NTT "51nod" 1514 wonderful sequence

Test instructions: In the full arrangement of the 1~n, how many permutations meet arbitrarily from the middle into two segments, the left segment maximum is greater than the right segment of the minimum value?

For example, there are 3 types when n is 3

2 3 1

3 1 2

3 2 1

Explanation: For example 2 3 1

(2) (3 1) 1:2 Small

(2 3) (1) 1:2 Small

All meet the above conditions.

3 2 1

(3) (2 1) 1:3 Small

(32) (1) 1:3 Small

All meet the above conditions.

and 2 1 3 is not satisfied, because (2 1) (3), 3 is larger than all the numbers on the left.

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First sequence for wonderful equivalence to non-existence (1<=i<n) makes the first I number of 1~i permutations

Make F[n] The answer to the length n

The

F[0]=0

F[i]=i!-sigma (f[j]* (I-J)!) 0<=j<i

We will deform it

Sigma (f[j]* (i-j)!) = i! I > 0, j=0~i

Sigma (f[j]* (i-j)!) = 0 i = 0, j=0~i

Make g (x) =sigma (i!*x^i), F (x) =sigma (f[i]*x^i)

F (x) *g (x) =g (x)-1 (minus one for i = 0)

F (x) = (G (x)-1)/g (x) =1-1/g (x)

Polynomial inversion can be

1#include <cstdio>2#include <iostream>3typedefLong Longll;4 using namespacestd;5 Const intN =262144, K = -;6 intN, M, I, K;7 inta[n+Ten], b[n+Ten], tmp[n+Ten], tmp2[n+Ten];8 intP =998244353, G =3, g[k+1], ng[k+Ten], inv[n+Ten], inv2;9 intPowintAintb) {intt=1; for(; b;b>>=1, A= (LL) a*a%p)if(b&1) t= (ll) t*a%p;returnt;}Ten voidNTT (int*a,intNintt) { One      for(intI=1, j=0; i<n-1; i++){ A          for(ints=n;j^=s>>=1,~j&s;); -         if(i<j) Swap (A[i], a[j]); -     } the      for(intD=0;(1&LT;&LT;D) <n;d++){ -         intm=1<<d,m2=m<<1, _w=t==1?G[d]:ng[d]; -          for(intI=0; i<n;i+=m2) for(intw=1, j=0; j<m;j++){ -             int&a=a[i+j+m],&b=a[i+j],t= (LL) w*a%P; +A=b-t;if(a<0) a+=P; -B=b+t;if(b>=p) b-=P; +w= (LL) w*_w%P; A         } at     } -     if(t==-1) for(intI=0, j=inv[n];i<n;i++) a[i]= (LL) a[i]*j%P; - } - //given a, find the inverse of a B - voidGETINV (int*a,int*b,intN) { -     if(n==1) {b[0]=pow (a[0],p-2);return;} inGETINV (a,b,n>>1); -     intk=n<<1, I; to      for(i=0; i<n;i++) tmp[i]=A[i]; +      for(i=n;i<k;i++) tmp[i]=b[i]=0; -NTT (Tmp,k,1), NTT (B,k,1); the      for(i=0; i<k;i++){ *b[i]= (LL) b[i]* (2-(LL) tmp[i]*b[i]%p)%P; $     if(b[i]<0) b[i]+=P;Panax Notoginseng     } -NTT (b,k,-1); the      for(i=n;i<k;i++) b[i]=0; + } A intMain () { the      for(G[k]=pow (g, (p1)/N), Ng[k]=pow (g[k],p-2), i=k-1; ~i;i--) +g[i]= (LL) g[i+1]*g[i+1]%p,ng[i]= (LL) ng[i+1]*ng[i+1]%P; -      for(inv[1]=1, i=2; i<=n;i++) inv[i]= (LL) (p-inv[p%i]) * (p/i)%p;inv2=inv[2]; $     intLen =1; $      while(Len <=100000) Len <<=1; -a[0] =1; -      for(inti =1; i < Len; i++) theA[i] = (LL) i*a[i-1]%P; - Getinv (A, b, Len);Wuyi      for(inti =0; i < Len; i++) theB[i] = (-b[i]+p)%P; -b[0]++; Wu     intT, N; scanf"%d", &t); -      while(t--){ Aboutscanf"%d", &n); $printf"%d\n", B[n]); -     } -     return 0; -}

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NTT "51nod" 1514 wonderful sequence