LGP7511题解

首先会被分成若干个环，可以对于每个环分别考虑。

考虑令 \(g_{u,m}\) 表示 \(u\) 这个环有恰好 \(m\) 个位置上升的方案数，那么只需要搞个堆把所有的 \(g\) 卷起来即可。

对于一个长度为 \(m\) 的环，从其权值最小处断开（这里一定会产生一个上升），剩下的相当于是长度为 \(m-1\) 的序列有多少个位置恰好上升，这个东西是欧拉数。

然后，每个本质不同的环被算到了一次，所以答案需要乘上 \(m\)。

也就是：

\[g_{u,k}=|u|\times\left\langle\begin{matrix}|u|-1\\k-1\end{matrix}\right\rangle \]

每次算一行欧拉数即可，复杂度 \(O(n\log^2n)\)。

#include<algorithm>
#include<cstdio>
#include<cctype>
#include<vector>
#include<queue>
#define IMP(lim,act) for(int qwq=(lim),i=0;i^qwq;++i)act
const int M=1<<20|5,mod=998244353;
int Inv[M],buf[M<<2];int*now=buf,*w[23];
inline void swap(int&a,int&b){
	int c=a;a=b;b=c;
}
inline int Add(const int&a,const int&b){
	return a+b>=mod?a+b-mod:a+b;
}
inline int Del(const int&a,const int&b){
	return b>a?a-b+mod:a-b;
}
inline int max(const int&a,const int&b){
	return a>b?a:b;
}
inline void write(int n){
	static char s[10];int top(0);while(s[++top]=n%10^48,n/=10);while(putchar(s[top--]),top);
}
inline int read(){
	int n(0);char s;while(!isdigit(s=getchar()));while(n=n*10+(s&15),isdigit(s=getchar()));return n;
}
inline int pow(int a,int b=mod-2){
	int ans(1);for(;b;b>>=1,a=1ull*a*a%mod)if(b&1)ans=1ull*ans*a%mod;return ans;
}
inline int Getlen(const int&n){
	int len(0);while((1<<len)<n)++len;return len;
}
inline void init(const int&n){
	const int&m=Getlen(n);w[m]=now;now+=1<<m;Inv[1]=1;
	for(int i=2;i<n;++i)Inv[i]=1ull*(mod-mod/i)*Inv[mod%i]%mod;
	w[m][0]=1;w[m][1]=pow(3,mod-1>>m+1);for(int i=2;i<(1<<m);++i)w[m][i]=1ull*w[m][i-1]*w[m][1]%mod;
	for(int k=m-1;k>=0&&(w[k]=now,now+=1<<k);--k)IMP(1<<k,w[k][i]=w[k+1][i<<1]);
}
struct Poly{
	std::vector<int>F;
	Poly(const Poly&G){F=G.F;}
	Poly(const std::vector<int>G){F=G;}
	Poly(const int&x=0){if(x)F=std::vector<int>(x);}
	inline Poly&resize(const int&len){
		F.resize(len);return*this;
	}
	inline int size()const{
		return F.size();
	}
	inline int&operator[](const int&id){
		return F[id];
	}
	inline void push_back(const int&x){
		F.push_back(x);
	}
	inline Poly&reverse(){
		std::reverse(F.begin(),F.end());return*this;
	}
	inline Poly operator>>(const int&x){
		int i;Poly G;IMP(F.size()-x,G.push_back(F[i+x]));return G;
	}
	inline Poly operator<<(const int&x){
		int i;Poly G;G.resize(x);IMP(F.size(),G.push_back(F[i]));return G;
	}
	inline int operator()(const int&x){
		int i,y(1),ans(0);IMP(F.size(),ans=(ans+1ull*F[i]*y)%mod),y=1ull*y*x%mod;return ans;
	}
	inline void px(Poly G){
		F.resize(max(F.size(),G.size()));G.resize(F.size());
		for(int i(0);i^F.size();++i)F[i]=1ull*F[i]*G[i]%mod;
	}
	inline Poly&Der(){
		for(int i(1);i^F.size();++i)F[i-1]=1ull*F[i]*i%mod;F.pop_back();return*this;
	}
	inline Poly&Int(){
		F.push_back(0);
		for(int i(F.size()-1);i;--i)F[i]=1ull*F[i-1]*::Inv[i]%mod;F[0]=0;return*this;
	}
	inline void DFT(const int&M){
		int i,k,d,x,y,len,*W,*L,*R;F.resize(1<<M);
		for(len=F.size()>>1,d=M-1;len;--d,len>>=1)for(k=0;k^F.size();k+=len<<1){
			W=w[d];L=&F[k];R=&F[k|len];IMP(len,(x=*L,y=*R)),*L++=Add(x,y),*R++=1ull**W++*Del(x,y)%mod;
		}
	}
	inline void IDFT(const int&M){
		int i,k,d,x,y,len,*W,*L,*R;F.resize(1<<M);
		for(len=1,d=0;len^F.size();len<<=1,++d)for(k=0;k^F.size();k+=len<<1){
			W=w[d];L=&F[k];R=&F[k|len];IMP(len,(x=*L,y=1ull**W++**R%mod)),*L++=Add(x,y),*R++=Del(x,y);
		}
		k=::pow(F.size());IMP(F.size(),F[i]=1ull*F[i]*k%mod);for(i=1;(i<<1)<F.size();++i)swap(F[i],F[F.size()-i]);
	}
	inline Poly operator+(Poly G)const{
		Poly F=this->F;int i;F.resize(max(F.size(),G.size()));G.resize(F.size());
		IMP(F.size(),F[i]=Add(F[i],G[i]));return F;
	}
	inline Poly operator-(Poly G)const{
		Poly F=this->F;int i;F.resize(max(F.size(),G.size()));G.resize(F.size());
		IMP(F.size(),F[i]=Del(F[i],G[i]));return F;
	}
	inline Poly operator*(const int&x)const{
		Poly F=this->F;int i;
		IMP(F.size(),F[i]=1ull*F[i]*x%mod);return F;
	}
	inline Poly operator*(Poly G)const{
		Poly F=*this;const int&m=F.size()+G.size()-1,&len=Getlen(m);
		F.DFT(len);G.DFT(len);F.px(G);F.IDFT(len);return F.resize(m);
	}
	inline Poly operator/(Poly G){
		Poly F=*this,sav;const int&m=F.size()-G.size()+1;
		sav.resize(m);IMP(m,sav[i]=G.size()+i<m?0:G[G.size()-m+i]);
		sav.reverse().inv();sav*=F.reverse();
		return sav.resize(m).reverse();
	}
	inline Poly operator%(Poly G){
		return(*this-*this/G*G).resize(G.size()-1);
	}
	inline Poly&inv(){
		Poly b1,b2,b3;const int&m=Getlen(F.size());if(!F.empty())b1.push_back(::pow(F[0]));
		for(int len=1;len<=m;++len){
			b3=b1*2;(b2=F).resize(1<<len);
			b1.DFT(len+1);b1.px(b1);b2.DFT(len+1);b1.px(b2);b1.IDFT(len+1);
			b1=b3-b1.resize(1<<len);
		}
		return*this=b1.resize(F.size());
	}
	inline Poly&ln(){
		const int&m=F.size()-1;Poly G=*this;return(this->Der()*=G.inv()).resize(m).Int();
	}
	inline Poly&exp(){
		Poly b1,b2,b3;const int&m=Getlen(F.size());b1.push_back(1);
		for(int len=1;len<=m;++len){
			b3=b2=b1;b2.resize(1<<len).ln();b2=(*this-b2).resize(1<<len);++b2[0];
			b2.DFT(len);b3.DFT(len);b2.px(b3);b2.IDFT(len);b1.resize(1<<len);
			IMP(1<<len-1,b1[1<<len-1|i]=b2[1<<len-1|i]);
		}
		return*this=b1.resize(F.size());
	}
	inline Poly&sqrt(){
		Poly b1,b2;const int&m=Getlen(F.size());b1.push_back(1);
		for(int len=1;len<=m;++len){
			b2=(b1*2).resize(1<<len).inv();
			b1.DFT(len);b1.px(b1);b1.IDFT(len);
			b1=((*this+b1).resize(1<<len)*b2).resize(1<<len);
		}
		return*this=b1.resize(F.size());
	}
	inline Poly&pow(const int&k){
		int i;ln();IMP(F.size(),F[i]=1ull*F[i]*k%mod);return exp();
	}
	inline Poly&operator>>=(const int&x){
		return*this=operator>>(x);
	}
	inline Poly&operator<<=(const int&x){
		return*this=operator<<(x);
	}
	inline Poly&operator+=(const Poly&G){
		return*this=*this+G;
	}
	inline Poly&operator-=(const Poly&G){
		return*this=*this-G;
	}
	inline Poly&operator*=(const Poly&G){
		return*this=*this*G;
	}
	inline Poly&operator/=(const Poly&G){
		return*this=*this/G;
	}
	inline Poly&operator%=(const Poly&G){
		return*this=*this%G;
	}
};
inline Poly resize(Poly F,const int&n){
	return F.resize(n);
}
inline Poly reverse(Poly F){
	return F.reverse();
}
inline Poly Int(Poly F){
	return F.Int();
}
inline Poly Der(Poly F){
	return F.Der();
}
inline Poly px(Poly F,Poly G){
	return F.px(G),F;
}
inline Poly inv(Poly F){
	return F.inv();
}
inline Poly ln(Poly F){
	return F.ln();
}
inline Poly exp(Poly F){
	return F.exp();
}
inline Poly sqrt(Poly F){
	return F.sqrt();
}
inline Poly pow(Poly F,const int&k){
	return F.pow(k);
}
int n,m,k,top,x[M],y[M],p[M],pri[M],pos[M],idk[M],ifac[M];Poly F,G;bool vis[M];
struct cmp{
	inline bool operator()(const Poly&a,const Poly&b)const{
		return a.size()>b.size();
	}
};std::priority_queue<Poly,std::vector<Poly>,cmp>q;
inline void sieve(const int&n){
	ifac[0]=idk[1]=1;
	for(int i=2;i<=n;++i){
		if(!pos[i])pri[pos[i]=++top]=i,idk[i]=pow(i,n);
		for(int x,j=1;j<=pos[i]&&(x=i*pri[j])<=n;++j)pos[x]=j;
	}
	for(int i=1;i<=n;++i)ifac[i]=1ll*ifac[i-1]*Inv[i]%mod;
	pri[0]=1;for(int i=1;i<=n;++i)x[i]=pri[pos[i]],y[i]=i/x[i];
}
inline void Get(const int&n){
	idk[1]=1;for(int i=2;i<=n;++i)idk[i]=x[i]==i?pow(i,n):1ll*idk[x[i]]*idk[y[i]]%mod;
}
inline Poly GetF(const int&n){
	static Poly F,G;std::vector<int>().swap(F.F);std::vector<int>().swap(G.F);F.resize(n+1);G.resize(n+1);Get(n);
	F[0]=1;for(int i=1;i<=n;++i)F[i]=1ll*(n-i+1)*Inv[i]%mod*F[i-1]%mod;
	for(int i=0;i<=n;++i)F[i]=i&1?mod-F[i]:F[i],G[i]=idk[i];F*=G;F.resize(n+1);
	for(int i=n;i>=1;--i)F[i]=Del(F[i],F[i-1]);return F>>1;
}
signed main(){
	int ans(1);n=read();k=read();for(int i=1;i<=n;++i)p[i]=read();init(n+1<<1);sieve(n);F.push_back(1);q.push(F);
	for(int i=1;i<=n;++i)if(!vis[i]){
		int u(i),S(0);while(!vis[u])vis[u]=true,++S,u=p[u];ans=1ll*ans*ifac[S]%mod;
		if(S==1)continue;F=GetF(S-1)<<1;IMP(S,F[i]=1ll*F[i]*S%mod);q.push(F);
	}
	while(q.size()^1)F=q.top(),q.pop(),G=q.top(),q.pop(),q.push(F*G);F=q.top();ans=1ll*ans*(k<F.size()?F[k]:0)%mod;
	for(int i=1;i<=n;++i)ans=1ll*ans*i%mod;write(ans);
}