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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?

Thomas Wening
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  • Could you define how you're using the terms "cyclic permutation" and "anti-cyclic permutation", and what your notation $(1023)$ means? I've never seen the term "anti-cyclic permutation" before, and I get the impression that your other terminology and notation may also not be the conventional ones. – joriki Jun 16 '18 at 01:24
  • A permutation is cyclic, if it is the product of an even number of transpositions. It is anti-cyclic, if it is the product of an odd number of transpositions. I have amended my post to distinguish the actual images (the tupels) of the permutations from the transpositions in cycle notation. – Thomas Wening Jun 16 '18 at 08:01

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