#include<bits/stdc++.h>
#define Gc() getchar()
#define Me(x,y) memset(x,y,sizeof(x))
#define Mc(x,y) memcpy(x,y,sizeof(x))
#define d(x,y) ((m)*(x-1)+(y))
#define R(n) (rnd()%(n)+1)
#define Pc(x) putchar(x)
#define LB lower_bound
#define UB upper_bound
#define fi first
#define se second
#define eb emplace_back
#define all(x) x.begin(),x.end()
using namespace std;using ll=long long;using db=double;using lb=long db;using ui=unsigned;using ull=unsigned long long;using pii=pair<int,int>;
const int N=100+5,M=N*200+5,K=1000+5,mod=998244353,Mod=mod-1;const db eps=1e-9;const int INF=1e9+7;mt19937 rnd(28382);
#define Tp template<typename T>
#define Ts template<typename T,typename... Ar>
namespace Debug{
	Tp void _debug(char* f,T t){cerr<<f<<'='<<t<<endl;}
	Ts void _debug(char* f,T x,Ar... y){while(*f!=',') cerr<<*f++;cerr<<'='<<x<<",";_debug(f+1,y...);}
	#ifdef LOCAL
	#define gdb(...) _debug((char*)#__VA_ARGS__,__VA_ARGS__)
	#else 
	#define gdb(...) void()
	#endif
}using namespace Debug;

using LL=__uint128_t;
namespace Encoder{
	int flag=0;
	LL dp[N];
	void init(){
		const int n=70;
		dp[1]=1;
		for(int i=1;i<=n;i++){
			for(int j=1;j<i;j++) dp[i]+=dp[j]*dp[i-j];
		}
		flag=1;
	}
	vector<int> S[N];
	int siz[N];
	void make(int x,int La){
		siz[x]=1;
		for(int i:S[x]) if(i^La) make(i,x),siz[x]+=siz[i];
	}
	LL getans(int x,int La){
		int sum=siz[x];
		LL ans=0;
		for(int i:S[x]){
			for(int j=1;j<siz[i];j++) ans+=dp[j]*dp[sum-j];
			ans+=getans(i,x)*dp[sum-siz[i]];
			sum-=siz[i];
		}
		return ans;
	}
	LL calc(int n,const int *p){
		if(!flag) init();
		for(int i=1;i<=n;i++) S[i].clear();
		for(int i=2;i<=n;i++) S[p[i]].push_back(i);
		make(1,0);
		return getans(1,0);
	}
}



LL encoder(int n, const int *p){
	return Encoder::calc(n,p);
}

namespace Decoder{
	int flag=0;
	LL dp[N];
	void init(){
		const int n=70;
		dp[1]=1;
		for(int i=1;i<=n;i++){
			for(int j=1;j<i;j++) dp[i]+=dp[j]*dp[i-j];
		}
		flag=1;
	}
	int cnt=0;
	void build(int n,LL k,int rt,int *p){
		while(n>1){
			for(int i=1;;i++){
				gdb(n,i);
				LL w=dp[i]*dp[n-i];
				if(k>=w) k-=w;
				else{
					p[++cnt]=rt;
					build(i,k/dp[n-i],cnt,p);
					k%=dp[n-i];
					n-=i;break;
				}
			}
		}
	}
	void calc(int n,LL k,int *p){
		if(!flag) init();
		p[1]=0;cnt=1;
		build(n,k,1,p);
	}
}

void decoder(int n, LL M, int *p){
	Decoder::calc(n,M,p);
}