【UOJ #424】【集訓隊作業2018】count（矩陣快速冪 / 生成函數 / NTT）

傳送門

考慮實際上就是深度不超過$m$的$n$個點的笛卡爾樹計數
設$f_i(x)$爲深度不超過$i$的生成函數
那麼有$f_i(x)=f_{i-1}(x)xf_i(x)+1$
$f_i(x)=\frac{1}{1-xf_{i-1}(x)}$

考慮直接多項式矩乘複雜度大常數$O(nlog^2n)$而且顯然有毒

考慮帶入點值計算，帶入$w_n$作爲點值
這樣轉回多項式只需要$\mathrm{IDFT}$即可
如果看做線性變換需要求逆元複雜度是$O(nlog^2n)$的
考慮分子分母分別維護，最後轉回多項式後再多項式求逆即可
複雜度$O(nlogn)$

#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 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 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();
	return top;
}
template<typename tp>inline void chemx(tp &a,tp b){a<b?a=b:0;}
template<typename tp>inline void chemn(tp &a,tp b){a>b?a=b:0;}
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(int x){return (x<0)?x+mod:x;}
typedef vector<int> poly;
namespace Poly{
	cs int C=18,M=(1<<C)|5,G=3;
	int *w[C+1],rev[M];
	inline void init_w(){
		for(int i=1;i<=C;i++)w[i]=new int[(1<<(i-1))|1];
		int wn=ksm(G,(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];
	}
	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 ntt(int *f,int lim,int kd){
		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);
		if(kd==-1){
			reverse(f+1,f+lim);
			for(int i=0,iv=Inv(lim);i<lim;i++)Mul(f[i],iv);
		}
	}
	inline poly operator +(poly a,cs 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,cs 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){
		int deg=a.size()+b.size()-1;
		if(a.size()<=16||b.size()<=16){
			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[0],lim,1);
		b.resize(lim),ntt(&b[0],lim,1);
		for(int i=0;i<lim;i++)Mul(a[i],b[i]);
		ntt(&a[0],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){
			init_rev(lim),c.resize(lim>>1);
			for(int i=0;i<(lim>>1);i++)c[i]=(i<a.size()?a[i]:0);
			b.resize(lim),ntt(&b[0],lim,1);
			c.resize(lim),ntt(&c[0],lim,1);
			for(int i=0;i<lim;i++)Mul(b[i],dec(2,mul(b[i],c[i])));
			ntt(&b[0],lim,-1),b.resize(lim>>1);
		}
		b.resize(deg);return b;
	}
}
using namespace Poly;
int n,m;
struct mat{
	int a[2][2];
	mat(){memset(a,0,sizeof(a));}
	friend inline mat operator *(cs mat &a,cs mat &b){
		mat c;
		Add(c.a[0][0],mul(a.a[0][0],b.a[0][0]));
		Add(c.a[0][1],mul(a.a[0][0],b.a[0][1]));
		Add(c.a[0][0],mul(a.a[0][1],b.a[1][0]));
		Add(c.a[0][1],mul(a.a[0][1],b.a[1][1]));
		Add(c.a[1][0],mul(a.a[1][0],b.a[0][0]));
		Add(c.a[1][1],mul(a.a[1][0],b.a[0][1]));
		Add(c.a[1][0],mul(a.a[1][1],b.a[1][0]));
		Add(c.a[1][1],mul(a.a[1][1],b.a[1][1]));
		return c;
	}
	friend inline mat operator ^(mat a,int b){
		mat c;c.a[0][0]=c.a[1][1]=1;
		for(;b;b>>=1,a=a*a)if(b&1)c=c*a;
		return c;
	}
};
int main(){
	#ifdef Stargazer
	freopen("lx.in","r",stdin);
	#endif
	int L=1<<17;init_w();
	int W=ksm(G,(mod-1)/L);
	n=read(),m=read();
	if(n<m)return puts("0"),0;
	poly f(L),g(L);
	for(int mt=1,i=0;i<L;i++,Mul(mt,W)){
		mat p;
		p.a[0][0]=0,p.a[0][1]=dec(0,mt),p.a[1][0]=p.a[1][1]=1;
		p=p^m;
		f[i]=add(p.a[0][0],p.a[1][0]),g[i]=add(p.a[0][1],p.a[1][1]);
	}init_rev(L);
	ntt(&f[0],L,-1),ntt(&g[0],L,-1);
	f.resize(n+1),g.resize(n+1);
	f=f*Inv(g,n+1);
	cout<<f[n]<<'\n';
	return 0;
}