【Codeforces 1349 F2】Slime and Sequences (Hard Version)（生成函数 / 多项式Exp / 拉格朗日反演）

传送门

考虑一个排列和那个是一一对应的
$p_{a_i}$的权值是 $a_i$前 $<$的个数+1

而一个排列 $<$的个数即欧拉数，设为 $f[i][j]$
计算可以利用二项式反演至少
至少钦定组内递增就是斯特林数
$f[i][j]=\frac 1 {j!}\sum_{k=j}^{i}k!\frac{(-1)^{k-j}}{(k-j)!}[x^i](e^x-1)^{i-k}$

$ans_j=\sum_{i=1}^n{n\choose i}f[i][j-1](n-i)!$
整理得
$ans_j= n!\frac{1}{(j-1)!}\sum_{k=j-1}^n\frac{(-1)^{k-j+1}}{(k-j+1)!}k!\sum_{i=k+1}^{n}[x^i](e^x-1)^{i-k}$

现在只考虑求 $R_k=\sum_{i=k+1}^{n}[x^i](e^x-1)^{i-k}$，剩下翻转卷积即可

$R_k=\sum_{i=k+1}^n[x^k](\frac{e^x-1}{x})^{i-k}\\ =[x^k]\sum_{i=1}^{n-k}f^i,f=\frac{e^x-1}{x}$
$R_k=[x^k]\frac{f}{1-f}-[x^k]\frac{f^{n-k+1}}{1-f}$

只用考虑后面怎么求
$res=[x^{n+1}]\frac{(fx)^{n-k+1}}{1-f}$

设 $g=xf$
$t(x)$满足 $\frac{g}{t(g)}=x$
$res=[x^{n+1}y^{n-i+1}]\sum_{i=0}^{\infty}\frac{(xyf)^k}{1-f}=[x^{n+1}y^{n-i+1}]\frac{1}{1-t(g)}*\frac{1}{1-yg}$
则 $f=t(g),t=\frac{x}{\ln(x+1)}$
$p(x)=\frac{x}{t(x)},p(g)=x$
然后利用拓展拉格朗日反演得到
$res=[y^{n-i+1}]\frac 1{n+1}[x^n]((\frac{1}{1-t(x)}*\frac{1}{1-yx})'t(x)^{n+1})$
求导裂项分母展开
$=[y^{n-i+1}]\frac 1{n+1}[x^n](t(x)^{n+1}(\frac{1}{1-t(x)}\sum_{i}(i+1)x^iy^{i+1}+\frac{t(x)'}{(1-t(x))^2}\sum_{i=0}^{\infty}x^iy^i))\\ =\frac 1{n+1}([x^i](\frac{t(x)^{n+1}}{1-t(x)})(n-i+1)+[x^{i-1}](\frac{t'(x)t(x)^{n+1}}{(1-t(x))^2}))$

直接快速幂即可

复杂度 $O(nlogn)$
注意 $1-t,1-f,\ln(x+1)$等作为分母常数项会为 $0$
需要除 $x$求逆后加回去

#include<bits/stdc++.h>
using namespace std;
#define cs const
#define re register
#define pb push_back
#define pii pair<int,int>
#define ll long long
#define y1 shinkle
#define fi first
#define se second
#define bg begin
cs int RLEN=1<<20|1;
inline char gc(){
    static char ibuf[RLEN],*ib,*ob;
    (ib==ob)&&(ob=(ib=ibuf)+fread(ibuf,1,RLEN,stdin));
    return (ib==ob)?EOF:*ib++;
}
inline int read(){
    char ch=gc();
    int res=0;bool f=1;
    while(!isdigit(ch))f^=ch=='-',ch=gc();
    while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
    return f?res:-res;
}
inline ll readll(){
    char ch=gc();
    ll res=0;bool f=1;
    while(!isdigit(ch))f^=ch=='-',ch=gc();
    while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
    return f?res:-res;
}
inline char readchar(){
	char ch=gc();
	while(isspace(ch))ch=gc();
	return ch;
}
inline int readstring(char *s){
	int top=0;char ch=gc();
	while(isspace(ch))ch=gc();
	while(!isspace(ch)&&ch!=EOF)s[++top]=ch,ch=gc();
	s[top+1]='\0';return top;
}
template<typename tp>inline void chemx(tp &a,tp b){a=max(a,b);}
template<typename tp>inline void chemn(tp &a,tp b){a=min(a,b);}
cs int mod=998244353;
inline int add(int a,int b){return (a+b)>=mod?(a+b-mod):(a+b);}
inline int dec(int a,int b){return (a<b)?(a-b+mod):(a-b);}
inline int mul(int a,int b){static ll r;r=(ll)a*b;return (r>=mod)?(r%mod):r;}
inline void Add(int &a,int b){a=(a+b)>=mod?(a+b-mod):(a+b);}
inline void Dec(int &a,int b){a=(a<b)?(a-b+mod):(a-b);}
inline void Mul(int &a,int b){static ll r;r=(ll)a*b;a=(r>=mod)?(r%mod):r;}
inline int ksm(int a,int b,int res=1){for(;b;b>>=1,Mul(a,a))(b&1)&&(Mul(res,a),1);return res;}
inline int Inv(int x){return ksm(x,mod-2);}
inline int fix(ll x){x%=mod;return (x<0)?x+mod:x;}
cs int N=200051;
int fac[N],ifac[N];
typedef vector<int> poly;
inline void init_fac(){
	fac[0]=ifac[0]=1;
	for(int i=1;i<N;i++)fac[i]=mul(fac[i-1],i);
	ifac[N-1]=Inv(fac[N-1]);
	for(int i=N-2;i;i--)ifac[i]=mul(ifac[i+1],i+1);
}
inline int Cb(int n,int m){return (n<0||m<0||n<0)?0:mul(fac[n],mul(ifac[m],ifac[n-m]));}
namespace Poly{
	cs int C=18,M=(1<<C)|5;
	int *w[C+1],rev[M],iv[M];
	inline void init_rev(int lim){
		for(int i=0;i<lim;i++)rev[i]=(rev[i>>1]>>1)|((i&1)*(lim>>1));
	}
	inline void init_w(){
		for(int i=1;i<=C;i++)w[i]=new int[(1<<(i-1))|1];
		int wn=ksm(3,(mod-1)/(1<<C));w[C][0]=1;
		for(int i=1,l=1<<(C-1);i<l;i++)w[C][i]=mul(w[C][i-1],wn);
		for(int i=C-1;i;i--)
		for(int j=0,l=1<<(i-1);j<l;j++)w[i][j]=w[i+1][j<<1];
		iv[0]=iv[1]=1;
		for(int i=2;i<M;i++)iv[i]=mul(mod-mod/i,iv[mod%i]);
	}
	inline void dft(int *f,int lim){
		for(int i=0;i<lim;i++)if(i>rev[i])swap(f[i],f[rev[i]]);
		for(int mid=1,l=1,a0,a1;mid<lim;mid<<=1,l++)
		for(int i=0;i<lim;i+=mid<<1)
		for(int j=0;j<mid;j++)
		a0=f[i+j],a1=mul(w[l][j],f[i+j+mid]),f[i+j]=add(a0,a1),f[i+j+mid]=dec(a0,a1);
	}
	inline void ntt(poly &f,int lim,int kd){
		dft(&f[0],lim);
		if(kd==-1){
			reverse(f.bg()+1,f.bg()+lim);
			for(int i=0;i<lim;i++)Mul(f[i],iv[lim]);
		}
	}
	inline poly operator +(poly a,poly b){
		if(a.size()<b.size())a.resize(b.size());
		for(int i=0;i<b.size();i++)Add(a[i],b[i]);
		return a;
	}
	inline poly operator -(poly a,poly b){
		if(a.size()<b.size())a.resize(b.size());
		for(int i=0;i<b.size();i++)Dec(a[i],b[i]);
		return a;
	}
	inline poly operator *(poly a,int b){
		for(int i=0;i<a.size();i++)Mul(a[i],b);return a;
	}
	inline poly operator *(poly a,poly b){
		if(!a.size()||!b.size())return poly(0);
		int deg=a.size()+b.size()-1;
		if(a.size()<=32||b.size()<=32){
			poly c(deg,0);
			for(int i=0;i<a.size();i++)
			for(int j=0;j<b.size();j++)
			Add(c[i+j],mul(a[i],b[j]));
			return c;
		}int lim=1;while(lim<deg)lim<<=1;
		init_rev(lim);
		a.resize(lim),ntt(a,lim,1);
		b.resize(lim),ntt(b,lim,1);
		for(int i=0;i<lim;i++)Mul(a[i],b[i]);
		ntt(a,lim,-1),a.resize(deg);
		return a;
	}
	inline poly Inv(poly a,int deg){
		poly b(1,::Inv(a[0])),c;
		for(int lim=4;lim<(deg<<2);lim<<=1){
			c.resize(lim>>1);init_rev(lim);
			for(int i=0;i<(lim>>1);i++)c[i]=(i<a.size()?a[i]:0);
			c.resize(lim),ntt(c,lim,1);
			b.resize(lim),ntt(b,lim,1);
			for(int i=0;i<lim;i++)Mul(b[i],dec(2,mul(b[i],c[i])));
			ntt(b,lim,-1),b.resize(lim>>1);
		}b.resize(deg);return b;
	}
	inline poly deriv(poly a){
		for(int i=0;i+1<a.size();i++)a[i]=mul(a[i+1],i+1);
		a.pop_back();return a;
	}
	inline poly integ(poly a){
		a.pb(0);
		for(int i=a.size()-1;i;i--)a[i]=mul(a[i-1],iv[i]);
		a[0]=0;return a;
	}
	inline poly Ln(poly a,int deg){
		a=integ(Inv(a,deg)*deriv(a)),a.resize(deg);return a;
	}
	inline poly Exp(poly a,int deg){
		poly b(1,1),c;
		for(int lim=2;lim<(deg<<1);lim<<=1){
			c=Ln(b,lim);
			for(int i=0;i<lim;i++)c[i]=dec(i<a.size()?a[i]:0,c[i]);
			Add(c[0],1);
			b=b*c,b.resize(lim);
		}b.resize(deg);return b;
	}
	inline poly ksm(poly a,int b,int deg){
		return Exp(Ln(a,deg)*b,deg);
	}
	inline poly Mulx(poly a){
		a.pb(0);
		for(int i=a.size()-1;i;i--)a[i]=a[i-1];
		a[0]=0;return a;
	}
	inline poly Divx(poly a){
		for(int i=0;i+1<a.size();i++)a[i]=a[i+1];
		a.pop_back();return a;
	}
}
using namespace Poly;
void write(poly a){
	for(int i=0;i<a.size();i++)cout<<a[i]<<" ";puts("");
}
inline poly calc_g(int n){
	int lm=n+3;
	poly f(lm);
	for(int i=0;i<f.size();i++)f[i]=ifac[i+1];
	poly res1=Divx(Inv(Divx(poly(1,1)-f),lm)*f);
	res1.resize(lm);
	poly t(2);t[0]=t[1]=1,t=Inv(Divx(Ln(t,lm)),lm);
	poly dn=Inv(Divx(poly(1,1)-t),lm),pt=ksm(t,n+1,lm);
	poly v1=pt*dn,v2;v1.resize(lm),dn=dn*dn,dn.resize(lm);
	pt=pt*deriv(t),pt.resize(lm),v2=pt*dn,v2.resize(lm);
	poly res2(n,0);
	for(int i=0;i<n;i++)res2[i]=dec(res1[i],mul(iv[n+1],add(mul(n-i+1,v1[i+1]),v2[i+1])));
	return res2;
}
int n;
int main(){
	#ifdef Stargazer
	freopen("lx.in","r",stdin);
	#endif
	init_w(),init_fac();
	n=read();
	poly g=calc_g(n);
	for(int i=0;i<g.size();i++)Mul(g[i],fac[i]);
	g.resize(n);
	poly f(n+1);
	for(int i=0;i<=n;i++)f[i]=(i&1)?dec(0,ifac[i]):ifac[i];
	reverse(f.bg(),f.end());
	f=f*g;
	for(int i=1;i<=n;i++)cout<<mul(fac[n],mul(ifac[i-1],f[n+i-1]))<<" ";
	return 0;
}