Help:=proc(): print(`Flip(pi,i,j) `):
print(`Sphere1(pi), SphereR(pi,R) , W(n)`):
end:

#Flip(p,i,j): Given a list pi returns
#the same list but op(i..j,pi) reversed:
Flip:=proc(p,i,j) local k:
[op(1..i-1,p),seq(p[j-k],k=0..j-i),op(j+1..nops(p),p)]:
end:


#Sphere1(pi): The sphere of distance 1 from the permutation
#pi (via one prefix reversal)
Sphere1:=proc(pi) local j:
option remember:
{seq(Flip(pi,1,j),j=2..nops(pi)) }:
end:

#SphereR(pi,R): The spehere of distance R from the permutation
#pi (via one prefix reversal)
SphereR:=proc(pi,R) local i,B,S:
option remember:
if R=0 then
  RETURN({pi}):
fi:

B:= {seq(op(SphereR(pi,i)),i=0..R-1)}:
S:=SphereR(pi,R-1):

{seq(op(Sphere1(S[i])),i=1..nops(S))} minus B:
end:

#W(n): the number of operations that it may
#take to sort any permutation of length n
#followed by the most diffucult layer
W:=proc(n) 
local C,N,pi,i:
pi:=[seq(i,i=1..n)]:

for i from 0 do
C:=SphereR(pi,i):
N:=SphereR(pi,i+1):
 if N={} then
   RETURN(i,C):
 fi:
od:
FAIL:
end: