洛谷 5825.排列计数

为新年的军队做一点铺垫。

考虑转为算概率,然后考虑一个有趣的映射:计算 \(n\) 个在 \((0,1)\) 上均匀分布的实数 \(\alpha_1,\alpha_2,\dots,\alpha_n\),然后通过考虑每个实数的排名来考虑原来的排列。
根据对称性,这样得到的概率是相等的。

\(\alpha_0=0\),考虑差分 \(\beta_i = (\alpha_i - \alpha_{i-1}) \bmod 1\),容易知道这样建立了 \(\alpha\)\(\beta\) 之间的双射。
进一步,若有 \(k\) 个位置 \(i\) 满足 \(\alpha_i < \alpha_{i+1}\),那么 \(\sum\limits_{i=1}^n \beta_i = a_n + n - 1 - k\)
所以我们的问题又转化为计算对 \(\beta_1,\beta_2,\dots,\beta_n \in (0,1)^n\) 的均匀分布,\(\sum\limits_{i=1}^n \beta_i \in (n-1-k,n-k)\) 的概率。

考虑差分成 \(\sum\limits_{i=1}^n \beta_i < n-k\),然后考虑对单个 \(\beta_i < 1\) 的限制进行容斥。有 \[ \sum\limits_{i=0}^{n-k} \binom ni (-1)^i \frac{(n-k-i)^n}{i!} = \sum\limits_{i=0}^{n-k} \frac{(-1)^i}{i!(n-i)!} (n-k-i)^n \]

卷积即可。

代码:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
#include <cstdio>
#include <vector>
#include <cstring>
#include <algorithm>
#define add(a,b) (a + b >= mod ? a + b - mod : a + b)
#define dec(a,b) (a < b ? a - b + mod : a - b)
using namespace std;
const int N = 2e5;
const int mod = 998244353;
int n;
int ans[N + 5];
inline int fpow(int a,int b)
{
int ret = 1;
for(;b;b >>= 1)
(b & 1) && (ret = (long long)ret * a % mod),a = (long long)a * a % mod;
return ret;
}
namespace Poly
{
const int LG = 18;
const int N = 1 << LG + 1;
const int G = 3;
int lg2[N + 5];
int rev[N + 5],fac[N + 5],ifac[N + 5],inv[N + 5];
int rt[N + 5];
inline void init()
{
for(register int i = 2;i <= N;++i)
lg2[i] = lg2[i >> 1] + 1;
rt[0] = 1,rt[1 << LG] = fpow(G,(mod - 1) >> LG + 2);
for(register int i = LG;i;--i)
rt[1 << i - 1] = (long long)rt[1 << i] * rt[1 << i] % mod;
for(register int i = 1;i < N;++i)
rt[i] = (long long)rt[i & i - 1] * rt[i & -i] % mod;
fac[0] = 1;
for(register int i = 1;i <= N;++i)
fac[i] = (long long)fac[i - 1] * i % mod;
ifac[N] = fpow(fac[N],mod - 2);
for(register int i = N;i;--i)
ifac[i - 1] = (long long)ifac[i] * i % mod;
for(register int i = 1;i <= N;++i)
inv[i] = (long long)ifac[i] * fac[i - 1] % mod;
}
struct poly
{
vector<int> a;
inline poly(int x = 0)
{
x && (a.push_back(x),1);
}
inline poly(const vector<int> &o)
{
a = o,shrink();
}
inline poly(const poly &o)
{
a = o.a,shrink();
}
inline void shrink()
{
for(;!a.empty() && !a.back();a.pop_back());
}
inline int size() const
{
return a.size();
}
inline void resize(int x)
{
a.resize(x);
}
inline int operator[](int x) const
{
if(x < 0 || x >= size())
return 0;
return a[x];
}
inline void clear()
{
vector<int>().swap(a);
}
inline poly rever() const
{
return poly(vector<int>(a.rbegin(),a.rend()));
}
inline void dif()
{
int n = size();
for(register int i = 0,len = n >> 1;len;++i,len >>= 1)
for(register int j = 0,*w = rt;j < n;j += len << 1,++w)
for(register int k = j,R;k < j + len;++k)
R = (long long)*w * a[k + len] % mod,
a[k + len] = dec(a[k],R),
a[k] = add(a[k],R);
}
inline void dit()
{
int n = size();
for(register int i = 0,len = 1;len < n;++i,len <<= 1)
for(register int j = 0,*w = rt;j < n;j += len << 1,++w)
for(register int k = j,R;k < j + len;++k)
R = add(a[k],a[k + len]),
a[k + len] = (long long)(a[k] - a[k + len] + mod) * *w % mod,
a[k] = R;
reverse(a.begin() + 1,a.end());
for(register int i = 0;i < n;++i)
a[i] = (long long)a[i] * inv[n] % mod;
}
inline void ntt(int type = 1)
{
type == 1 ? dif() : dit();
}
friend inline poly operator+(const poly &a,const poly &b)
{
vector<int> ret(max(a.size(),b.size()));
for(register int i = 0;i < ret.size();++i)
ret[i] = add(a[i],b[i]);
return poly(ret);
}
friend inline poly operator-(const poly &a,const poly &b)
{
vector<int> ret(max(a.size(),b.size()));
for(register int i = 0;i < ret.size();++i)
ret[i] = dec(a[i],b[i]);
return poly(ret);
}
friend inline poly operator*(poly a,poly b)
{
if(a.a.empty() || b.a.empty())
return poly();
if(a.size() < 40 || b.size() < 40)
{
if(a.size() > b.size())
swap(a,b);
poly ret;
ret.resize(a.size() + b.size() - 1);
for(register int i = 0;i < ret.size();++i)
for(register int j = 0;j <= i && j < a.size();++j)
ret.a[i] = (ret[i] + (long long)a[j] * b[i - j]) % mod;
ret.shrink();
return ret;
}
int lim = 1,tot = a.size() + b.size() - 1;
for(;lim < tot;lim <<= 1);
a.resize(lim),b.resize(lim);
a.ntt(),b.ntt();
for(register int i = 0;i < lim;++i)
a.a[i] = (long long)a[i] * b[i] % mod;
a.ntt(-1),a.shrink();
return a;
}
poly &operator+=(const poly &o)
{
resize(max(size(),o.size()));
for(register int i = 0;i < o.size();++i)
a[i] = add(a[i],o[i]);
return *this;
}
poly &operator-=(const poly &o)
{
resize(max(size(),o.size()));
for(register int i = 0;i < o.size();++i)
a[i] = dec(a[i],o[i]);
return *this;
}
poly &operator*=(poly o)
{
return (*this) = (*this) * o;
}
poly deriv() const
{
if(a.empty())
return poly();
vector<int> ret(size() - 1);
for(register int i = 0;i < size() - 1;++i)
ret[i] = (long long)(i + 1) * a[i + 1] % mod;
return poly(ret);
}
poly integ() const
{
if(a.empty())
return poly();
vector<int> ret(size() + 1);
for(register int i = 0;i < size();++i)
ret[i + 1] = (long long)a[i] * inv[i + 1] % mod;
return poly(ret);
}
inline poly modxn(int n) const
{
if(a.empty())
return poly();
n = min(n,size());
return poly(vector<int>(a.begin(),a.begin() + n));
}
inline poly inver(int m) const
{
poly ret(fpow(a[0],mod - 2)),f,g;
for(register int k = 1;k < m;)
{
k <<= 1,f.resize(k),g.resize(k);
for(register int i = 0;i < k;++i)
f.a[i] = (*this)[i],g.a[i] = ret[i];
f.ntt(),g.ntt();
for(register int i = 0;i < k;++i)
f.a[i] = (long long)f[i] * g[i] % mod;
f.ntt(-1);
for(register int i = 0;i < (k >> 1);++i)
f.a[i] = 0;
f.ntt();
for(register int i = 0;i < k;++i)
f.a[i] = (long long)f[i] * g[i] % mod;
f.ntt(-1);
ret.resize(k);
for(register int i = (k >> 1);i < k;++i)
ret.a[i] = dec(0,f[i]);
}
return ret.modxn(m);
}
inline pair<poly,poly> div(poly o) const
{
if(size() < o.size())
return make_pair(poly(),*this);
poly f,g;
f = (rever().modxn(size() - o.size() + 1) * o.rever().inver(size() - o.size() + 1)).modxn(size() - o.size() + 1).rever();
g = (modxn(o.size() - 1) - o.modxn(o.size() - 1) * f.modxn(o.size() - 1)).modxn(o.size() - 1);
return make_pair(f,g);
}
inline poly log(int m) const
{
return (deriv() * inver(m)).integ().modxn(m);
}
inline poly exp(int m) const
{
poly ret(1),iv,it,d = deriv(),itd,itd0,t1;
if(m < 70)
{
ret.resize(m);
for(register int i = 1;i < m;++i)
{
for(register int j = 1;j <= i;++j)
ret.a[i] = (ret[i] + (long long)j * operator[](j) % mod * ret[i - j]) % mod;
ret.a[i] = (long long)ret[i] * inv[i] % mod;
}
return ret;
}
for(register int k = 1;k < m;)
{
k <<= 1;
it.resize(k >> 1);
for(register int i = 0;i < (k >> 1);++i)
it.a[i] = ret[i];
itd = it.deriv(),itd.resize(k >> 1);
iv = ret.inver(k >> 1),iv.resize(k >> 1);
it.ntt(),itd.ntt(),iv.ntt();
for(register int i = 0;i < (k >> 1);++i)
it.a[i] = (long long)it[i] * iv[i] % mod,
itd.a[i] = (long long)itd[i] * iv[i] % mod;
it.ntt(-1),itd.ntt(-1),it.a[0] = dec(it[0],1);
for(register int i = 0;i < k - 1;++i)
itd.a[i % (k >> 1)] = dec(itd[i % (k >> 1)],d[i]);
itd0.resize((k >> 1) - 1);
for(register int i = 0;i < (k >> 1) - 1;++i)
itd0.a[i] = d[i];
itd0 = (itd0 * it).modxn((k >> 1) - 1);
t1.resize(k - 1);
for(register int i = (k >> 1) - 1;i < k - 1;++i)
t1.a[i] = itd[(i + (k >> 1)) % (k >> 1)];
for(register int i = k >> 1;i < k - 1;++i)
t1.a[i] = dec(t1[i],itd0[i - (k >> 1)]);
t1 = t1.integ();
for(register int i = 0;i < (k >> 1);++i)
t1.a[i] = t1[i + (k >> 1)];
for(register int i = (k >> 1);i < k;++i)
t1.a[i] = 0;
t1.resize(k >> 1),t1 = (t1 * ret).modxn(k >> 1),t1.resize(k);
for(register int i = (k >> 1);i < k;++i)
t1.a[i] = t1[i - (k >> 1)];
for(register int i = 0;i < (k >> 1);++i)
t1.a[i] = 0;
ret -= t1;
}
return ret.modxn(m);
}
inline poly sqrt(int m) const
{
poly ret(1),f,g;
for(register int k = 1;k < m;)
{
k <<= 1;
f = ret,f.resize(k >> 1);
f.ntt();
for(register int i = 0;i < (k >> 1);++i)
f.a[i] = (long long)f[i] * f[i] % mod;
f.ntt(-1);
for(register int i = 0;i < k;++i)
f.a[i % (k >> 1)] = dec(f[i % (k >> 1)],(*this)[i]);
g = (2 * ret).inver(k >> 1),f = (f * g).modxn(k >> 1),f.resize(k);
for(register int i = (k >> 1);i < k;++i)
f.a[i] = f[i - (k >> 1)];
for(register int i = 0;i < (k >> 1);++i)
f.a[i] = 0;
ret -= f;
}
return ret.modxn(m);
}
inline poly pow(int m,int k1,int k2 = -1) const
{
if(a.empty())
return poly();
if(k2 == -1)
k2 = k1;
int t = 0;
for(;t < size() && !a[t];++t);
if((long long)t * k1 >= m)
return poly();
poly ret;
ret.resize(m);
int u = fpow(a[t],mod - 2),v = fpow(a[t],k2);
for(register int i = 0;i < m - t * k1;++i)
ret.a[i] = (long long)operator[](i + t) * u % mod;
ret = ret.log(m - t * k1);
for(register int i = 0;i < ret.size();++i)
ret.a[i] = (long long)ret[i] * k1 % mod;
ret = ret.exp(m - t * k1),t *= k1,ret.resize(m);
for(register int i = m - 1;i >= t;--i)
ret.a[i] = (long long)ret[i - t] * v % mod;
for(register int i = 0;i < t;++i)
ret.a[i] = 0;
return ret;
}
};
}
using Poly::init;
using Poly::poly;
inline int C(int n,int m)
{
return n < m ? 0 : (long long)Poly::fac[n] * Poly::ifac[m] % mod * Poly::ifac[n - m] % mod;
}
poly f,g;
int main()
{
init();
scanf("%d",&n),f.resize(n + 1),g.resize(n + 1);
for(register int i = 0;i <= n;++i)
f.a[i] = (long long)(i & 1 ? mod - 1 : 1) * Poly::ifac[i] % mod * Poly::ifac[n - i] % mod,
i && (g.a[i] = (fpow(i,n) - fpow(dec(i,1),n) + mod) % mod);
f *= g;
for(register int i = 0;i <= n;++i)
ans[i] = (long long)Poly::fac[n] * f[n - i] % mod;
for(register int i = 0;i <= n;++i)
printf("%d%c",ans[i]," \n"[i == n]);
}