Is there any algorithm to systematically write out all the (anti-)cyclic permutations of, say the set $\{0,1,2,3\}$?
I've already tried taking $(0,1,2,3)$, permuting it cycically, then transposing two of the numbers, say $(01)$, which yields $(1,0,2,3)$. This is then anti-cyclic and if I permute that tuple, it will remain so. Then, adding another transposition $(01)(23)$, yielding $(1,0,3,2)$. This is again cyclic, because it came about by using an even amount of transpositions.
But this seems to me to be quite tedious. Maybe there is a more effective method?