#!/usr/local/bin/python3
from	itertools	import	permutations	as	P
from	collections	import	defaultdict	as	D
W=sorted(set(P('0011223')))
def	fd(w,u):
	a=[0]*(len(u)+1)
	a[0]=1
	for	ch	in	w:
		for	j	in	range(len(u)-1,-1,-1):
			if	ch==u[j]:a[j+1]+=a[j]
	return	a[-1]
def	S(w):return	tuple(fd(w,u)	for	u	in	('01','12','23','012','123','0123'))
G=D(list)
for	w	in	W:G[S(w)].append(w)
L=list(G.values())
idx={w:k	for	k,x	in	enumerate(L)	for	w	in	x}
def	rv(w):return	tuple(reversed(w))
key={}
for	k,x	in	enumerate(L):key[k]=idx[rv(x[0])]
print('mapok',all(key[key[k]]==k	for	k	in	key),'pairs',sum(key[k]!=k	for	k	in	key)//2+sum(key[k]==k	for	k	in	key))
cs=tuple('1322100')
print('classstar',[''.join(w)	for	w	in	sorted(G[S(cs)])])
print('revstarclass',[''.join(w)	for	w	in	sorted(G[S(tuple(reversed(cs)))])])
print('staridx',idx[cs],'revidx',idx[rv(cs)])
fixed=[''.join(x[0])	for	k,x	in	enumerate(L)	if	key[k]==k]
print('fixed',fixed)
print('revsigformula',S(rv(cs)),S(cs))