LOJ#2320 生成树计数

解：讲一个别的题解里我比较难以理解的地方，就是为什么可以把这两个东西合起来看成某一个连通块指数是2m而别的指数都是m。

其实很好理解，但是别人都略过了......把后面的∑提到∏的前面，然后展开，也可以理解成把∏塞到∑里面。

然后我们就发现对于每个生成树，我们其实有n种选择，分别把某个块的次数变成2m，且每种选择都作为一棵生成树计入贡献，且这回的贡献，一个树内部各个块全部是乘积的形式。

发现贡献与度数有关，又要求所有生成树，于是考虑prufer序列。

如何看待每个点是一个连通块？就是对于一种生成树，实际方案要乘上(点数^度数)。代表每条边连哪个点。

这样一个生成树的权值是这个东西：

接下来考虑钦定每个连通块的度数。在prufer序列中每个数的出现次数是度数-1，于是令d_i = d_i - 1

首先要把这些点在prufer中排列一下，于是有：

接下来一顿操作，把di！塞到∏里面，(n-2)!提出来，就有个式子。

然后考虑生成函数，推荐这个。

关于求等幂和这个式子，实在是不理解...

这东西还要我用vector写多项式......

然后搞来搞去，不管了...

  1 #include <cstdio>
  2 #include <algorithm>
  3 #include <cstring>
  4 #include <cmath>
  5 #include <vector>
  6 
  7 typedef long long LL;
  8 const int N = 100010;
  9 const LL MO = 998244353, G = 3;
 10 typedef LL arr[N << 2];
 11 typedef std::vector<LL> Poly;
 12 
 13 arr A, B, exp_t, inv_t, inv_t2, f, g, h, p, ex;
 14 int r[N << 2], n;
 15 LL m, a[N], pw[N];
 16 
 17 inline LL qpow(LL a, LL b) {
 18     a %= MO;
 19     LL ans = 1;
 20     while(b) {
 21         if(b & 1) ans = ans * a % MO;
 22         a = a * a % MO;
 23         b = b >> 1;
 24     }
 25     return ans;
 26 }
 27 
 28 inline void prework(int n) {
 29     static int R = 0;
 30     if(R == n) return;
 31     R = n;
 32     int lm = 1;
 33     while((1 << lm) < n) lm++;
 34     for(int i = 0; i < n; i++) r[i] = (r[i >> 1] >> 1) | ((i & 1) << (lm - 1));
 35     return;
 36 }
 37 
 38 inline void NTT(LL *a, int n, int f) {
 39     prework(n);
 40     for(int i = 0; i < n; i++) {
 41         if(i < r[i]) std::swap(a[i], a[r[i]]);
 42     }
 43     for(int len = 1; len < n; len <<= 1) {
 44         LL Wn = qpow(G, (MO - 1) / (len << 1));
 45         if(f == -1) Wn = qpow(Wn, MO - 2);
 46         for(int i = 0; i < n; i += (len << 1)) {
 47             LL w = 1;
 48             for(int j = 0; j < len; j++) {
 49                 LL t = a[i + len + j] * w % MO;
 50                 a[i + len + j] = (a[i + j] - t) % MO;
 51                 a[i + j] = (a[i + j] + t) % MO;
 52                 w = w * Wn % MO;
 53             }
 54         }
 55     }
 56     if(f == -1) {
 57         LL inv = qpow(n, MO - 2);
 58         for(int i = 0; i < n; i++) a[i] = a[i] * inv % MO;
 59     }
 60     return;
 61 }
 62 
 63 inline void mul(const LL *a, const LL *b, LL *c, int n) {
 64     int len = 1;
 65     while(len < n + n) len <<= 1;
 66     memcpy(A, a, n * sizeof(LL)); memset(A + n, 0, (len - n) * sizeof(LL));
 67     memcpy(B, b, n * sizeof(LL)); memset(B + n, 0, (len - n) * sizeof(LL));
 68     NTT(A, len, 1); NTT(B, len, 1);
 69     for(int i = 0; i < len; i++) c[i] = A[i] * B[i] % MO;
 70     NTT(c, len, -1);
 71     memset(c + n, 0, (len - n) * sizeof(LL));
 72     return;
 73 }
 74 
 75 inline Poly mul(const Poly &a, const Poly &b) {
 76     int len = 1, lena = a.size(), lenb = b.size();
 77     while(len < lena + lenb) len <<= 1;
 78     Poly ans(lena + lenb - 1);
 79     for(int i = 0; i < lena; i++) A[i] = a[i];
 80     for(int i = 0; i < lenb; i++) B[i] = b[i];
 81     memset(A + lena, 0, (len - lena) * sizeof(LL));
 82     memset(B + lenb, 0, (len - lenb) * sizeof(LL));
 83     NTT(A, len, 1); NTT(B, len, 1);
 84     for(int i = 0; i < len; i++) A[i] = A[i] * B[i] % MO;
 85     NTT(A, len, -1);
 86     for(int i = 0; i < lena + lenb - 1; i++) ans[i] = A[i];
 87     return ans;
 88 }
 89 
 90 void Inv(const LL *a, LL *b, int n) {
 91     if(n == 1) {
 92         b[0] = qpow(a[0], MO - 2);
 93         b[1] = 0;
 94         return;
 95     }
 96     Inv(a, b, n >> 1);
 97     /// ans = b * (2 - a * b);
 98     memcpy(A, a, n * sizeof(LL)); memset(A + n, 0, n * sizeof(LL));
 99     memcpy(B, b, n * sizeof(LL)); memset(B + n, 0, n * sizeof(LL));
100     NTT(A, n << 1, 1); NTT(B, n << 1, 1);
101     for(int i = 0; i < (n << 1); i++) b[i] = B[i] * (2 - A[i] * B[i] % MO) % MO;
102     NTT(b, n << 1, -1);
103     memset(b + n, 0, n * sizeof(LL));
104     return;
105 }
106 
107 inline void getInv(const LL *a, LL *b, int n) {
108     int len = 1;
109     while(len < n) len <<= 1;
110     Inv(a, b, len);
111     memset(b + n, 0, (len - n) * sizeof(LL));
112     return;
113 }
114 
115 inline Poly getInv(const Poly &a) {
116     int n = a.size(), len = 1;
117     while(len < n) len <<= 1;
118     for(int i = 0; i < n; i++) inv_t[i] = a[i];
119     memset(inv_t + n, 0, (len - n) * sizeof(LL));
120     getInv(inv_t, inv_t2, n);
121     Poly ans(n);
122     for(int i = 0; i < n; i++) ans[i] = inv_t2[i];
123     return ans;
124 }
125 
126 inline void der(const LL *a, LL *b, int n) {
127     for(int i = 0; i < n - 1; i++) {
128         b[i] = a[i + 1] * (i + 1) % MO;
129     }
130     b[n - 1] = 0;
131     return;
132 }
133 
134 inline void ter(const LL *a, LL *b, int n) {
135     for(int i = n - 1; i >= 1; i--) {
136         b[i] = a[i - 1] * qpow(i, MO - 2) % MO;
137     }
138     b[0] = 0;
139     return;
140 }
141 
142 inline void getLn(const LL *a, LL *b, int n) {
143     getInv(a, inv_t, n);
144     der(a, A, n);
145     int len = 1;
146     while(len < n + n) len <<= 1;
147     memset(A + n, 0, (len - n) * sizeof(LL));
148     memcpy(B, inv_t, n * sizeof(LL)); memset(B + n, 0, (len - n) * sizeof(LL));
149     NTT(A, len, 1); NTT(B, len, 1);
150     for(int i = 0; i < len; i++) b[i] = A[i] * B[i] % MO;
151     NTT(b, len, -1);
152     memset(b + n, 0, (len - n) * sizeof(LL));
153     ter(b, b, n);
154     return;
155 }
156 
157 void Exp(const LL *a, LL *b, int n) {
158     if(n == 1) {
159         b[0] = 1;
160         b[1] = 0;
161         return;
162     }
163     Exp(a, b, n >> 1);
164     /// ans = b * (1 + a - ln b)
165     getLn(b, exp_t, n);
166     for(int i = 0; i < n; i++) A[i] = (a[i] - exp_t[i]) % MO;
167     A[0] = (A[0] + 1) % MO;
168     memset(A + n, 0, n * sizeof(LL));
169     memcpy(B, b, n * sizeof(LL)); memset(B + n, 0, n * sizeof(LL));
170     NTT(A, n << 1, 1); NTT(B, n << 1, 1);
171     for(int i = 0; i < (n << 1); i++) b[i] = A[i] * B[i] % MO;
172     NTT(b, n << 1, -1);
173     memset(b + n, 0, n * sizeof(LL));
174     return;
175 }
176 
177 inline void getExp(const LL *a, LL *b, int n) {
178     int len = 1;
179     while(len < n) len <<= 1;
180     Exp(a, b, len);
181     memset(b + n, 0, (len - n) * sizeof(LL));
182     return;
183 }
184 
185 inline void out(const Poly &a) {
186     //printf("siz = %d ", a.size());
187     for(int i = 0; i < a.size(); i++) {
188         printf("%lld ", (a[i] + MO) % MO);
189     }
190     printf("\n");
191     return;
192 }
193 
194 Poly dvd(int l, int r) {
195     if(l == r) {
196         Poly res(2);
197         res[0] = 1; res[1] = -a[r];
198         return res;
199     }
200     int mid = (l + r) >> 1;
201     Poly ans = mul(dvd(l, mid), dvd(mid + 1, r));
202     return ans;
203 }
204 
205 inline void solve1() {
206     Poly q = dvd(1, n);
207     Poly p(n);
208     for(int i = 0; i < n; i++) {
209         p[i] = q[i] * (n - i) % MO;
210     }
211     p = mul(p, getInv(q));
212     for(int i = 0; i < n; i++) {
213         ex[i] = p[i];
214         //printf("ex %d = %lld \n", i, ex[i]);
215     }
216     return;
217 }
218 
219 int main() {
220 
221     scanf("%d%lld", &n, &m);
222     for(int i = 1; i <= n; i++) {
223         scanf("%lld", &a[i]);
224     }
225     ///
226     solve1();
227 
228     pw[0] = 1;
229     for(int i = 1; i <= n; i++) pw[i] = pw[i - 1] * i % MO;
230     for(int i = 0; i < n; i++) {
231         f[i] = qpow(i + 1, m) * qpow(pw[i], MO - 2) % MO;
232         h[i] = qpow(i + 1, m << 1) * qpow(pw[i], MO - 2) % MO;
233     }
234     getInv(f, p, n);
235     mul(p, h, h, n);
236 
237     getLn(f, g, n);
238     for(int i = 0; i < n; i++) {
239         g[i] = g[i] * ex[i] % MO;
240         h[i] = h[i] * ex[i] % MO;
241     }
242     getExp(g, f, n);
243 
244     mul(h, f, f, n);
245 
246     LL ans = f[n - 2];
247     for(int i = 1; i < n - 1; i++) ans = ans * i % MO;
248     for(int i = 1; i <= n; i++) ans = ans * a[i] % MO;
249 
250     if(ans < 0) ans += MO;
251     printf("%lld\n", ans);
252 
253     return 0;
254 }