- | - |
| testcase_16 | -- | - |
| testcase_17 | -- | - |
| testcase_18 | -- | - |
| testcase_19 | -- | - |
| testcase_20 | -- | - |
| testcase_21 | -- | - |
| testcase_22 | -- | - |
| testcase_23 | -- | - |
| testcase_24 | -- | - |
| testcase_25 | -- | - |
| testcase_26 | -- | - |
| testcase_27 | -- | - |
| testcase_28 | -- | - |
| testcase_29 | -- | - |
| testcase_30 | -- | - |
| testcase_31 | -- | - |
| testcase_32 | -- | - |
| testcase_33 | -- | - |
| testcase_34 | -- | - |
| testcase_35 | -- | - |
| testcase_36 | -- | - |
| testcase_37 | -- | - |
| testcase_38 | -- | - |
| testcase_39 | -- | - |
| testcase_40 | -- | - |
| testcase_41 | -- | - |
ソースコード
mod = 998244353
n = 10 ** 7
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]
# 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())
if N - 1 > K + 2:
# 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 = 0
f = 1
for i in range(K+2):
f *= (N - 1 - i)
f %= mod
for i in range(K+2):
l = f * inv(N - 1 - i)
l %= mod
l *= fact_inv[i] * fact_inv[K+1-i] % mod
l %= mod
if (K + 1 - i) % 2:
l = mod - l
p += (l*y[i] % mod)
p %= mod
# (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 = 0
f *= (N - 1 - (K + 2))
f %= mod
for i in range(K+3):
l = f * inv(N-1-i)
l %= mod
l *= fact_inv[i] * fact_inv[K+2-i]
l %= mod
if (K + 2 - i) % 2:
l = mod - l
q += l*y[i] % mod
q %= mod
else:
p = 0
q = 0
for i in range(1,N):
p += pow(i,K,mod)
p %= mod
q += pow(i,K+1,mod)
q %= mod
ans = N*p - q
ans %= mod
ans *= (N - 1)
ans %= mod
ans *= 2
ans = ans * get_fact(N - 2) % mod
print(ans)